noncommutative

Transcription

1 noncommutative Lieven Le Bruyn (999)

2 2 Motivation. This book is all about smooth noncommutative algebras and combinatorial tools to study them. It is an old result, due to A. Grothendieck, that when A is a commutative affine C-algebra, A is the coordinate ring of a smooth affine variety if and only if A satisfies the following lifting property. For every test-object (B, I) where B is a commutative C-algebra and I R is a nilpotent ideal and any C-algebra morphism φ : A there is a C-algebra lift φ B I A... φ B φ B I making the diagram commute. As this is a purely categorical characterization of smooth affine commutative algebras, it can be extended to more general settings where we restrict the test-objects and morphisms to a specified category of C- algebras. In this book we focuss on two such settings. If we take the category alg with objects all (not necessarily commutative) C- algebras and as morphisms all C-algebra morphisms, then algebras satisfying the above lifting property with respect to test-objects in alg are called (formally) smooth algebras or Quillen-smooth algebras. The importance of this class of algebras is that they are often associated to natural families of moduli problems. Examples of Quillen-smooth algebras include : semi-simple C-algebras (as we can lift idempotents modulo nilpotent ideals), free algebras C x,..., x k, path algebras CQ of quivers (oriented graphs) as well as more exotic algebras constructed from these by universal constructions such as the free product A A 2, the n-th root n A and so on. However, the coordinate ring of an affine smooth variety is Quillen-smooth if and only if the variety is a curve. To see at least one implication of this equivalence it suffices by reasoning locally to verify the lifting property for the formal power series ring C[[x,..., x k ]]. Take the 4-dimensional noncommutative algebra T = C x, y (x 2, y 2 = C Cx Cy Cxy, xy + yx) then the quotient modulo the nilpotent ideal I = (xy yx) is a 3-dimensional commutative ring and we have an algebra map x x, x 2 y and x i 0 for i 3 which is verified not to allow a lift unless k =, the curve case. The second category we will consider is CH(n), the category of all Cayley- Hamilton algebras of degree n. Its objects are C-algebras A equipped with a linear trace map tr : A A satisfying tr(a)b = btr(a), tr(ab) = tr(ba) and tr(tr(a)b) = tr(a)tr(b) for all a, b A. The formal Cayley-Hamilton polynomial of degree n of an element a A is defined by expressing n i= (t x i) as a polynomial in t with coefficients which (being symmetric functions) can be expressed as polynomial functions in the Newton functions x m i. Replacing x m i by tr(a m ) we obtain

3 the Cayley-Hamilton polynomial χ (n) a (t) A[t]. A is said to be a Cayley-Hamilton algebra of degree n provided tr() = n and χ (n) a (a) = 0 for all a A. Naturally, morphisms in CH(n) must be trace preserving. A C-algebra A CH(n) with the above lifting property for test-objects (B, I), where B CH(n) and I B nilpotent such that tr(i) I (making B I CH(n)) and where φ : A B I as well as its lift φ : A B must be trace preserving, is called a Cayley-smooth algebra (of degree n). Examples include semi-simple algebras M n (C)... M nk (C) with n = i n i (equipped with the sum of the natural traces) as well as many nice C[X]-orders in central simple C(X)-algebras of dimension n 2 where X is an affine normal variety. An important subclass are the so called Azumaya algebras over X, such as M n (C[X]), when X is smooth. The more interesting examples include ramified orders where the central variety X is allowed to have singularities. We will give a complete étale local classifications of Cayley-smooth orders and of the central singularities. In the next two chapters we motivate the study of these noncommutative smooth algebras. In chapter we study Calogero particles which is a classical n-particle system in C with positions x i C and velocities y i C and Hamiltonian H = yi 2 2 i i<j (x i x j ) 2 This is a completely integrable dynamical system via the equations of motion. If the n particles are distinct (that is x i x j for i j) the corresponding point in the phase space is the 2n-tuple (x, y ;... ; x n, y n ) under the proviso that two such tuples are the same when we permute the n couples (x i, y i ). As the system is attractive, collisions will occur and one wants to extend the phase space analytically. This was done by G. Wilson in [33] who showed that the extended phase space Calo n is the 2n-dimensional connected manifold of GL n -orbits of quadruples (X, Y, u, v) M n (C) M n (C) C n C n such that [X, Y ] + u.v = n where the action is given by g.(x, Y, u, v) = (gxg, gy g, gu, vg ). This phase space Calo n should be compared with the 2n-dimensional Hilbert scheme Hilb n of n points in C 2. Whereas Hilb n decomposes in strata according to the multiplicities of the points, Calo n was shown by V. Ginzburg [8] to be a coadjoint orbit for an infinite dimensional Lie algebra, which is independent of n. The Lie algebra is naturally associated to a Quillen-smooth algebra M which is the path algebra of the quiver x e y 3 u v f We will define noncommutative functions, differential forms and symplectic structures on Quillen-smooth algebras. They have the characteristic property of inducing corresponding GL m -invariant classical structures on the representation varieties rep m A of these algebras. In the case of path algebras such as M, these representation spaces decompose according to the dimensions of the vertex spaces. In the special case of dimension vector α = (n, ) we have rep α M = M n (C) M n (C) C n C n

4 4 with the GL(α) = GL n C -action given by (g, λ).(x, Y, u, v) = (gxg, gy g, guλ, λvg ). To this action we associate the moment map rep α M µ Mn (C) C (X, Y, u, v) ([X, Y ] + u.v, v.u) and one proves that Calo n is equal to µ ( n, n)/gl(α) and that Hilb n is an open subvariety of µ (0 n, 0)/GL(α). The infinite dimensional Lie algebra mentioned before is associated to the group of C-algebra morphisms of M preserving the moment-map element [x, y] + [u, v]. In analogy with the commutative case, one would expect that the difference in behaviour between Calo n and Hilb n is caused by the fact that the fiber-algebra associated to Calo n is Quillen-smooth whereas that associated to Hilb n is not (contains noncommutative singularities). Calon Hilbn sing repα M/GL(α) (n, n) (0n, 0) A natural definition of these noncommutative fiber algebras are the deformed preprojective algebras introduced and studied by W. Crawley-Boevey and M.P. Holland in [6]. In the case under consideration, the fiber algebra of Calo n (resp. Hilb n ) is M (resp. M 0 ) where for any λ C we define M λ = M ([x, y] + [u, v] λ(e nf)) One verifies that for every λ C the fiber-algebra M λ is not Quillen-smooth, so apparently there is no difference between M and M 0. However, if one focusses on Calo n or Hilb n for a specific value of n, these fiberalgebras are a bit too big and it is more natural to consider algebras associated to M λ and the dimension vector α = (n, ). First, we define the algebra M(n) to be the C-subalgebra of M n+ (C[rep α M]) generated by the polynomial invariants C[rep α M] GL(α) (embedded as scalar matrices) and the following matrices x... x n 0... e n =. 0 f n = x n =... x n... x nn y... y n u y n =... y n... y nn 0 u n = u n v n = v... v n 0 The usual trace map on M n+ (C[rep α M]) makes M(n) a Cayley-Hamilton algebra of degree n + and it is even a Cayley-smooth algebra of degree n +. The closed

5 5 affine subscheme π (λ n, nλ) has as its defining ideal of relations I λ the entries in the n + n + matrix [ λ ] [x n, y n ] + [u n, v n ] n 0 0 nλ By invariant theory we have that the defining ideal of the quotient scheme µ (λ n, nλ)/gl(α) is J λ = I λ C[rep α M] GL(α). The restricted fiber-algebra we are interested in is M(n) M λ (n) = M(n)J λ M(n) Again, M λ (n) is a Cayley-Hamilton algebra of degree n+. The distinction between Calo n and Hilb n is a consequence of the fact that { M (n) is Cayley-smooth of degree n +, M 0 (n) is not Cayley-smooth of degree n + and the last fact holds even when we restrict to the open subvariety Hilb n of µ (0 n, 0). The homogeneous character of Calo n follows from the fact that M (n) is an Azumaya algebra over Calo n. That is, to every point in the extended phase space corresponds a simple n + -dimensional representation. In chapter 2 we will generalize these results to fiber-algebras for the moment map of arbitrary quiver settings. In chapter we will give an outline of the main ingredients going into the proof of Ginzburg s result on coadjointness of Calo n, in particular the acyclicity of the Karoubi complex and noncommutative symplectic geometry identifying tangent vectors (derivations) with noncommutative -forms. More details will be given in chapters 9,0 and 2. The investigation of Cayley-smooth algebras has also a more classical motivation as we will recall in chapter 2. Let X be a projective normal variety with function field C(X). An important birational invariant of X is the Brauer group Br C(X). The elements of Br C(X) are equivalence classes of central simple C(X)-algebras. That is, the center of is C(X), has no proper twosided ideals and dim C(X) = n 2 for some integer n. Two such algebras and are equivalent if M k ( ) M l ( ) for certain k, l and the isomorphism is as C(X)-algebras. Then, tensor product over C(X) induces a group structure on Br C(X). In chapter 2 we will recall the necessary ingredients from étale cohomology to outline the proof of the coniveau spectral sequence which gives us a handle on the n-torsion part of Br C(X). This result also shows that the collection of all central simple C(X)-algebras of degree n is huge. For example, a subset of the n-torsion part of Br C(x, y) is given by the following geometrical data. Let C and C be two smooth irreducible projective curves in P 2, intersecting each other transversally in the points {P,..., P k }. Let a i Z/nZ for every i k such that k i= a i = 0. Now, take a cyclic Z/nZ-cover of smooth curves D C and D C such that D (resp. D ) are ramified only in the points P i with ramification determined by the class a i Z/nZ (resp. a i Z/nZ). One can control such coverings using the fundamental group of the Riemann surfaces C (and C ) with k punctures. For example, if C has genus g, then the collection of such covers is in one-to-one correspondence with group morphisms π (C {P,..., P k }) = u,v,...,u g,v g,x,...,x k (u v u v...ugvgu g vg x...x k ) Aut Z/nZ

6 6 mapping x i to the multiplication by a i (and similarly for C mapping the x i to multiplication by a i ). To every such data corresponds a class of order n in Br C(x, y) which often corresponds to a central simple C(x, y)-algebra of degree n, that is, of dimension n 2 over C(x, y). Returning to the general setting of a projective normal variety X, let be a central simple C(X)-algebra of degree n. If K is the algebraic closure of C(X), then C(X) K M n (K) and by Galois descent the usual trace on M n (K) induces a trace map on making it a Cayley-Hamilton algebra of degree n. An important class of Cayley-Hamilton algebras of degree n is given by section algebras of O X - orders in. That is, let A be a sheaf of O X -algebras over X such that for every affine open subset U X we have Γ(U, A)Γ(U, O X ) = Because Γ(U, O X ) is integrally closed, the trace map on determines a trace map on the section algebra Γ(U, A) making it a Cayley-Hamilton algebra of degree n. We call a sheaf A of O X -orders in to be a noncommutative smooth model for if all affine section algebras are Cayley-smooth of degree n. The archetypical example of such a noncommutative smooth model is given by the Artin-Mumford counterexamples to the Lüroth problem [3], that is the construction of certain unirational non-rational threefolds. Let C and C be two smooth elliptic curves in P 2 intersecting transversally in {P,..., P 9 }, let all a i = 0 and consider two unramified Z/2Z-covers D C and D C. Then, D and D are again elliptic curves and the covers are given by dividing out a point of order two. Let be the corresponding central simple algebra over C(x, y) which is a quaternion algebra. Next, let X P 2 be the rational projective surface obtained by blowing up the P i and let A be a maximal O X -order in. M. Artin and D.Mumford are able to calculate the local description of A. If x X not lying on C C then A x is an Azumaya algebra and if x C C (the strict transforms), then i 2 = a A x = O X,x O X,x i O X,x j O X,x ij with j 2 = bt ji = ij where a and b are units in O X,x and t = 0 is a local equation for C C near x. Extending the classical notion of Brauer-Severi varieties of central simple algebras to these orders they define BS(A) which is a projective space bundle over X BS(A) π X Using the local description of A they show that BS(A) is a smooth variety, π is a flat morphisms and the geometric fibers are isomorphic to P (resp. to P P ) whenever x / C C (resp. x C C ). For specific starting configurations they then show that the threefold BS(A) is unirational but non-rational. With hindsight, the characteristic property of A allowing a local description, a smooth Brauer-Severi scheme and a description of the fibers is that A is a noncommutative smooth model for. In this book we will generalize these computations both to higher degree central simple algebras and higher dimensional base varieties. In chapter 6 we will give a complete characterization of the central simple algebras over C(S) where S is a projective smooth surface such that allows a noncommutative smooth model. For example, among the subclass of Br n C(x, y) described before by two curves and ramified covers those allowing a smooth model are precisely the configurations where all a i = 0, that is, such that the covers are unramified. In fact, for such a an explicit smooth model is obtained by taking

7 7 a maximal order A in where X is the surface obtained after blowing up all the intersection points P i. In this generality we will be able in chapter 5 and 6 to determine the étale local structure of A z in z C C (in all other points it is an Azumaya algebra). If is of degree n it is determined by combinatorial data consisting of a circuit on k n vertices ordered starting in the vertex having an extra loop. and an unordered partition p = (p,..., p k ) of n having exactly k parts. The m z - adic completion of A z is then isomorphic to the subalgebra of M n (C[[x, y]]) for local coordinates (x, y) near z having the following block form M p (C[[x, y]]) M p p 2 ((x))... M p p k ((x)) M p2 p (C[[x, y]]) M p2 (C[[x, y]])... M p2 p k ((x)) Â z..... M pk p (C[[x, y]]) m pk p 2 (C[[x, y]])... M pk (C[[x, y]]) The combinatorial data is constant along C and a possibly different data is constant along C. We will show in chapter 2 that for such orders the Brauer-Severi scheme BS(A) is a smooth variety. In chapter 8 we will give a combinatorial method to describe the fibers of the structural morphism BS(A) X. Consider the quiver-data p p2 p3 pk p k 2 p k That is, we add a vertex v 0 and connect it to vertex v i with p i arrows. We will prove in chapter 8 that the fiber is the moduli space of θ-semistable representations in the nullcone of this quiver with dimension vector (,,..., ) and where θ = (n, p, p 2,..., p k ). That is, we have to classify isomorphism classes of representations such that at least one of the arrows in the circuit is zero and such that the representation contains no proper subrepresentation of dimension vector

8 8 (, n, n 2,..., n k ) with all n i = 0 or such that n i p in i > 0. In chapter 7 we will give more details on such moduli spaces and the combinatorial aspects of θ-semistable representations.

9 Chapter Calogero Systems. One motivation to study noncommutative geometry comes from physics. One wants to understand the behaviour of n-particle systems when n. In this chapter we will give an illustrative example : collisions of Calogero particles. We will first describe the phase space of collisions of n Calogero particles Calo n using invariant theory. Its description is closely related to that of the Hilbert scheme of n points in the plane, Hilb n. In fact, Nakajima [2] and G. Wilson [33] have shown that there is a diffeomorphism of C (real) manifolds between the two spaces. However, this diffeomorphism does not respect the complex structure and, in fact, both spaces have fundamentally different properties. Recent work of Y. Berest and G. Wilson [4] relating the phase space of Calogero particles to the study of isomorphism classes of right ideals in the Weyl algebra suggests that Calo n is a coadjoint orbit of some infinite dimensional algebraic group. This conjecture was recently proved by V. Ginzburg [8] using noncommutative symplectic geometry. We will briefly indicate the main steps in Ginzburg s proof (more details will be given in chapter 2). The phase space Calo n turns out to be a fiber of the moment map on the representation space of a noncommutative smooth algebra. Following the lead of M. Karoubi [] and J. Cuntz and D. Quillen [7] one defines differential forms and de Rham cohomology for noncommutative algebras. In the case of the smooth algebra M under consideration the de Rham complex is shown to be acyclic, a result first proved by M. Kontsevich [3] for the free algebra. Moreover, there is a natural symplectic structure on M giving a natural one-to-one correspondence between -forms and derivations (vector fields). Ginzburg s result then follows from a noncommutative version of the classical exact sequence in symplectic geometry describing Hamiltonian vector fields. Similarly, the Hilbert scheme Hilb n is (part of) a fiber of the moment map but one knows that this space cannot be a coadjoint orbit. The distinction between the two cases comes from investigating noncommutative algebras, finite modules over their centers, associated to these fibers. It turns out that the algebra corresponding to Calo n is a smooth order (in fact it is even an Azumaya algebra over Calo n ), whereas that corresponding to Hilb n is not. The description and study of such smooth orders will be one of the main goals of this book. In chapter 2, we will be able to extend the above results to general quiver varieties. 9

10 0 CHAPTER. CALOGERO SYSTEMS.. Calogero particles. The Calogero system is a classical particle system of n particles on the real line with inverse square potential. x x 2 x n That is, if the i-th particle has position x i and velocity (momentum) y i, then the Hamiltonian is equal to H = n yi i= i<j (x i x j ) 2 The Hamiltonian equations of motions is the system of 2n differential equations dx i dt = H y i dy i dt = H x i This defines a dynamical system which is integrable. A convenient way to study this system is as follows. Assign to a position defined by the 2n vector (x, y ;..., x n, y n ) the couple of Hermitian or self-adjoint n n matrices i i y x x x x n i. x x 2 x y.. 2. X = and Y = i x n x n x i i n x n x x n x n y n Physical quantities are given by invariant polynomial functions under the action of the unitary group U n (C) under simultaneous conjugation. In particular one considers the functions F j = tr Xj j For example, { tr(x) = yi the total momentum 2 tr(x2 ) = 2 y 2 i i<j (x i x j) the Hamiltonian 2 We can now consider the U n (C)-translates of these matrix couples. This is shown to be a manifold with a free action of U n (C) such that the orbits are in one-toone correspondence with points (x, y ;... ; x n, y n ) in the phase space (that is, we agree that two such 2n tuples are determined only up to permuting the couples (x i, y i ). The n-functions F j give a completely integrable system on the phase space via Liouville s theorem, see for example []. In the classical case, all points are assumed to lie on the real axis and the potential is repulsive so that collisions do not appear. G. Wilson [33] considered an alternative where the points are assumed to lie in the complex numbers and such

11 .. CALOGERO PARTICLES. that the potential is attractive (to allow for collisions), that is, the Hamiltonian is of the form H = yi 2 2 (x i i<j i x j ) 2 giving again rise to a dynamical system via the equations of motion. One recovers the classical situation back if the particles are assumed only to move on the imaginary axis. x x 2 x n In general, we want to extend the phase space of n distinct points analytically to allow for collisions. When all the points are distinct, that is, if all eigenvalues of Y are distinct we will see in a moment that there is a unique GL n (C)-orbit of couples of n n matrices (up to permuting the n couples (x i, y i )). y x x x x n. x 2 x y.. 2. X = x n x n x n x x n x n y n For matrix couples in this standard form one verifies that... [X, Y ] = n... and Y = x... This equality suggests an approach to extend the phase space of n distinct complex Calogero particles to allow for collisions. Consider the 2n 2 + 2n-dimensional vectorspace (the notation will be explained later) rep α M = M n (C) M n (C) C n C n (where C n is the space of row-vectors). Consider the subvariety CALO n of quadruples (X, Y, u, v) such that [X, Y ] + u.v = n There is an action of GL n (C) on the vectorspace defined by g.(x, Y, u, v) = (gxg, gy g, gu, vg ) which preserves CALO n and which we will show to be free. We can define the phase space for Calogero collisions of n particles to be the orbit space Calo n = CALO n /GL n (C) the space of orbits of GL n (C) on CALO n. In a moment we will show : x n

12 2 CHAPTER. CALOGERO SYSTEMS. Theorem. The phase space Calo n of Calogero collisions of n-particles is a connected complex manifold of dimension 2n..2 The moment map. The moment map for the GL n (C)-action on M n (C) M n (C) C n C n is defined to be M n (C) M n (C) C n C n µ Mn (C) (X, Y, u, v) [X, Y ] + u.v We will be interested in the differential dµ of this map which we can compute by the ɛ-method : [(X + ɛa), (Y + ɛb)] + (u + ɛc).(v + ɛd) is equal to ([X, Y ] + u.v) + ɛ([x, B] + [A.Y ] + u.d + c.v) whence the differential dµ in the point (X, Y, u, v) is equal to dµ (X,Y,u,v) (A, B, c, d) = [X, B] + [A, Y ] + u.d + c.v. We say that u is a cyclic vector for the matrix-couple (X, Y ) M n (C) M n (C) if there is no proper subspace of C n containing u which is stable under left multiplication by X and Y. Lemma.2 The differential dµ is surjective in (X, Y, u, v) if u is a cyclic vector for (X, Y ). Proof. Consider the nondegenerate symmetric bilinear form on M n (C) M n (C) M n (C) C (M, N) tr(mn) Nondegeneracy means that tr(mn) = 0, N M n (C) is equivalent to M = 0. With respect to this inproduct on M n (C) the space orthogonal to the image of dµ (X,Y,u,v) is equal to {M M n (C) tr([x, B]M + [A, Y ]M + u.dm + c.vm) = 0, (A, B, c, d)} Because the trace does not change under cyclic permutations and is nondegenerate we see that this space is equal to {M M n (C) [M, X] = 0 [Y, M] = 0 Mu = 0 and vm = 0} But then, the kernel ker M is a subspace of C n containing u and stable under left multiplication by X and Y. By the cyclicity assumption this implies that ker M = C n or equivalently that M = 0. As dµ (X,Y,u,v) = 0 and tr is nondegenerate, this implies that the differential is surjective. It follows from the implicit function theorem that the image of the moment map is open in M n (C). If we denote by rep s α M (again, we will explain the terminology later) the open submanifold of M n (C) M n (C) C n C n consisting of quadruples (X, Y, u, v) such that u is a cyclic vector for (X, Y ) then we obtain

13 .3. HILBERT STAIRS. 3 Proposition.3 For every matrix M M n (C) in the image of the map rep s α M µ Mn (C) the inverse image µ (M) is a submanifold of M n (C) M n (C) C n C n dimension n 2 + 2n. of.3 Hilbert stairs. For the investigation of the GL n (C)-orbits on rep s α M we introduce a combinatorial gadget : the Hilbert n-stair. This is the lower triangular part of a square n n array of boxes n n filled with go-stones according to the following two rules : each row contains exactly one stone, and each column contains at most one stone of each color. For example, the set of all possible Hilbert 3-stairs is given below. To every Hilbert stair σ we will associate a sequence of monomials W (σ) in the free noncommutative algebra C x, y, that is W (σ) is a sequence of words in x and y. At the top of the stairs we place the identity element. Then, we descend the stairs according to the following rule. Every go-stone has a top word T which we may assume we have constructed before and a side word S and they are related as indicated below T T T S xt yt For example, for the Hilbert 3-stairs we have the following sequences of noncommutative words x x x y y y y x 2 yx x y 2 xy

14 4 CHAPTER. CALOGERO SYSTEMS. We will evaluate a Hilbert n-stair σ with associated sequence of non-commutative words W (σ) = {, w 2 (x, y),..., w n (x, y)} on rep α M = M n (C) M n (C) C n C n For a quadruple (X, Y, u, v) we replace every occurrence of x in the word w i (x, y) by X and every occurrence of y by Y to obtain an n n matrix w i = w i (X, Y ) M n (C) and by left multiplication on u a column vector w i.v. The evaluation of σ on (X, Y, u, v) is the determinant of the n n matrix σ(x, Y, u, v) = det u w 2.u w 3.u... w n.u For a fixed Hilbert n-stair σ we denote with rep(σ) the subset of quadruples (X, Y, u, v) in rep α M such that the evaluation σ(v, X, Y ) 0. Theorem.4 For every Hilbert n-stair, rep (σ) Proof. Let u be the basic column vector 0 e =. 0 Let every black stone in the Hilbert stair σ fix a column of X by the rule 0... i i 0 0 n j n X =... 0 j That is, one replaces every black stone in σ by at the same spot in X and fills the remaining spots in the same column by zeroes. The same rule applies to Y for white stones. We say that such a quadruple (X, Y, u, v) is in σ-standard form. With these conventions one easily verifies by induction that w i (X, Y )e = e i for all 2 i n. Hence, filling up the remaining spots in X and Y σ(x, Y, u, v) 0 proving the claim. arbitrarily one has that Hence, rep (σ) is an open subset of rep α M (and even of rep s α M) for every Hilbert n-stair σ. Further, for every word (monomial) w(x, y) and every g GL n (C) we have that w(gxg, gy g )gv = gw(x, Y )v and therefore the open sets rep (σ) are stable under the GL n (C)-action on rep α M. We will give representatives of the orbits in rep (σ).

15 .3. HILBERT STAIRS. 5 Let W n = {, x,..., x n, xy,..., y n } be the set of all words in the non-commuting variables x and y of length n, ordered lexicographically. For every quadruple (X, Y, u, v) rep α M consider the n m matrix ψ(x, Y, u, v) = [ u Xu X 2 u... Y n u ] where m = 2 n+ and the j-th column is the column vector w(x, Y )v with w(x, y) the j-th word in W n. Hence, (X, Y, u, v) rep (σ) if and only if the n n minor of ψ(x, Y, u, v) determined by the word-sequence {, w 2,..., w n } of σ is invertible. Moreover, as ψ(gxg, gy g, gu, vg ) = gψ(v, X, Y ) we deduce that the GL n (C)-orbit of (X, Y, u, v) rep α M contains a unique quadruple (X, Y, u, v ) such that the corresponding minor of ψ(x, Y, u, v ) = n. Hence, each GL n (C)-orbit in rep (σ) contains a unique representant in σ- standard form. Therefore, Proposition.5 The action of GL n (C) on rep (σ) is free and the orbit space is an affine space of dimension n 2 + 2n. rep (σ)/gl n (C) Proof. The dimension is equal to the number of non-forced entries in X, Y and v. As we fixed n columns in X or Y this dimension is equal to k = 2n 2 (n )n + n = n 2 + 2n. The argument above shows that every GL n (C)-orbit contains a unique quadruple in σ-standard form so the orbit space is an affine space. Theorem.6 The orbit space rep s α M/GL n (C) is a complex manifold of dimension n 2 + 2n and is covered by the affine spaces rep (σ). Proof. Recall that rep s α M is the open submanifold consisting of quadruples (x, Y, u, v) such that u is a cyclic vector of (X, Y ) or equivalently such that C X, Y u = C n where C X, Y is the not necessarily commutative subalgebra of M n (C) generated by the matrices X and Y. Hence, clearly rep (σ) rep n M for any Hilbert n-stair σ. Conversely, we claim that a quadruple (X, Y, u, v) rep s α M belongs to at least one of the open subsets rep (σ). Indeed, either Xu / Cu or Y u / Cu as otherwise the subspace W = Cu would contradict the cyclicity assumption. Fill the top box of the stairs with the corresponding stone and define the 2-dimensional subspace V 2 = Cu + Cu 2 where u = u and u 2 = w 2 (X, Y )u with w 2 the corresponding word (either x or y). Assume by induction we have been able to fill the first i rows of the stairs with stones leading to the sequence of words {, w 2 (x, y),..., w i (x, y)} such that the subspace V i = Cu Cu i with u i = w i (X, Y )v has dimension i.

16 6 CHAPTER. CALOGERO SYSTEMS. Then, either Xu j / V i for some j or Y u j / V i (if not, V i would contradict cyclicity). Then, fill the j-th box in the i + -th row of the stairs with the corresponding stone. Then, the top i+ rows of the stairs form a Hilbert i+-stair as there can be no stone of the same color lying in the same column. Define w i+ (x, y) = xw i (x, y) (or yw i (x, y)) and u i+ = w i+ (X, Y )u. Then, V i+ = Cu Cu i+ has dimension i +. Continuing we end up with a Hilbert n-stair σ such that (X, Y, u, v) rep (σ). This concludes the proof. Example.7 Orbits when n = 3. Representatives for the GL 3 (C)-orbits in rep (σ) are given by the following quadruples for σ a Hilbert 3-stair : X Y u v a b 3 2 c d a b5 4 0 a b 3 2 c d5 4 0 a b 3 0 c d5 0 e f 0 c 0 e f e f g h i j5 4 d e f 3 2 g h i 5 4 g 0 h 3 2 i 0 j5 4 0 g h 3 i j5 k l j k l k l 0 k l 2 4 a b c 3 2 d e f5 4 a 0 b 3 c 0 d5 g h i e f j k5 4 0 g h 3 i j5 0 l 0 k l ˆm n o ˆm n o ˆm n o ˆm n o ˆm n o ˆm n o Let λ = λ n be a scalar matrix in M n (C) and hence fixed under the action by conjugation of GL n (C). Then, the subvariety µ (λ) of rep α M is GL n -stable. Because the GL n (C)-action is free on rep s α M we have the following situation µ (λ) rep s α M rep s α M (µ (λ) rep s α M)/GL n (C) rep s α M/GL n (C) and we obtain : Theorem.8 For a scalar matrix λ M n (C) lying in the image of µ, the orbit space (µ (λ) rep s α M)/GL n (C) is a submanifold of rep s α M/GL n (C) of dimension 2n. We will now investigate two of these manifolds : the Hilbert scheme of n points in the plane and the phase space of collisions of n Calogero particles..4 The Hilbert scheme Hilb n. Consider n distinct points in the complex plane C 2. Identifying C 2 with R 4 we can view these points as particles in (flat) space-time. Particles have a tendency to

17 .4. THE HILBERT SCHEME HILB N. 7 collide with each other and we want to construct a manifold describing all possible collisions. To a point p = (a, b) C 2 corresponds a maximal ideal m = (x a, y b) of the polynomial algebra C[x, y]. To a set of n distinct points P = {p,..., p n } we can associate a codimension n ideal i P = m... m n C[x, y] Now, consider an arbitrary codimension n ideal i C[x, y] and fix a basis {v,..., v n } in the quotient space V i = C[x, y] i = Cv Cv n. Multiplication by x on C[x, y] induces a linear operator on the quotient V i and hence determines a matrix X i M n (C) with respect to the chosen basis {v,..., v n }. Similarly, multiplication by y determines a matrix Y i M n (C). Moreover, the image of the unit element C[x, y] in V i determines with respect to the basis {v,..., v n } a column vector u C n = V i. Clearly, this vector and matrices satisfy : [X i, Y i ] = 0 and C[X i, Y i ]u = C n. Here, C[X i, Y i ] is the n-dimensional subalgebra of M n (C) generated by the two matrices X i and Y i. In particular, u is a cyclic vector for the matrix-couple (X, Y ). We have seen that to a codimension n ideal i corresponds a cyclic triple (u i, X i, Y i ) such that X i and Y i commute with each other. Conversely, if (X, Y, u) M n (C) M n (C) C n is a cyclic triple such that [X, Y ] = 0, then C X, Y = C[X, Y ] is an n-dimensional commutative subalgebra of M n (C) and the kernel of the natural epimorphism C[x, y] C[X, Y ] x X y Y is a codimension n ideal i of C[x, y]. Indeed, the linear map C[x, y] φ C n defined by sending a polynomial f(x, y) C[x, y] to the vector f(x, Y ).u is surjective by the cyclicity assumption. However, there is some redundancy in the assignment i (Xi, Y i, u i ) as it depends on the choice of basis of V i. If we choose a different basis {v,..., v n} with basechange matrix g GL n (C), then the corresponding triple is (X i, Y i, u i) = (g.x i.g, g.y i.g, gu i ) That is, we have an action of GL n (C) on the space of all triples GL n (C) (M n (C) M n (C) C n ) Mn (C) M n (C) C n (g, (X, Y, u)) (g.x.g, g.y.g, gu) The above discussion shows that there is a one-to-one correspondence between codimension n ideals i of C[x, y], and GL n (C)-orbits of cyclic triples (X, Y, u) in M n (C) M n (C) C n such that [X, Y ] = 0.

18 8 CHAPTER. CALOGERO SYSTEMS. Example.9 The Hilbert scheme Hilb. The space of all triples when n = is (X, Y, u) C C C. Clearly, they are all commuting and u is a cyclic vector for (X, Y ) if and only if u 0. The basechange group in this case is GL (C) = C and it acts on the space of triples via g.(x, Y, u) = (X, Y, gu) Therefore, the GL (C) orbits of the cyclic (commuting) triples are parameterized by the points {(X, Y, ) C C C} C 2 The ideal i of codimension one corresponding to (X, Y, ) are the polynomials vanishing in the point p = (X, Y ) C 2. Hence, Hilb C 2. Example.0 The Hilbert scheme Hilb 2. Consider a triple (X, Y, u) M 2 (C) M 2 (C) C 2 and assume that either X or Y has distinct eigenvalues (type a). As»»» ν 0 a b 0 (ν [, ] = ν 2 )b 0 ν 2 c d (ν 2 ν )c 0 we have a representant in the orbit of the form» λ 0 (, 0 λ 2» µ 0, 0 µ 2» u ) u 2 where cyclicity of the column vector implies that u u 2 0. The stabilizer subgroup of the matrix-pair is the group of diagonal matrices C C GL 2 (C), hence the orbit has a unique representant of the form» λ 0 ( The corresponding ideal i C[x, y] is then 0 λ 2,» µ 0, 0 µ 2» ) i = {f(x, y) C[x, y] f(λ, µ ) = 0 = f(λ 2, µ 2 )} hence these orbits correspond to sets of two distinct points in C 2. The situation is slightly more complicated when X and Y have only one eigenvalue (type b). If (X, Y, u) is a cyclic commuting triple, then either X or Y is not diagonalizable. But then, as»» ν a b [ 0 ν, c ] = d» c d a 0 c we have a representant in the orbit of the form»»» λ α µ β u (,, ) 0 λ 0 µ u 2 with [α : β] P and u 2 0. The stabilizer of the matrixpair is the subgroup» c d { c 0} 0 c GL 2 (C) and hence we have a unique representant of the form»»» λ α µ β 0 (,, ) 0 λ 0 µ The corresponding ideal i C[x, y] is i = {f(x, y) C[x, y] f(λ, µ) = 0 and α f f (λ, µ) + β (λ, µ) = 0} x y as one proves by verification on monomials because» k» l»» λ α µ β 0 kαλ = 0 λ 0 µ k µ l + lβλ k µ l λ k µ l Therefore, i corresponds to the set of two points at (λ, µ) C 2 infinitesimally attached to each other in the direction α x + β y. For each point in C2 there is a P family of such fat points.

19 .4. THE HILBERT SCHEME HILB N. 9 Thus, points of Hilb 2 correspond to either of the following two situations : C 2 C 2 p p p type a type b π The Hilbert-Chow map Hilb 2 S 2 C 2 (where S 2 C 2 is the symmetric power of C 2, that is S 2 = Z/2Z orbits of couples of points from C 2 ) sends a point of type a to the formal sum [p] + [p ] and a point of type b to 2[p]. Over the complement of (the image of) the diagonal, this map is a one-to-one correspondence. However, over points on the diagonal the fibers are P corresponding to the directions in which two points can approach each other in C 2. As a matter of fact, the symmetric power S 2 C 2 has π singularities and the Hilbert-Chow map Hilb 2 S 2 C 2 is a resolution of singularities. Theorem. Let rep α M µ Mn (C) be the moment map, then Hilb n (µ (0) rep s α M)/GL n (C) and is therefore a complex manifold of dimension 2n. Proof. We identify the triples (X, Y, u) M n (C) M n (C) C n such that u is a cyclic vector of (X, Y ) and [X, Y ] = 0 with the subspace {(X, Y, u, 0) [X, Y ] = 0 and u is cyclic } rep s α M which is clearly contained in µ (0). To prove the converse inclusion assume that (X, Y, u, v) is a cyclic quadruple such that [X, Y ] + uv = 0. Let m(x, y) be any word in the noncommuting variables x and y. We claim that v.m(x, Y ).u = 0. We will prove this by induction on the length l(m) of the word m(x, y). l(m) = 0 then l(x, y) = and we have When v.l(x, Y ).u = v.u = tr(u.v) = tr([x, Y ]) = 0. Assume we proved the claim for all words of length < l and take a word of the form m(x, y) = m (x, y)yxm 2 (x, y) with l(m ) + l(m 2 ) + 2 = l. Then, we have wm(x, Y ) = wm (X, Y )Y Xm 2 (X, Y ) = wm (X, Y )([Y, X] + XY )m 2 (X, Y ) = (wm (X, Y )v).wm 2 (X, Y ) + wm (X, Y )XY m 2 (X, Y ) = wm (X, Y )XY m 2 (X, Y ) where we used the induction hypotheses in the last equality (the bracketed term vanishes).

20 20 CHAPTER. CALOGERO SYSTEMS. Hence we can reorder the terms in m(x, y) if necessary and have that wm(x, Y ) = wx l Y l2 with l + l 2 = l and l the number of occurrences of x in m(x, y). Hence, we have to prove the claim for X l Y l2. wx l Y l2 v = tr(x l Y l2 vw) = tr(x l Y l2 [X, Y ]) = tr([x l Y l2, X]Y ) = tr(x l [Y l2, X]Y ) = l 2 i=0 tr(xl Y i [Y, X]Y l2 i ) = l 2 i=0 tr(y l2 i X l Y i [Y, X] = l 2 i=0 tr(y l2 i X l Y i v.w = l 2 i=0 wy m2 i X l Y i v But we have seen that wy l2 i X l Y i = wx l Y l2 hence the above implies that wx l Y l2 v = l 2 wx l Y l2 v. But then wx l Y l2 v = 0, proving the claim. Consequently, w.c X, Y.v = 0 and by the cyclicity condition we have w.c n = 0 hence w = 0. Finally, as v.w + [X, Y ] = 0 this implies that [X, Y ] = 0 and we can identify the fiber µ (0) with the indicated subspace. From this the result follows. We can cover the subset (X, Y, u, 0) such that [X, Y ] = 0 and u a cyclic vector by their intersections with the rep (σ) for σ a Hilbert n-stair. In particular, we can cover Hilb n by open subsets Hilb n (σ) = {(X, Y, u, 0) in σ-standard form such that [X, Y ] = 0}. Example.2 The Hilbert scheme Hilb 2. Consider Hilb 2 ( ). Because»»» 0 a c d ae d af ac bd [, ] = b e f c + be f d ae this subset can be identified with C 4 using the equalities Similarly, Hilb 2 ( ) C 4. d = ar Example.3 The Hilbert scheme Hilb 3. andf = c + be. Up to change of colors there are three 3-stairs to consider. We claim that Hilb 3 ( ) C 6. For consider the commutator 2 [ 4 0 a b 3 2 c d5, 4 0 g h b g ai + bk cg eh aj + bl dg fh 0 i j5 ] = 4 d i g + dk ej h + cj + dl di fj5 0 e f k k f k a ck el + ei + fk b dk + ej Taking the Groebner basis of these relations one finds the following relations 8 f = k g = ej ik >< d = i h = i 2 cj + jk il b = g >: a = ei ck + k 2 el

21 .4. THE HILBERT SCHEME HILB N. 2 from which the claim follows. In a similar manner one proves that However, the situation for Hilb 3 ( ) C 6. is more complicated. Hilb 3 ( ) Observe that some of these intersections may be empty. For example, for the Hilbert 5-stair Hilb 5 ( ) = Indeed, the associated series of words is {, x, y, xy, yx} whence σ(x, Y, u, 0) = 0 whenever [X, Y ] = 0. Hence all Hilbert stairs σ containing this stair (that is, if we recover the 5-stair after removing certain rows and columns) satisfy Hilb n (σ) =. We have shown that Hilb n is a manifold of dimension 2n. A priori it may have many connected components (all of dimension 2n). We will now show that Hilb n is connected. Theorem.4 The Hilbert scheme Hilb n of n points in C 2 is a complex connected manifold of dimension 2n. Proof. The symmetric power S n C parametrizes sets of n-points on the line C and can be identified with C n. Consider the map Hilb n π S n C defined by mapping a cyclic triple (X, Y, u) with [X, Y ] = 0 in the orbit corresponding to the point of Hilb n to the set {λ,..., λ n } of eigenvalues of X. Observe that this map does not depend on the point chosen in the orbit. Let be the big diagonal in S n C, that is, S n C is the space of all sets of n distinct points from C. Clearly, S n C is a connected n-dimensional manifold. We claim that π (S n C ) (S n C ) C n and hence is connected. Indeed, take a matrix X with n distinct eigenvalues {λ,..., λ n }. We may diagonalize X. But then, as y... y n (λ λ )y... (λ λ n )y n [ λ...,.. ] =.. λ n y n... y nn (λ n λ )y n... (λ n λ n )y nn we see that also Y must be a diagonal matrix with entries (µ,..., µ n ) C n where µ i = y ii. But then the cyclicity condition implies that all coordinates of v must be non-zero. Now, the stabilizer subgroup of the commuting (diagonal) matrix-pair (X, Y ) is the maximal torus T n = C... C of diagonal invertible n n matrices.

22 22 CHAPTER. CALOGERO SYSTEMS. Using its action we may assume that all coordinates of v are equal to. That is, the points in π ({λ,..., λ n }) with λ i λ j have unique (up to permutation as before) representatives of the form λ λ 2 (... λn µ, µ 2...,. ) µn that is π ({λ,..., λ n } can be identified with C n, proving the claim. Next, we claim that all the fibers of π have dimension at most n. Let {λ,..., λ n } S n C then there are only finitely many X in Jordan normalform with eigenvalues {λ,..., λ n }. Fix such an X, then the subset T (X) of cyclic triples (X, Y, u) with [X, Y ] = 0 has dimension at most n + dim C(X) where C(X) is the centralizer of X in M n (C), that is, C(X) = {Y M n (C) XY = Y X}. The stabilizer subgroup Stab(X) = {g GL n (C) gxg = X} is an open subset of the vectorspace C(X) and acts freely on the subset T (X) because the action of GL n (C) on µ (0) rep s α M has trivial stabilizers. But then, the orbitspace for the Stab(X)-action on T (X) has dimension at most n + dim C(X) dim Stab(X) = n. As we only have to consider finitely many X this proves the claim. The diagonal has dimension n in S n C and hence by the foregoing we know that the dimension of π ( ) is at most 2n. Let H be the connected component of Hilb n containing the connected subset π (S n C ). If π ( ) were not entirely contained in H, then Hilb n would have a component of dimension less than 2n, which we proved not to be the case. This finishes the proof..5 The phase space Calo n. Recall that Calogero quadruples were defined to be CALO n = {(X, Y, u, v) [X, Y ] + u.v = n } and that the phase space of collisions of n Calogero particles is the orbit space Calo n = CALO n /GL n (C). Theorem.5 Let rep α M µ Mn (C) be the moment map, then Calo n µ ( n )/GL n (C) = (µ ( n ) rep s α M)/GL n (C) and is therefore a complex manifold of dimension 2n. Proof. The result will follow if we can prove that any Calogero quadruple (X, Y, u, v) has the property that u is a cyclic vector, that is, lies in rep s α M. Assume that U is a subspace of C n stable under X and Y and containing u. U is then also stable under left multiplication with the matrix A = [Y, X] + n

23 .5. THE PHASE SPACE CALO N. 23 and we have that tr(a U) = tr( n U) = dim U. On the other hand, A = u.v and therefore u A. c. =.. [ u ] n v... v n. c. = ( v i c i ). c n u n c i= n u n Hence, if we take a basis for U containing u, then we have that tr(a U) = a where A.u = au, that is a = u i v i. But then, tr(a U) = dim U is independent of the choice of U. Now, C n is clearly a subspace stable under X and Y and containing u, so we must have that a = n and so the only subspace U possible is C n proving cyclicity of u with respect to the matrix-couple (X, Y ). Again, it follows that we can cover the phase space Calo n by open subsets Calo n (σ) = {(X, Y, u, v) in σ-standard form such that [X.Y ] + u.v = n } where σ runs over the Hilbert n-stairs. Example.6 The phase space Calo 2. Consider Calo 2 ( ). Because»» 0 a c d [, ] + b e f» 0. ˆg h 2 =» g d + ae h + af ac bd c f + be d ae We obtain after taking Groebner bases that the defining equations are 8 g = 2 >< h = b f = c + eh >: d = + ae In particular we find»» 0 a c + ae Calo 2 ( ) = {(, b e c + be,» 0, ˆ2 b ) a, b, c, e C} C 4 and a similar description holds for Calo 2 ( ). Example.7 The phase space Calo 3. We claim that Calo 3 ( ) C 6 For, if we compute the 3 3 matrix 2 [ 4 0 a b 3 2 c d5, 4 0 g h i j5] ˆm n o 3 0 e f k l 0 then the Groebner basis for its entries gives the following defining equations 8 m = 3 n = c + k o = i + l >< f = k d = o l g = 2 + b l = g ej kl + ko h = 2jk + 2l >: 2 jn 3lo + o 2 a = 2k 2 2el kn + eo

24 24 CHAPTER. CALOGERO SYSTEMS. In a similar manner one can show that Calo 3 ( is again more difficult to describe. ) C 6 but Calo 3 ( ) We will prove that Calo n is connected by a strategy similar to that used for Hilb n. Proposition.8 Let (X, Y, u, v) CALO n and suppose that X is diagonalizable. Then. all eigenvalues of X are distinct, and 2. the GL n (C)-orbit contains a representative such that α λ λ λ λ n. λ X = λ... 2 λ α.. 2. Y = λ n λ n λ n λ n λ λ n λ n α n u = v = [... ]. and this representative is unique up to permutation of the n couples (λ i, α i ). Proof. Choose a representative with X a diagonal matrix as indicated. Equating the diagonal entries in [X, Y ] + u.v = n we obtain that for all i n we have u i v i =. Hence, none of the entries of [Y, X] + n = u.v is zero. Consequently, by equating the (i, j)-entry it follows that λ i λ j for i j. The representative with X a diagonal matrix is therefore unique up to the action of a diagonal matrix D and of a permutation. The freedom in D allows us to normalize u and v as indicated, the effect of the permutation is described in the last sentence. Finally, the precise form of Y can be calculated from the normalized forms of X, u and v and the equation [X, Y ] + u.v = n. Consider the map Calo n π S n C by mapping a point in CALO n to the set of the eigenvalues of X (and as this does not depend on the point in the orbit this map factors through Calo n ). The foregoing proposition describe π (S n C ) where is the big diagonal and hence the subset of Calo n with X diagonalizable is connected as it coincides with (S n C ) C n. The identification is made through the parameters λ i and α i of the proposition. Theorem.9 The phase space Calo n is a complex connected manifold of dimension 2n.

25 .6. THE NONCOMMUTATIVE SMOOTH ALGEBRA M. 25 Proof. Reasoning as in the proof for the Hilbert scheme, the result will follow once we prove that all the fibers of π have dimension at most n. Consider the projection to the last two factors CALO n p C 2n For a fixed matrix X M n (C) we claim that the subset CALO n (X) (Calogero quadruples with fixed X) maps under p into an n-dimensional subvariety of C 2n. Because p is GL n (C)-equivariant we may assume that X is in Jordan normal form. Consider first the case of one block, that is λ X = λ λ Then, in [Y, X] every diagonal below the main diagonal has entries that add up to 0 as one verifies. On the other hand, for a rank one matrix, the lowest nonvanishing diagonal can have just one non-zero entry. Therefore, the rank one matrix [Y, X] + n = u.v is upper triangular and there is just one non-zero entry (which must be equal to n by taking traces) on the main diagonal. If the first non-zero entry of v occurs at place i then the last non-zero entry of u must also occur in place i and the product of both must be equal to n. In this way, the possible pairs (u, v) fall into n families (indexed by i) each of dimension n. This proves the claim in the case of one Jordan block. For the general case X = j X j one writes Y, u and v in the corresponding block forms, and one sees that [Y jj, X j ] + n = u j.v j and one repeats the above argument for each of the blocks, proving the claim. Now, define Calo n (O) to be the subset of Calo n represented by quadruples (X, Y, u, v) such that X belongs to a fixed conjugacy class O in M n (C). We claim that Calo n (O) has dimension at most n. Fix a matrix X O, then Calo n (O) = CALO n (X)/G where G is the centralizer of X in GL n (C). Now, the part of CALO n (X) lying over a fixed (u, v) C 2n is parametrized by the Lie algebra Lie(G) and so by the foregoing claim dim CALO n (X) n + dim G Finally, the action of G is free and we have proved that all the fibers of π have dimension at most n..6 The noncommutative smooth algebra M. We will now bring in some noncommutative algebras. Consider the following quiver (that is, directed graph) on two vertices x e y u v f

26 26 CHAPTER. CALOGERO SYSTEMS. We define M to be the path algebra of this quiver. That is, as a C-vectorspace M has as basis the oriented paths in the quiver, including those of length zero corresponding to the two vertices. We agree that we write paths from right to left (as we do with compositions of morphisms). To each vertex there is a path of length zero. An associative algebra structure on M is induced by concatenation of paths when possible and zero otherwise. That is, M is the algebra on 6 noncommuting generators { e, f the paths of length zero x, y, u, v the paths of length one Concatenation of paths induces the following defining relations for M e 2 = e f 2 = f e + f = e.x = x e.y = y e.u = u e.v = 0 x.e = x y.e = y u.e = 0 v.e = v f.x = 0 f.y = 0 f.u = 0 f.v = v x.f = 0 y.f = 0 u.f = u v.f = 0 x.v = 0 u.x = 0 y.v = 0 u.y = 0 u.u = 0 v.v = 0 Horrible as these relations may seem, the algebra M has one important property, it is (formally) smooth. That is, if A is any C-algebra having a twosided ideal I with I 2 = 0, then we can lift C-algebra morphisms M... φ A φ π A I from the quotient to A. The relevance of this notion will become clear when we study differential forms and connections on noncommutative manifolds. This lifting property can be seen as follows. Let a A be such that π(a) = φ(e), then one verifies that E = (2 a) 2 a 2 is an idempotent of A such that π(e) = φ(e). Define a lift φ by sending e E, f F = E and x any element of E.(φ(x) + I).E, y any element of E.(φ(y) + I).E, u any element of E.(φ(u) + I).F, v any element of F.(φ(v) + I).E. and one immediately verifies that all the relations holding in M are preserved under φ whence φ is an algebra morphism lifting φ. A representation of a quiver assigns to each vertex a finite dimensional vectorspace and to each arrow a linear map between the corresponding vertex-spaces. The dimension-vector of a representation is then the integral vector containing the

27 .6. THE NONCOMMUTATIVE SMOOTH ALGEBRA M. 27 dimensions of the vertex-spaces. Fixing a dimension-vector and a basis in each vertex-space we see that the representations form an affine space. For the quiver described above, the set of representations of dimension vector α = (m, n) can be identified with the affine space rep α (M) = M n (C) M n (C) M n m (C) M m n (C) the factors corresponding respectively to the arrows x, y, u and v. On this affine space there is a natural action of the algebraic group GL(α) = GL n (C) GL m (C) given by base-change in the vertex-spaces. That is, (g, h) GL(α) acts as (g, h).(x, Y, U, V ) = (gxg, gy g, guh, hv g ). Two representations are said to be isomorphic if and only if they belong to the same GL(α)-orbit. We will now relate the vectorspaces rep α M and the action of GL(α) on it to the study of finite dimensional representations of M. For an integer n N, an n-dimensional representation of M is a C-algebra morphism M φ M n (C) and two n-dimensional representations φ and φ 2 are said to be isomorphic (or equivalent) iff there is a g GL n (C) such that the diagram below commutes M M n (C) φ c g φ2 M n (C) where c g is conjugation by g on M n (C). Because M is an affine C-algebra, the set of all n-dimensional representations is an affine variety rep n M. Indeed, any representation is determined by the images of the generators e, f, u, v, x and y. That is, rep n M M n (C) 6 is the closed subvariety where the ideal of relations is generated by the entries of the matrix identities determined by the defining relations for M. Clearly, conjugation on M n (C) defines an action of GL n (C) on the affine variety rep n M. For α = (n, m) let n = n + m and consider the diagonal embedding of GL(α) in GL n (C) [ ] GLn (C) 0 GLn (C). 0 GL m (C) Using this embedding there is a natural GL(α) action on the product GL n (C) rep α M given by [ ] g 0 (g, h).(g, X, Y, U, V ) = (G 0 h, gxg, gy g, guh, hv g ) and the space of GL(α)-orbits is called the associated fiber bundle and denoted by GL n (C) GL(α) rep α M Left multiplication defines a GL n (C)-action on the product GL n (C) rep α M which commutes with the action of GL(α) and hence induces an action on the associated fiber bundle.

28 28 CHAPTER. CALOGERO SYSTEMS. Lemma.20 The representation spaces rep n M are affine manifolds and rep n M as manifolds with GL n (C)-action. α=(n,m) n+m=n GL n (C) GL(α) rep α M Proof. Given a representation M φ Mn (C), the images φ(e) and φ(f) give an orthogonal decomposition of n into idempotents. If the rank of φ(e) is n, then the rank of φ(f) is m = n n and under simultaneous conjugation by an element G GL n (C) we reduce to the case that φ(e) = [ n ] and φ(f) = [ ] m But then using equations such as exe = x and euf = u we deduce that the images of the other generators are of the following matrix shape in M n (C) φ(x) = [ ] X φ(y) = [ ] Y φ(u) = [ ] 0 U 0 0 φ(v) = [ ] 0 0 V 0 and hence correspond to a point in rep α M for α = (n, m). The assertion is now easy to verify. In chapter 5 we will prove that the representation spaces rep n A of any formally smooth algebra are affine manifolds and even have an analytic local description by quiver representations. For applications to Calogero particles and Hilbert schemes, we specialize to the case m =, that is, α = (n, ) and n = n + and recover the vectorspace rep α M of the previous section. Lemma.2 There is a natural one-to-one correspondence between GL n C -orbits in rep α M = M n (C) M n (C) C n C n GL n -orbits in M n (C) M n (C) C n C n as in the previous section Proof. One implication is obvious by taking (g, ) GL(α) = GL n C. Conversely, assume that the quadruples (X, Y, u, v) and (X, Y, u, v ) belong to the same GL(α)-orbit in rep α M. Then there is a g GL n and λ C such that gxg = X gy g = Y guλ = u and λvg = v Then, taking the matrix g = g.(λ n ) GL n we see that these quadruples also belong to the same GL n -orbit. Therefore, we would like to construct an orbit space for the GL(α)-action on rep α M. However, such an orbit-space would have horrible topological properties (such as being non-hausdorff) as there are GL(α)-orbits which are not closed. For example, if α = (2, ) consider the representation X = [ ] Y = [ ] [ 0 u = 0] v = [ 0 0 ]

29 .7. INVARIANT THEORY AND REP α M. 29 [ ] t 0 and consider the elements G t = (, ) GL(α), then the action on the above 0 quadruple gives the representation [ ] [ ] [ 0 t X = Y = u = v = ] [ 0 0 ] whence if t GL(α)-orbit. 0 we see that the zero representation lies in the closure of the.7 Invariant theory and rep α M. Invariant theory provides us with the best Hausdorff approximation to the orbit space problem, that is a classifying space for the closed orbits. We will prove in chapter 4 that closed GL n (C)-orbits in the representation space rep n A for an arbitrary affine C-algebra A are in one-to-one correspondence with isomorphism classes of semi-simple n-dimensional representations of A. Recall that a representation is simple if it has no proper (non-zero) subrepresentations and is semi-simple if it is the direct sum of simple representations. Our first job is to find a criterium on the dimension vector α = (n, m) to ensure that rep α M contains simple representations. A necessary condition is m n m Indeed, if m > n then any representation (X, Y, U, V ) contains as non-trivial subrepresentation the trivial representation (all matrices zero-matrices) of dimension vector (dim Ker U, 0). In chapter 6 we will give a combinatorial description of the dimension vectors of simple representations of quivers which implies that for the quiver under consideration this necessary condition is also sufficient. In particular, for α = (, n) there is an open submanifold of rep α M consisting of simple representations. Invariant theory learns us that closed orbits can be separated by invariant polynomial functions. We will focuss here on the special case of interest α = (n, ) although the arguments hold more generally as we will prove in chapter 4. The coordinate ring of rep α M is the polynomial algebra C[rep α M] = C[x,..., x nn, y,..., y nn, u,..., u n, v,..., v n ] where the x ij, y ij, u i and v j with i, j n are the coordinate functions of the matrices (X, Y, u, v). The group GL(α) acts on this algebra by automorphisms as follows, let G = (g, λ) GL(α) = GL n C then ψ G is the automorphism sending x... x n x ij to the (i, j) entry of the matrix g.. x n... x nn y... y n y ij to the (i, j) entry of the matrix g.. y n... y nn n g, g,

30 30 CHAPTER. CALOGERO SYSTEMS. u i to the (i, ) entry of the matrix g u. u n λ, v j to the (, j) entry of the matrix λ [ v... v n ] g. The ring of polynomial invariants C[rep α M] GL(α) is the subalgebra consisting of polynomials P such that ψ G (P ) = P for all G GL(α). We will prove in chapter 5 that this algebra is generated by traces along oriented cycles in the quiver. That is, consider all necklace words w w where each bead is one of the following n n matrices x... x n y... y n =.. =.. x n... x nn y n... y nn = u v... u v n.. u n v... u n v n Multiplying these bead-matrices cyclicly in the indicated orientation and taking the trace of the n n matrix obtained gives a polynomial tr(w) of C[rep α M] which is clearly left invariant under the GL(α)-action. The assertion is that these invariants generate all the invariant functions. We will even show that it suffices to take necklace words having at most (n + ) 2 + beads. Assume there are s distinct necklace words of length (n + ) 2 +, then we can evaluate tr(w) at a representation (X, Y, u, v) rep α M by substituting the entries for the coordinate functions and obtain a map rep α M π C s The image of π will be shown to be the affine variety corresponding to the ring of invariant polynomials. It is called the quotient variety and is denoted rep α M/GL(α). If ξ Im π then π (ξ) contains a unique closed orbit. In particular, if two semisimple representations (X, Y, u, v ) and (X 2, Y 2, u 2, v 2 ) have all their necklaceinvariants equal then they belong to the same orbit. These quotients varieties are in general not manifolds. In fact, we will give in chapter 6 combinatorial tools to determine the singularities and to describe the analytic local structure of the quotient variety near these singularities. Applying these results we will see that rep α M/GL(α) always has singularities except in the trivial case α = (, ) where the quotient variety is easily seen to be C 3. When studying rep α M for a specific dimension vector α = (n, ), working with M is overdoing things. We will construct another noncommutative algebra M(n) which is a finite module over the ring of polynomial invariants C[rep α M] GL(α). We have a canonical morphism M tn M(n) with the property that α-dimensional

31 .8. DE RHAM COHOMOLOGY FOR M. 3 representation of M (with the images of e and f fixed) factors through t n. One way to define M(n) is as the ring of equivariant maps from rep α M to M n+ (C). Embed GL(α) diagonally in GL n+ (C), that is [ ] GLn (C) 0 GL(α) = 0 C GLn+ (C) then GL(α) acts on M n+ (C) via the conjugation action of GL n+ (C). A polynomial map rep α M p Mn+ (C) is said to be GL(α)-equivariant if it is compatible with the GL(α)-action on source and target space. That is, for all (g, λ) GL(α) and all (X, Y, u, v) rep α (C) we have [ ] [ ] p(gxg, gy g, guλ, λvg g 0 g 0 ) = p(x, Y, u, v) 0 λ 0 λ Addition and multiplication in the target space M n+ (C) define a C-algebra structure on the equivariant maps. A concrete realization of this ring M(n) can be given as follows. Consider the matrix algebra M n+ (C[rep α M]), then M(n) is the C-subalgebra generated by the polynomial invariants C[rep α M] GL(α) (embedded as scalar matrices) and the following matrices x... x n 0... e n =. 0 f n = x n =... x n... x nn y... y n u y n = y n... y nn u n = u n v n = v... v n 0 Clearly, C[rep α M] GL(α) is a central subring of M(n) and from the generation of the polynomial invariants by traces of necklace words we see that the restriction of the usual trace tr on M n+ (C[rep α M]) to the subring M(n) defines a trace map on M(n), that is a map t M(n) tr t C[rep α M] GL(α) M(n) satisfying t(ab) = t(ba), t(a)b = bt(a), t(t(a)b) = t(a)t(b) and t() = n. It is a rather straightforward consequence of the Cayley-Hamilton equation that M(n) is a finite module over its center which is equal to C[rep α M] GL(α). The factorizing property for representations in rep α M mentioned above follows..8 de Rham cohomology for M. Formally smooth algebras should be viewed as the coordinate rings of affine noncommutative manifolds. Associated to them is a well developed theory of differential

32 32 CHAPTER. CALOGERO SYSTEMS. forms, de Rham cohomology groups, connections and so on. We will introduce and study all these concepts in chapter 9 in great detail. Here, we merely sketch these notions for the formally smooth algebra M as they are crucial in the proof of Ginzburg s result. Recall that M (resp. C C) is the path algebra of the quiver x e y e u v f resp. f As the images of S = C C is fixed in any representation of rep α M, we are interested in relative differential forms of M with respect to the subalgebra C C. Let M be the S-bimodule cokernel of the inclusion S M, then we define the space of relative differential forms of degree n to be Ω n rel M = M S M S... S M }{{} n We will denote a tensor a 0 a... a n Ω n rel M with all a i M by a 0 da... da n. It is easy to see that a basis for Ω n rel M is given by the elements p 0 dp... dp n where p i is an oriented path in the quiver such that length p 0 0 and length p i for i n and such that the starting point of p i is the endpoint of p i+ for all i n. We define an algebra structure on Ω rel M = n Ω n rel M by the product rule (a 0 da... da n )(a n+ da n+2... da m ) = n i=0 ( )n a 0 da... da i d(a i a i+ )da i+2... da m and we make this into a differential graded C-algebra by defining a differential of degree one... d Ω n rel M d Ω n rel M d Ω n+ rel M d... by the rule that d(a 0 da... da n ) = da 0 da... da n. Clearly d d = 0 and d is a super-derivation meaning that d(rs) = (dr)s + ( ) i r(ds) when r Ω i rel M. For ω Ωi rel M and ω Ω j rel M we define the super-commutator to be [ω, ω ] = ωω ( ) ij ω ω and following M. Karoubi [] we define the space of noncommutative differential forms of degree i on M to be the quotient space dr i rel M = i j=0 Ω i rel M [ Ωj rel M, Ωi j rel M ]

33 .8. DE RHAM COHOMOLOGY FOR M. 33 and one verifies that the differential d induces a differential on the Karoubi complex of M dr 0 rel M d dr rel M d dr 2 rel M d... An important result we will prove in chapter 9 is that this complex is acyclic, that is, if we define the i-th de Rham cohomology group of M to be the i-th homology of the Karoubi complex HdR i M = Ker dri rel M d dr i+ rel M Im dr i rel M dr i d rel M then we will prove that H i dr M = { C C when i = 0, 0 when i. We will compute the first few terms in the Karoubi complex. Noncommutative functions on M are the 0-forms, which is by definition the quotient space dr 0 rel M = M [ M, M ] If p is an oriented path of length in the quiver with different begin- and endpoint, then we can write p as a concatenation p = p p 2 with p i an oriented path of length 0 such that p 2 p = 0 in M. As [p, p 2 ] = p p 2 p 2 p = 0 in drrel 0 M we deduce that the space of noncommutative functions on M has as C-basis the necklace words w w where each bead is this time one of the elements = x = y and = uv together with the necklace words of length zero e and f. Each necklace word w corresponds to the equivalence class of the words in M obtained from multiplying the beads in the indicated orientation and and two words in {x, y, u, v} in M are said to be equivalent if they are identical up to cyclic permutation of the terms. Substituting each bead with the n n matrices specified before and taking traces we get a map drrel 0 M M = tr C[repα M] [ M, M ] Hence, noncommutative functions on M induce ordinary functions on all the representation spaces rep α M and these functions are GL(α)-invariant. Moreover, the image of this map generates the ring of polynomial invariants as we mentioned before.

34 34 CHAPTER. CALOGERO SYSTEMS. Next, we consider noncommutative -forms on M which are by definition elements of the space drrel Ω rel M = M [ M, Ω rel M ] Recall that Ω rel M is spanned by the expressions p 0dp with p 0 resp. p oriented paths in the quiver of length 0 resp. and such that the starting point of p 0 is the end point of p. To form drrel M we have to divide out expressions such as [ p, p 0 dp ] = pp 0 dp + p 0 p dp p 0 d(p p) That is, if we have connecting oriented paths p 2 and p both of length we have in dr rel M p 0 d(p p 2 ) = p 2 p 0 dp + p 0 p dp 2 and by iterating this procedure whenever the differential term is a path of length 2 we can represent each class in drrel M as a combination from Me dx + Me dy + Me du + Mf dv Now, Me = eme + fme and let p fme. Then, we have in dr rel M d(xp) = p dx + x dp but by our description of Ω M the left hand term is zero as is the second term on the right, whence p dx = 0. A similar argument holds replacing x by y. As for u, let p eme, then we have in dr rel M d(up) = p du + u dp and again the left-hand and the second term on the right are zero whence p du = 0. An analogous result holds for v and p fmf. Therefore, we have the description of noncommutative -forms on M That is, in graphical terms dr rel M = eme dx + eme dy + fme du + emf dv x y drrel M = e d e + e d e + f e d e u + f e f d f v e.9 Symplectic geometry on M. Recall that a symplectic structure on a (commutative) manifold M is given by a closed differential 2-form. The non-degenerate 2-form ω gives a canonical isomorphism T M T M that is, between vector fields on M and differential -forms. Further, there is a unique C-linear map from functions f on M to vectorfields ξ f by the requirement that df = i ξf ω where i ξ is the contraction of n-forms to n -forms using the vectorfield ξ. We can make the functions on M into a Poisson algebra by defining {f, g} = ω(ξ f, ξ g )

35 .9. SYMPLECTIC GEOMETRY ON M. 35 and one verifies that this bracket satisfies the Jacobi and Leibnitz identities. The Lie derivative L ξ with respect to ξ is defined by the Cartan homotopy formula L ξ ϕ = i ξ dϕ + di ξ ϕ for any differential form ϕ. A vectorfield ξ is said to be symplectic if it preserves the symplectic form, that is, L ξ ω = 0. In particular, for any function f on M we have that ξ f is symplectic. Moreover the assignment f ξf defines a Lie algebra morphism from the functions O(M) on M equipped with the Poisson bracket to the Lie algebra of symplectic vectorfields, V ect ω M. Moreover, this map fits into the exact sequence 0 C O(M) V ect ω M H dr M 0 It is this sequence that we will generalize to the noncommutative algebra M. We say that a noncommutative symplectic structure on M is given by a 2-form ω dr 2 rel M such that d ω = 0 dr 3 rel M Given the shape of the defining quiver, a natural choice of symplectic structure is taking ω = dx dy + du dv By a (relative) vectorfield on M we understand a C C-derivation on M. That is, a linear map M θ M such that θ(ab) = θ(a)b + aθ(b) and θ(e) = 0 = θ(f). For a given θ we define a degree preserving derivation L θ and a degree superderivation i θ on Ω M d Ω n M Ω n M Ω n+ M d L θ i θ L θ i θ L θ defined by the rules { L θ (a) = θ(a) i θ (a) = 0 L θ (da) = d θ(a) i θ (da) = θ(a) for all a M. We will prove in chapter 0 an analog for the Cartan homotopy formula L θ = i θ d + d i θ and that these operators induce operators on the Karoubi complex dr M. analog of the isomorphism T M T M is the isomorphism The Der C C M i.ω dr rel M as for any C C-derivation θ we have i θ ω = i θ (dx)dy dxi θ (dy) + i θ (du)dv dui θ (dv) = θ(x)dy dxθ(y) + θ(u)dv duθ(v) θ(x)dy θ(y)dx + θ(u)dv θ(v)du

36 36 CHAPTER. CALOGERO SYSTEMS. and using the relations in M we can easily prove that any C C derivation on M must satisfy θ(x) eme θ(y) eme θ(u) emf θ(v) fme so the above expression belongs to drrel M. Conversely, any θ defined by its images on the generators x, y, u and v by θ(y)dx + θ(x)dy θ(v)du + θ(u)dv dr rel M induces a derivation on M. In analogy with the classical case we define a derivation θ to be symplectic if and only if L θ ω = 0 in dr 2 rel M. We denote these derivations by Der ω M. From the homotopy formula it follows that θ Der ω M d(i θ ω) = 0 in dr 2 rel M But then, using the above identification Der C C M drrel M and the fact that HdR M = 0 we obtain an analog of the map f ξf from functions to symplectic vectorfields in the classical case M [ M, M ] = dr0 rel M d (dr i.ω rel M) closed Derω M which fits into the exact sequence, using our knowledge of the de Rham cohomology of M 0 C C M Derω M [ M, M ] 0 which we claim to be an exact sequence of Lie algebras. Hence we need to define a M Poisson bracket on the noncommutative functions [ M,M ]. We want to mimic the Poisson bracket on C[x, y, u, v] determined by dx dy + du dv which is {f, g} = ( f x. g y f y. g x ) + ( f u. g v f v. g u ) but then we need a substitute for these partial derivatives. Using our description of drrel M we have for any f dr0 rel M == uniquely defined partial derivatives M [M,M] a x, a y, a u, a v : M [ M, M ] M by the formula da = a x x + a y y + a u u + a v v. We have to specify these on necklace words w. Using the calculation rules in dr rel M one verifies that the partial derivatives of w are the sums of the oriented paths in the quiver one obtains by cyclicly running through the remaining path by deleting occurrences of the differentiating arrow. That is, w x = w w y = w

37 .0. FIBERS OF THE MOMENT MAP. 37 w u = w w v v = w u Using these partial derivatives one can then define a Poisson product on dr 0 rel M by {w, w 2 } K ( w x. w2 y w y. w2 x ) + ( w u. w2 v w v. w2 u ) modulo [ M, M ] In particular, if the w i are necklace words, the Poisson bracket {w, w 2 } K is a sum of necklace words. Using the above graphical description we have that {w, w 2 } is equal to... w w2... w w w u v w2... w v u w2.0 Fibers of the moment map. After this excursion to the symplectic geometry of the noncommutative manifold M it is time to return to the study of the phase space Calo n. Observe that the center C = (λ n, λ) GL(α) acts trivially on rep α M so the relevant acting group is rather P GL(α) = GL(α) C ( n, ).

38 38 CHAPTER. CALOGERO SYSTEMS. There is an open subset in rep α M of representations such that the stabilizer subgroup of P GL(α) is trivial. This is the case for all simple representations by Schur s lemma. However, there are others. For example any representation (X, Y, u, 0) Hilb n has trivial stabilizer, but none of them are simple representations of M as they always have a nontrivial subrepresentation of dimension vector (, 0) determined by a common eigenvector of X and Y (which exists because X and Y commute with each other). We will denote the open set of representations with trivial stabilizer by rep s α M and call any (X, Y, u, v) rep s α M a Schur representation of the quiver. Remark that the Lie algebra of P GL(α) is the vectorspace Lie P GL(α) = M 0 α(c) = { (M, c) M n (C) C tr(m) + c = 0 } The relevant moment map for the action of (P )GL(α) on the representation space rep α M is µ rep α M Lie P GL(α) (X, Y, u, v) ([X, Y ] + uv, vu) Fix λ C, then λ = (λ n, nλ) M 0 α(c) and is fixed under conjugation by GL(α). Therefore, its preimage π (λ) = {(X, Y, u, v) rep α M [X, Y ] + uv = λ n and vu = nλ } is a closed affine subscheme of rep α M stable under the action of GL(α). In particular we have that CALO n = π (). We will now associate noncommutative algebras to the fibers π (λ). These are special examples of the deformed preprojective algebras introduced by W. Crawley- Boevey and M.P. Holland in [6]. For λ C we define M λ = M ([x, y] + [u, v] λ(e nf)) and by an argument similar to that of M we see that µ (λ) is the space of n + - dimensional representations M φ λ M n+ (C) such that [ n ] [ ] φ(e) = and φ(f) = The closed affine subscheme π (λ) has as its defining ideal of relations I λ the entries in the n + n + matrix [ λ ] [x n, y n ] + [u n, v n ] n 0 0 nλ We consider the ring of polynomial invariants C[µ (λ)] GL(α) and its corresponding affine scheme µ (λ)/gl(α). The natural algebra morphisms give the following geometric picture µ (λ) rep α M pi µ (λ)/gl(α) rep α M/GL(α)

39 .0. FIBERS OF THE MOMENT MAP. 39 Invariant theory tells us that the defining ideal of the closed subscheme µ (λ)/gl(α) is J λ = I λ C[rep α M] GL(α). Again. points in µ (λ)/gl(α) are in one-to-one correspondence with isomorphism classes of n + -dimensional semi-simple representations of the algebra M λ with specified ranks of the images of e and f. Analogous to the construction of the algebra M(n) we construct an algebra M λ (n) which is a finite module over its center by dividing out the centrally generated ideal of M determined by J λ M λ (n) = M(n) M(n) J λ M(n) Again, one verifies that the α-dimensional representations of M λ factor through M λ (n). The trace map t on M(n) introduced before defines a trace map on M λ (n). Clearly, this trace satisfies the Cayley-Hamilton identities for n + n + matrices and t() = n +. A We define a category CH(n + ) of all C-algebras A equipped with a trace map t A A such that ta () = n + and t A satisfies all Cayley-Hamilton identities holding for n + n + matrices. Morphisms A φ B in CH(n + ) are C- algebra morphisms compatible with the trace maps, that is, making the diagram below commute φ A B t B t A A φ B We say that an algebra S in CH(n + ) is smooth in CH(n + ) if it has the lifting property with respect to nilpotent ideals in CH(n + ). That is, for all A CH(n + ), I A a nilpotent ideal in A such that t A (I) I and a morphism ψ in CH(n + ) we can complete the diagram below by a morphism ψ in CH(n + ) ψ S... A ψ We will prove in chapter 5 that this property is equivalent to the geometric formulation that the space of trace preserving n + -dimensional representations of S is a smooth variety. Here, a trace preserving representation is a morphism S φ M n+ (C) in CH(n + ) where M n+ (C) is equipped with the usual trace. For example, the algebra M(n) is smooth in CH(n + ) as its space of trace preserving n+-dimensional representations is GL n+ (C) GL(α) rep α M. Similarly, the space of trace preserving n + -dimensional representations of the fiber algebra M λ (n) is equal to the fiber bundle A I GL n+ (C) GL(α) µ (λ) and is therefore smooth in CH(n + ) if and only if the fiber µ (λ) is a smooth submanifold of rep α M. In particular, because CALO n = µ () we deduce

40 40 CHAPTER. CALOGERO SYSTEMS. Theorem.22 The fiber algebra M (n) corresponding to the phase space Calo n is a smooth algebra in CH(n + ). In fact, it is an Azumaya algebra over Calo n = µ ()/GL(α). Proof. The first statement follows from the fact that CALO n is a submanifold of rep α M. Further, we know that the GL n -action on CALO n is free, hence the GL(α)-action has all its stabilizer subgroups equal to the center C. But then, CALO n Calon is a principal P GL(α)-fibration. We will see in chapter 5 that then the ring of P GL(α)-equivariant maps CALO n Mn+ (C) (which is equal to the fiber algebra M (n) is an Azumaya algebra with a specified embedding of the idempotents. This result gives a ringtheoretical interpretation of the uniformity of the phase space Calo n. Each point in the phase space corresponds to a simple n + - dimensional representation of M (n). In contrast, the fiber algebra M 0 (n) corresponding to Hilb n is not smooth in CH(n + ). Indeed, the fiber µ (0) is not even irreducible but even the irreducible component determined by Hilb n contains singularities.. Ginzburg s theorem for Calo n. We have now all the necessary ingredients to sketch the proof of Ginzburg s result that Calo n i s the coadjoint orbit of some infinite dimensional Lie algebra. To begin, we equip rep α M with the symplectic structure induced by the 2-form ω α = i,j n dx ij dy ij + n du i dv i The induced Poisson bracket {. } α on the ring of polynomial functions C[rep α M] is defined to be {f, g} α = ( f x ij i,j n g y ij f y ij i= g x ij ) + n i= ( f u i The action of GL(α) on rep α M is symplectic meaning that ω α (t, t ) = ω α (gt, gt ) g v i f v i g u i ) for all t, t T (X,Y,u,v) rep α M in all points (X, Y, u, v) and for all g GL(α). The infinitesimal GL(α) action gives a Lie algebra homomorphism Lie P GL(α) V ectωα rep α M which factorizes through a Lie algebra morphism H to the coordinate ring making the diagram below commute Lie P GL(α) H=µ C[rep α M] f ξ f V ect ωα rep α M

41 .. GINZBURG S THEOREM FOR CALO N. 4 We say that the action of GL(α) on rep α M is Hamiltonian. This makes the ring of polynomial invariants C[rep α M] GL(α) into a Poisson algebra and we will write Lie = (C[rep α M] GL(α), {, } α ) for the corresponding abstract infinite dimensional Lie algebra. The dual space of this Lie algebra Lie is then a Poisson manifold equipped with the Kirillov-Kostant bracket. Evaluation at a point in the quotient variety rep α M/GL(α) defines a linear function on Lie and therefore evaluation gives an embedding rep α M/GL(α) Lie as Poisson varieties. That is, the induced map on the polynomial functions is a morphism of Poisson algebras. Let λ C and let λ = (λ n, nλ) Lie P GL(α). Then, we have Theorem.23 Assume that µ (λ) is a smooth variety on which P GL(α) acts freely. Then, the quotient variety µ (λ)/gl(α) is an affine symplectic manifold and the Poisson embeddings µ (λ)/gl(α) rep α M/GL(α) Lie makes each connected component of µ (λ)/gl(α) a closed coadjoint orbit of Lie. Proof. (sketch) Because the action of the reductive group P GL(α) is free on the smooth affine variety π (λ), the quotient variety µ (λ)/gl(α) is smooth and affine. Moreover, the infinitesimal coadjoint action of Lie on Lie preserves µ (λ)/gl(α) and factors through the quotient Lie algebra Lie J λ. In general, if X is a smooth affine variety, then the differentials of polynomial functions on X span the tangent spaces at all points x of X. Therefore, if X is in addition symplectic, the infinitesimal Hamiltonian action of the Lie algebra C[X] (with the natural Poisson bracket) on X is infinitesimally transitive. But then, the evaluation map makes X a coadjoint orbit of the dual Lie algebra C[X]. Hence, µ (λ)/gl(α) is a coadjoint orbit in C[µ (λ)/gl(α)]. Therefore, the infinite dimensional group Ham generated by all Hamiltonian flows on µ (λ)/gl(α) acts with open orbits. Being connected, each irreducible component of µ (λ)/gl(α) is a single Ham-orbit finishing the proof. Observe that the conditions hold for λ =, that is, Calo n is a coadjoint orbit in Lie. In contrast, the fiber corresponding to Hilb n, that is, λ = 0 does not satisfy the requirements. The drawback is that the Lie algebra Lie still depends on n and we want a similar result holding for all n. We will now show that all Calo n are coadjoint orbits in the dual of a central extension of the Lie algebra Der ω M. The central extension in question is given by the exact sequence of Lie algebras we found when investigating the noncommutative derham cohomology of M 0 C C M [ M, M ] Derω M 0 M Moreover, we have seen that both [ M,M ] and Lie are generated by necklace words. the crucial point to note is now that Lie = M [ M, M ] tr C[rep α M] GL(α) = Lie

42 42 CHAPTER. CALOGERO SYSTEMS. obtained by mapping a necklace word w to its trace tr(w) using the n + n + matrices x n, y n, u n and v n introduced before, is a Lie algebra morphism where we equip Lie with the Poisson bracket determined by the partial derivations determined from our description of dr M and Lie is equipped with the Poisson bracket {, } α. That is, we have for all necklace words w and w 2 the identity {tr w, tr w 2 } α = tr {w, w 2 } K Because the tr w generate the ring of polynomial invariants C[rep α M] GL(α) it M follows that the elements tr [ M,M ] separate points of Calo n = µ ()/GL(α) as subset of rep α M/GL(α). That is, the composition Calo n repα M/GL(α) tr M ( [ M, M ] ) is injective. Analogously, the differentials of functions on Calo n obtained by restricting traces of necklace words viewed as linear functions on Lie span the cotangent spaces at all points of Calo n, concluding the proof of Theorem.24 For all n, the phase space Calo n is a coadjoint orbit in the dual of M the Lie algebra [ M,M ] which is a central extension of the Lie algebra Der ω M.

43 Chapter 2 Brauer-Severi Varieties. Let K be a field and = (a, b) K the quaternion algebra determined by a, b K. That is, = K. K.i K.j K.ij with i 2 = a j 2 = b and ji = ij The norm map on defines a conic in P 2 K BS( ) = V (x 2 ay 2 bz 2 ) called the Brauer-Severi variety of P 2 K = P roj K[x, y, z]. Its characteristic property is that a fieldextension L of K admits an L-rational point on BS( ) if and only if K L admits zero-divisors and hence is isomorphic to M 2 (L). More generally, let K be the algebraic closure of K. We will see that the Galois cohomology pointed set H (Gal(K/K), P GL n (K)) classifies at the same time the isomorphism classes of the following geometric and algebraic objects Brauer-Severi K-varieties BS, which are smooth projective K-varieties such that BS K P n K. Central simple K-algebras, which are K-algebras of dimension n 2 such that K K M n (K). The one-to-one correspondence between these two sets is given by associating to a central simple K-algebra its Brauer-Severi variety BS( ) which represents the functor associating to a fieldextension L of K the set of left ideals of K L which have L-dimension equal to n. In particular, BS( ) has an L-rational point if and only if K L M n (L) and hence the geometric object BS( ) encodes the algebraic splitting behaviour of. Now restrict to the case when K is the functionfield C(X) of a projective variety X and let be a central simple C(X)-algebra of dimension n 2. Let A be a sheaf of O X -orders in then we will see that there is a Brauer-Severi scheme BS(A) which is a projective space bundle over X with general fiber isomorphic to P n (C) embedded in P N (C) where N = ( ) n + k k. Over an arbitrary point of x the fiber may degenerate, for example if n = 2 the P (C) embedded as a conic in P 2 (C) can degenerate into a pair of P (C) s. The special case when BS(A) is a P n (C)-bundle corresponds to the case when A is a sheaf of Azumaya algebras over X. For arbitrary orders, the geometric structure of BS(A) can be fairly complicated. However, when A is a sheaf of smooth orders we will prove in chapter 8 that BS(A) 43

44 44 CHAPTER 2. BRAUER-SEVERI VARIETIES. is a smooth variety and indicate how one can compute the fibers over X explicitly. The motivating class of such examples are the unirational non-rational threefolds constructed by M. Artin and D. Mumford [3] in 972 as Brauer-Severi varieties of maximal orders in specified quaternion algebras (the restrictions impose that these maximal orders are indeed smooth orders). A major step in their construction is the description of the Brauer group of a simply connected projective surface using étale cohomology. We will outline this result in some detail as it gives us the opportunity to introduce some basic results on étale extentions, étale cohomology and étale descent. Roughly speaking, étale extensions give us an algebraic alternative for the implicit function theorem in differential geometry. In this book we will give several applications of étale descent. For example, we will give an étale local description of smooth orders which will allow us to deduce from the Artin-Mumford exact sequence which central simple algebras over a smooth projective surface allow a noncommutative smooth model, see chapter Unirational non-rational threefolds. In this section we will outline the major steps in the Artin-Mumford construction of unirational non-rational threefolds. We use this class of examples as motivation for introducing étale cohomology and smooth orders. For more details we refer to the original paper [3]. Consider P 2 = P 2 (C). We want to describe all central simple algebras over the functionfield C(x, y). In this chapter we will prove that this is a huge collection. The Artin-Mumford result describes them by a certain geo-combinatorial package which we call a Z n -wrinkle over P 2 = P 2 (C). A Z n -wrinkle is determined by A finite collection C = {C,..., C k } of irreducible curves in P 2, that is, C i = V (F i ) for an irreducible form in C[X, Y, Z] of degree d i. A finite collection P = {P,..., P l } of points of P 2 where each P i is either an intersection point of two or more C i or a singular point of some C i. For each P P the branch-data b P = (b,..., b ip ) with b i Z n = Z/nZ and {,..., i P } the different branches of C in P. These numbers must satisfy the admissibility condition b i = 0 Z n for every P P i for each C C we fix a cyclic Z n -cover of smooth curves D C of the desingularization C of C which is compatible with the branch-data. That is, if Q C corresponds to a C-branch b i in P, then D is ramified in Q with stabilizer subgroup Stab Q = b i Z n

45 2.. UNIRATIONAL NON-RATIONAL THREEFOLDS. 45 For example, a portion of a Z 4 -wrinkle can have the following picture D 0 2 C It is clear that the cover-data is the most untractable part of a Z n -wrinkle, so we want to have some control on the covers D C. Let {Q,..., Q z } be the points of C where the cover ramifies with branch numbers {b,..., b z }, then D is determined by a continuous module structure (that is, a cofinite subgroup acts trivially) of π ( C {Q,..., Q z }) on Z n where the fundamental group of the Riemann surface C with z punctures is known (topologically) to be equal to the group u, v,..., u g, v g, x,..., x z /([u, v ]... [u g, v g ]x... x z ) where g is the genus of C. The action of x i on Z n is determined by multiplication with b i. In fact, we need to use the étale fundamental group, see [20], but this group has the same finite continuous modules as the topological fundamental group. Example 2. Covers of P and elliptic curves.. If C = P then g = 0 and hence π (P {Q,..., Q z } is zero if z (whence no covers exist) and is Z if z = 2. Hence, there exists a unique cover D P with branch-data (, ) in say (0, ) namely with D the normalization of P in C( n x). 2. If C = E an elliptic curve, then g =. Hence, π (C) = Z Z and there exist unramified Z n -covers. They are given by the isogenies E E where E is another elliptic curve and E = E / τ where τ is an n-torsion point on E. We will show that any n-fold cover D C is determined by a function f C(C) /C(C) n. This allows us to put a group-structure on the equivalence classes of Z n -wrinkles. In particular, we call a wrinkle trivial provided all coverings D i Ci are trivial (that is, D i is the disjoint union of n copies of C i ). One of the main results we will prove in this chapter is the Artin-Mumford exact sequence for Brauer groups of simply connected surfaces. In the case of C(x, y) this result can be phrased as Theorem 2.2 If is a central simple C(x, y)-algebra of dimension n 2, then determines uniquely a Z n -wrinkle on P 2. Conversely, any Z n -wrinkle on P 2 determines a unique division C(x, y)- algebra whose class in the Brauer group has order n.

46 46 CHAPTER 2. BRAUER-SEVERI VARIETIES. Specialize to the case of quaternion algebras, that is n = 2. Consider E and E 2 two elliptic curves in P 2 and take C = {E, E 2 } and P = {P,... P 9 } the intersection points and all the branch data zero. Let E i be a twofold unramified cover of E i as in the example above given by modding our a 2-torsion point from E i. By the Artin-Mumford result there is a quaternion algebra corresponding to this Z 2 -wrinkle. Next, blow up the intersection points to get a surface S with disjoint elliptic curves C and C 2. Now take a maximal O S order in then we will see that the relevance of the curves C i is that they are the locus of the points s S where A s M 2 (C), the so called ramification locus of the order A. The local structure of A in a point s S is when s / C C 2, then A s is an Azumaya O S,s -algebra in, when s C i, then A s = O S,s. O S,s.i O S,s.j O S,s.ij with i 2 = a j 2 = bt ji = ij where t = 0 is a local equation for C i and a and b are units in O S,s. In chapter 6 we will see that this is the local description of a smooth order over a smooth surface in a quaternion algebra. Artin and Mumford then define the Brauer- Severi scheme of A as representing the functor which assigns to an S-scheme S the set of left ideals of A OS O S which are locally free of rank 2. Using the local description of A they show that BS(A) is a projective space bundle over S with the properties that BS(A) is a smooth variety and the projection morphism BS(A) π S is flat, all of the geometric fibers being isomorphic to P (resp. to P P ) whenever s / C C 2 (resp. s C C 2 ). Finally, for specific starting configurations E and E 2, they prove that the obtained Brauer-Severi variety BS(A) cannot be rational because there is torsion in H 4 (BS(A), Z 2 ), whereas BS(A) can be shown to be unirational for these specific configurations.

47 2.2. ETALE MORPHISMS AND SHEAVES Etale morphisms and sheaves. In the next few sections we will introduce the basic tools from étale topology leading to a sketch of the Artin-Mumford exact sequence. Definition 2.3 A finite morphism A be étale if and only if f B of commutative C-algebras is said to B = A[t,..., t k ]/(f,..., f k ) such that det ( f i x j ) i,j B C[x, x ][ n x] and the in- Example 2.4 Consider the morphism C[x, x ] duced map on the affine varieties V ar C[x, x ][ n x] ψ V ar C[x, x ] = C {0}. Clearly, every point λ C {0} has exactly n preimages λ i = ζ i n λ. Moreover, in a neighborhood of λ i, the map ψ is a diffeomorphism. Still, we do not have an inverse map in algebraic geometry as n x is not a polynomial map. However, C[x, x ][ n x] is an étale extension of C[x, x ]. That is, étale morphisms can be seen as an algebraic substitute for the nonexistence of an inverse function theorem in algebraic geometry. Proposition 2.5 Etale morphisms satisfy sorite, that is A et... A A B et B A et B et et A... C (basechange) (composition) f.f. A A et A A B et... B et A et B... A alg et B (descent) (morphisms) Here et means an étale morphism and f.f. stands for a faithfully flat morphism. Definition 2.6 The étale site of A, which we will denote by A et is the category with objects : the étale extensions A f B of A morphisms : compatible A-algebra morphisms A f f2 B φ B 2

48 48 CHAPTER 2. BRAUER-SEVERI VARIETIES. Observe that by the foregoing proposition all morphisms in A et are étale. We can put on A et a (Grothendieck) topology by defining cover : a collection C = {B fi B i } in A et such that Spec B = i Im (Spec B i f Spec B ) where Spec is the prime spectrum of a commutative algebra, that is the collection of all its prime ideals equipped with the Zariski topology. An étale presheaf of groups on A et is a functor G : A et Groups In analogy with usual (pre)sheaf notation we denote for each object B A et : Γ(B, G) = G(B) morphism B φ C in A et : Res B C = G(φ) : G(B) G(C) and g C = G(φ)(g). A presheaf G is a sheaf provided for every B A et and every cover {B B i } we have exactness of the equalizer diagram 0 G(B) i G(B i ) G(B i B B j ) i,j Example 2.7 Constant sheaf. If G is a group, then G : A et Groups B G π0(b) is a sheaf where π 0 (B) is the number of connected components of Spec B. Example 2.8 Multiplicative group G m. The functor G m : A et Ab B B is a sheaf on A et. A sequence of sheaves of Abelian groups on A et is said to be exact G f G g G if for every B A et and s G(B) such that g(s) = 0 G (B) there is a cover {B B i } in A et and sections t i G (B i ) such that f(t i ) = s B i. Example 2.9 Roots of unity µ n. We have a sheaf morphism G m ( ) n G m and we denote the kernel with µ n. As A is a C-algebra we can identify µ n with the constant sheaf Z n = Z/nZ via the isomorphism ζ i i after choosing a primitive n-th root of unity ζ C.

49 2.2. ETALE MORPHISMS AND SHEAVES. 49 Lemma 2.0 The (Kummer)-sequence of sheaves of Abelian groups 0 µn Gm ( ) n Gm 0 is exact on A et (but not necessarily on A Zar ). Proof. We only need to verify surjectivity. Let B A et and b G m (B) = B. Consider the étale extension B = B[t]/(t n b) of B, then b has an n-th root over in G m (B ). Observe that this n-th root does not have to belong to G m (B). If p is a prime ideal of A we will denote with k p the algebraic closure of the field of fractions of A/p. An étale neighborhood of p is an étale extension B A et such that the diagram below is commutative A nat kp et B The analogue of the localization A p for the étale topology is the strict Henselization A sh p = lim B where the limit is taken over all étale neighborhoods of p. Recall that a local algebra L with maximal ideal m and residue map π : L L/m = k is said to be Henselian if the following condition holds. Let f L[t] be a monic polynomial such that π(f) factors as g 0.h 0 in k[t], then f factors as g.h with π(g) = g 0 and π(h) = h 0. If L is Henselian then tensoring with k induces an equivalence of categories between the étale A-algebras and the étale k-algebras. An Henselian local algebra is said to be strict Henselian if and only if its residue field is algebraically closed. Thus, a strict Henselian ring has no proper finite étale extensions and can be viewed as a local algebra for the étale topology. Example 2. The algebraic functions C{x,..., x d } Consider the local algebra of C[x,..., x d ] in the maximal ideal (x,..., x d ), then the Henselization and strict Henselization are both equal to C{x,..., x d } the ring of algebraic functions. That is, the subalgebra of C[[x,..., x d ]] of formal power-series consisting of those series φ(x,..., x d ) which are algebraically dependent on the coordinate functions x i over C. In other words, those φ for which there exists a non-zero polynomial f(x i, y) C[x,..., x d, y] with f(x,..., x d, φ(x,..., x d )) = 0. These algebraic functions may be defined implicitly by polynomial equations. Consider a system of equations f i (x,..., x d ; y,..., y m ) = 0 for f i C[x i, y j ] and i m Suppose there is a solution in C with x i = 0 and y j = y o j

50 50 CHAPTER 2. BRAUER-SEVERI VARIETIES. such that the Jacobian matrix is non-zero det ( f i y j (0,..., 0; y o,..., y 0 m)) 0 Then, the system can be solved uniquely for power series y j (x,..., x d ) with y j (0,..., 0) = yj o by solving inductively for the coefficients of the series. One can show that such implicitly defined series y j (x,..., x d ) are algebraic functions and that, conversely, any algebraic function can be obtained in this way. If G is a sheaf on A et and p is a prime ideal of A, we define the stalk of G in p to be G p = lim G(B) where the limit is taken over all étale neighborhoods of p. One can verify monoepi- or isomorphisms of sheaves by checking it in all the stalks. If A is an affine algebra defined over an algebraically closed field, then it suffices to verify in the maximal ideals of A. 2.3 Etale cohomology Before we define cohomology of sheaves on A et let us recall the definition of derived functors. Let A be an Abelian category. An object I of A is said to be injective if the functor A Ab M Hom A (M, I) is exact. We say that A has enough injectives if, for every object M in A, there is a monomorphism M I into an injective object. If A has enough injectives and f : A B is a left exact functor from A into a second Abelian category B, then there is an essentially unique sequence of functors R i f : A B i 0 called the right derived functors of f having the following properties R 0 f = f R i I = 0 for I injective and i > 0 For every short exact sequence in A 0 M M M 0 there are connecting morphisms δ i : R i f(m ) R i+ f(m ) for i 0 such that we have a long exact sequence... R i f(m) R i f(m ) δ i R i+ f(m ) R i+ f(m)... For any morphism M N there are morphisms R i f(m) R i f(n) for i 0 In order to compute the objects R i f(m) define an object N in A to be f-acyclic if R i f(m) = 0 for all i > 0. If we have a resolution of M 0 M N0 N N2...

51 2.3. ETALE COHOMOLOGY 5 by f-acyclic object N i, then the objects R i f(m) are canonically isomorphic to the cohomology objects of the complex 0 f(n0 ) f(n ) f(n2 )... One can show that all injectives are f-acyclic and hence that derived objects of M can be computed from an injective resolution of M. Now, let S ab (A et ) be the category of all sheaves of Abelian groups on A et. This is an Abelian category having enough injectives whence we can form right derived functors of left exact functors. In particular, consider the global section functor Γ : S ab (A et ) Ab G G(A) which is left exact. The right derived functors of Γ will be called the étale cohomology functors and we denote R i Γ(G) = H i et(a, G) In particular, if we have an exact sequence of sheaves of Abelian groups 0 G G G 0, then we have a long exact cohomology sequence... H i et (A, G) H i et (A, G ) H i+ et (A, G )... If G is a sheaf of non-abelian groups (written multiplicatively), we cannot define cohomology groups. Still, one can define a pointed set Het(A, G) as follows. Take an étale cover C = {A A i } of A and define a -cocycle for C with values in G to be a family g ij G(A ij ) with A ij = A i A A j satisfying the cocycle condition (g ij A ijk )(g jk A ijk ) = (g ik A ijk ) where A ijk = A i A A j A A k. Two cocycles g and g for C are said to be cohomologous if there is a family h i G(A i ) such that for all i, j I we have g ij = (h i A ij )g ij (h j A ij ) This is an equivalence relation and the set of cohomology classes is written as H et(c, G). It is a pointed set having as its distinguished element the cohomology class of g ij = G(A ij ) for all i, j I. We then define the non-abelian first cohomology pointed set as H et(a, G) = lim H et(c, G) where the limit is taken over all étale coverings of A. It coincides with the previous definition in case G is Abelian. A sequence G G G of sheaves of groups on Aet is said to be exact if for every B A et we have G (B) = Ker G(B) G (B) For every g G (B) there is a cover {B g i G(B i ) such that g i maps to g B. Bi } in A et and sections

52 52 CHAPTER 2. BRAUER-SEVERI VARIETIES. Proposition 2.2 For an exact sequence of groups on A et G G G there is associated an exact sequence of pointed sets G (A) G(A) G (A) δ H et (A, G ) H et (A, G) H et (A, G )... H 2 et (A, G ) where the last map exists when G is contained in the center of G (and therefore is Abelian whence H 2 is defined). Proof. The connecting map δ is defined as follows. Let g G (A) and let C = {A Ai } be an étale covering of A such that there are g i G(A i ) that map to g A i under the map G(A i ) G (Ai ). Then, δ(g) is the class determined by the one cocycle g ij = (g i A ij ) (g j A ij ) with values in G. The last map can be defined in a similar manner, the other maps are natural and one verifies exactness. The main applications of this non-abelian cohomology to non-commutative algebra is as follows. Let Λ be a not necessarily commutative A-algebra and M an A-module. Consider the sheaves of groups Aut(Λ) resp. Aut(M) on A et associated to the presheaves B Aut B alg (Λ A B) resp. B Aut B mod (M A B) for all B A et. A twisted form of Λ (resp. M) is an A-algebra Λ (resp. an A-module M ) such that there is an étale cover C = {A A i } of A such that there are isomorphisms Λ A A i φ i Λ A A i resp. M A A i ψ i M A A i of A i -algebras (resp. A i -modules). The set of A-algebra isomorphism classes (resp. A-module isomorphism classes) of twisted forms of Λ (resp. M) is denoted by T w A (Λ) (resp. T w A (M)). To a twisted form Λ one associates a cocycle on C α Λ = α ij = φ i with values in Aut(Λ). Moreover, one verifies that two twisted forms are isomorphic as A-algebras if their cocycles are cohomologous. That is, there is an embedding T w A (Λ) H et(a, Aut(Λ)) and similarly T w A (M) H et(a, Aut(M)) In favorable situations one can even show bijectivity. In particular, this is the case if the automorphisms group is a smooth affine algebraic group-scheme. For example, consider Λ = M n (A), then the automorphism group is P GL n and twisted forms of Λ are classified by elements of the cohomology group H et(a, P GL n ) These twisted forms are precisely the Azumaya algebras of rank n 2 with center A. When A is an affine commutative C-algebra and B is an A-algebra with center A, then B is an Azumaya algebra of rank n 2 if and only if φ j B BmB M n(c) for every maximal ideal m of A. For example, the fiber algebra M (n) introduced in the foregoing chapter is an Azumaya algebra of rank (n + ) 2 over its center which is C[Calo n ].

53 2.4. CENTRAL SIMPLE ALGEBRAS Central simple algebras Let K be a field of characteristic zero, choose an algebraic closure K with absolute Galois group G K = Gal(K/K). Lemma 2.3 The following are equivalent. K A is étale 2. A K K K... K 3. A = L i where L i /K is a finite field extension Proof. Assume (), then A = K[x,..., x n ]/(f,..., f n ) where f i have invertible Jacobian matrix. Then A K is a smooth algebra (hence reduced) of dimension 0 so (2) holds. Assume (2), then Hom K alg (A, K) Hom K alg (A K, K) has dim K (A K) elements. On the other hand we have by the Chinese remainder theorem that A/Jac A = L i i with L i a finite field extension of K. However, dim K (A K) = i dim K (L i ) = dim K (A/Jac A) dim K (A) and as both ends are equal A is reduced and hence A = i L i whence (3). Assume (3), then each L i = K[x i ]/(f i ) with f i / x i invertible in L i. But then A = L i is étale over K whence (). To every finite étale extension A = L i we can associate the finite set rts(a) = Hom K alg (A, K) on which the Galois group G K acts via a finite quotient group. If we write A = K[t]/(f), then rts(a) is the set of roots in K of the polynomial f with obvious action by G K. Galois theory, in the interpretation of Grothendieck can now be stated as Proposition 2.4 The functor K et rts( ) finite GK sets is an anti-equivalence of categories. We will now give a similar interpretation of the Abelian sheaves on K et. Let G be a presheaf on K et. Define M G = lim G(L) where the limit is taken over all subfields L K that are finite over K. The Galois group G K acts on G(L) on the left through its action on L whenever L/K is Galois. Hence, G K acts an M G and M G = MG H where H runs through the open subgroups of G K. That is, M G is a continuous G K -module. Conversely, given a continuous G K -module M we can define a presheaf G M on K et such that

54 54 CHAPTER 2. BRAUER-SEVERI VARIETIES. G M (L) = M H where H = G L = Gal(K/L). G M ( L i ) = G M (L i ). One verifies that G M is a sheaf of Abelian groups on K et. Theorem 2.5 There is an equivalence of categories S(K et ) G K mod induced by the correspondences G M G and M G M. Here, G K mod is the category of continuous G K -modules. Proof. A G K -morphism M M induces a morphism of sheaves G M GM. Conversely, if H is an open subgroup of G K with L = K H, then if G φ G is a sheafmorphism, φ(l) : G(L) G (L) commutes with the action of G K by functoriality of φ. Therefore, lim φ(l) is a G K -morphism M G MG. One verifies easily that Hom GK (M, M ) Hom(GM, G M ) is an isomorphism and that the canonical map G G MG is an isomorphism. In particular, we have that G(K) = G(K) G K for every sheaf G of Abelian groups on K et and where G(K) = M G. Hence, the right derived functors of Γ and ( ) G coincide for Abelian sheaves. The category G K mod of continuous G K -modules is Abelian having enough injectives. Therefore, the left exact functor ( ) G : G K mod Ab admits right derived functors. They are called the Galois cohomology groups and denoted R i M G = H i (G K, M) Therefore, we have. Proposition 2.6 For any sheaf of Abelian groups G on K et isomorphism Het(K, i G) H i (G K, G(K)) we have a group Therefore, étale cohomology is a natural extension of Galois cohomology to arbitrary algebras. The following definition-characterization of central simple algebras is classical Proposition 2.7 Let A be a finite dimensional K-algebra. equivalent : The following are. A has no proper twosided ideals and the center of A is K. 2. A K = A K K M n (K) for some n. 3. A L = A K L M n (L) for some n and some finite Galois extension L/K. 4. A M k (D) for some k where D is a division algebra of dimension l 2 with center K.

55 2.4. CENTRAL SIMPLE ALGEBRAS 55 The last part of this result suggests the following definition. Call two central simple algebras A and A equivalent if and only if A M k ( ) and A M l ( ) with a division algebra. From the second characterization it follows that the tensorproduct of two central simple K-algebras is again central simple. Therefore, we can equip the set of equivalence classes of central simple algebras with a product induced from the tensorproduct. This product has the class [K] as unit element and [ ] = [ opp ], the opposite algebra as K opp End K ( ) = M l 2(K). This group is called the Brauer group and is denoted Br(K). We will quickly recall its cohomological description, all of which is classical. GL r is an affine smooth algebraic group defined over K and is the automorphism group of a vectorspace of dimension r. It defines a sheaf of groups on K et that we will denote by GL r. Using the general results on twisted forms of the foregoing chapter we have Lemma 2.8 H et(k, GL r ) = H (G K, GL r (K)) = 0 In particular, we have Hilbert s theorem 90 H et(k, G m ) = H (G K, K ) = 0 Proof. The cohomology group classifies K-module isomorphism classes of twisted forms of r-dimensional vectorspaces over K. There is just one such class. P GL n is an affine smooth algebraic group defined over K and it is the automorphism group of the K-algebra M n (K). It defines a sheaf of groups on K et denoted by PGL n. By the proposition we know that any central simple K-algebra of dimension n 2 is a twisted form of M n (K). Therefore, Lemma 2.9 The pointed set of K-algebra isomorphism classes of central simple algebras of dimension n 2 over K coincides with the cohomology set H et(k, PGL n ) = H (G K, P GL n (K)) Theorem 2.20 There is a natural inclusion H et(k, PGL n ) H 2 et(k, µ n ) = Br n (K) where Br n (K) is the n-torsion part of the Brauer group of K. Moreover, Br(K) = H 2 et(k, G m ) is a torsion group.

56 56 CHAPTER 2. BRAUER-SEVERI VARIETIES. Proof. Consider the exact commutative diagram of sheaves of groups on K et. µ n G m ( ) n G m SL n det GL n G m PGL n = PGL n Taking cohomology of the second exact sequence we obtain GL n (K) det K H et (K, SL n ) H et (K, GL n ) where the first map is surjective and the last term is zero, whence H et(k, SL n ) = 0 Taking cohomology of the first vertical exact sequence we get H et(k, SL n ) H et(k, PGL n ) H 2 et(k, µ n ) from which the first claim follows. As for the second, taking cohomology of the first exact sequence we get H et(k, G m ) H 2 et (K, µ n ) H 2 et (K, G m ) n. H 2 et (K, G m ) By Hilbert 90, the first term vanishes and hence H 2 et(k, µ n ) is equal to the n-torsion of the group H 2 et(k, G m ) = H 2 (G K, K ) = Br(K) where the last equality follows from the crossed product result, see for example [23]. So far, the field K was arbitrary. If K is of transcendence degree d, this will put restrictions on the size of the Galois group G K. In particular this will enable us to show that H i (G K, µ n ) = 0 for i > d. Before we can prove this we need to refresh our memory on spectral sequences. 2.5 Spectral sequences Let A, B and C be Abelian categories such that A and B have enough injectives and consider left exact functors A f B g C

57 2.5. SPECTRAL SEQUENCES 57 Let the functors be such that f maps injectives of A to g-acyclic objects in B, that is R i g(f I) = 0 for all i > 0. Then, there are connections between the objects R p g(r q f(a)) and R n gf(a) for all objects A A. These connections can be summarized by giving a spectral sequence Theorem 2.2 Let A, B, C be Abelian categories with A, B having enough injectives and left exact functors A f B g C such that f takes injectives to g-acyclics. Then, for any object A A there is a spectral sequence In particular, there is an exact sequence E p,q 2 = R p g(r q f(a)) = R n gf(a) 0 R g(f(a)) R gf(a) g(r f(a)) R 2 g(f(a))... Moreover, if f is an exact functor, then we have R p gf(a) R p g(f(a)) A spectral sequence E p.q 2 = E n (or E p,q = E n ) consists of the following data. A family of objects Er p,q in an Abelian category for p, q, r Z such that p, q 0 and r 2 (or r ). 2. A family of morphisms in the Abelian category d p.q r : E p.q r E p+r,q r+ r satisfying the complex condition d p+r,q r+ r d p,q r = 0 and where we assume that d p.q r = 0 if any of the numbers p, q, p+r or q r + is <. At level one we have the following q E p,q = p

58 58 CHAPTER 2. BRAUER-SEVERI VARIETIES. At level two we have the following q E p,q 2 = p 3. The objects E p,q r+ on level r + are derived from those on level r by taking the cohomology objects of the complexes, that is, E p r+ = Ker dp,q r / Im d p r,q+r r At each place (p, q) this process converges as there is an integer r 0 depending on (p, q) such that for all r r 0 we have d p.q r = 0 = d p r,q+r r. We then define E p,q = Er p,q 0 (= E p,q r =...) 0+ Observe that there are injective maps E 0,q E 0,q A family of objects E n for integers n 0 and for each we have a filtration 0 E n n E n n... E n E n 0 = E n such that the successive quotients are given by E n p / E n p+ = E p,n p That is, the terms E p,q are the composition terms of the limiting terms E p+q. Pictorially, q E p,q = E 0 E E 2 E 3 E 4 p For small n one can make the relation between E n and the terms E p,q 2 explicit. First note that E 0,0 2 = E 0,0 = E 0 Also, E = E,0 = E,0 2 and E /E = E 0, = Ker d 0, 2. This gives an exact sequence 0 E,0 2 E E 0, d 0, 2 2 E 2,0 2

59 2.6. TSEN AND TATE FIELDS 59 Further, E 2 E 2 E 2 2 where E2 2 = E 2,0 = E 2,0 2 / Im d 0, 2 and E 2 /E 2 2 = E, = Ker d, 2 whence we can extend the above sequence to... E 0, 2 d 0, 2 E 2,0 2 E 2 E, 2 d, 2 E 3,0 2 as E 2 /E 2 = E 0,2 E 0,2 2 we have that E 2 = Ker (E 2 E 0,2 2 ). If we specialize to the spectral sequence E p,q 2 = R p g(r q f(a)) = R n gf(a) we obtain the exact sequence 0 R g(f(a)) R gf(a) g(r f(a)) R 2 g(f(a)) E 2 where E 2 = Ker (R 2 gf(a) R g(r f(a)) g(r 2 f(a))). R 3 g(f(a)) An important example of a spectral sequence is the Leray spectral sequence. Assume we have an algebra morphism A f A and a sheaf of groups G on A et. We define the direct image of G under f to be the sheaf of groups f G on A et defined by f G(B) = G(B A A ) for all B A et (recall that B A A A et so the right hand side is well defined). This gives us a left exact functor f : S ab (A et) S ab (A et ) and therefore we have right derived functors of it R i f. If G is an Abelian sheaf on A et, then R i f G is a sheaf on A et. One verifies that its stalk in a prime ideal p is equal to (R i f G) p = Het(A i sh p A A, G) where the right hand side is the direct limit of cohomology groups taken over all étale neighborhoods of p. We can relate cohomology of G and f G by the following Theorem 2.22 (Leray spectral sequence) If G is a sheaf of Abelian groups on A et and A f A an algebra morphism, then there is a spectral sequence E p,q 2 = H p et(a, R q f G) = H n et(a, G) In particular, if R j f G = 0 for all j > 0, then for all i 0 we have isomorphisms H i et(a, f G) H i et(a, G) 2.6 Tsen and Tate fields Definition 2.23 A field K is said to be a T sen d -field if every homogeneous form of degree deg with coefficients in K and n > deg d variables has a non-trivial zero in K. For example, an algebraically closed field K is a T sen 0 -field as any form in n-variables defines a hypersurface in P n K. In fact, algebraic geometry tells us a stronger story Lemma 2.24 Let K be algebraically closed. If f,..., f r are forms in n variables over K and n > r, then these forms have a common non-trivial zero in K.

60 60 CHAPTER 2. BRAUER-SEVERI VARIETIES. Proof. Each f i defines a hypersurface V (f i ) P n K. The intersection of r hypersurfaces has dimension n r from which the claim follows. We want to extend this fact to higher Tsen-fields. The proof of the following result is technical unenlightening inequality manipulation, see for example [30]. Proposition 2.25 Let K be a T sen d -field and f,..., f r forms in n variables of degree deg. If n > rdeg d, then they have a non-trivial common zero in K. For our purposes the main interest in Tsen-fields comes from : Theorem 2.26 Let K be of transcendence degree d over an algebraically closed field C, then K is a T sen d -field. Proof. First we claim that the purely transcendental field C(t,..., t d ) is a T sen d - field. By induction we have to show that if L is T sen k, then L(t) is T sen k+. By homogeneity we may assume that f(x,..., x n ) is a form of degree deg with coefficients in L[t] and n > deg k+. For fixed s we introduce new variables y (s) ij with i n and 0 j s such that x i = y (s) i0 + y(s) i t y(s) is ts If r is the maximal degree of the coefficients occurring in f, then we can write f(x i ) = f 0 (y (s) ij ) + f (y (s) ij )t f deg.s+r(y (s) ij )tdeg.s+r where each f j is a form of degree deg in n(s + )-variables. By the proposition above, these forms have a common zero in L provided n(s + ) > deg k (ds + r + ) (n deg i+ )s > deg i (r + ) n which can be satisfied by taking s large enough. the common non-trivial zero in L of the f j, gives a non-trivial zero of f in L[t]. By assumption, K is an algebraic extension of C(t,..., t d ) which by the above argument is T sen d. As the coefficients of any form over K lie in a finite extension E of C(t,..., t d ) it suffices to prove that E is T sen d. Let f(x,..., x n ) be a form of degree deg in E with n > deg d. Introduce new variables y ij with x i = y i e +... y ik e k where e i is a basis of E over C(t,..., t d ). Then, f(x i ) = f (y ij )e f k (y ij )e k where the f i are forms of degree deg in k.n variables over C(t,..., t d ). Because C(t,..., t d ) is T sen d, these forms have a common zero as k.n > k.deg d. Finding a non-trivial zero of f in E is equivalent to finding a common non-trivial zero to the f,..., f k in C(t,..., t d ), done. A direct application of this result is Tsen s theorem : Theorem 2.27 Let K be the functionfield of a curve C defined over an algebraically closed field. Then, the only central simple K-algebras are M n (K). That is, Br(K) =.

61 2.6. TSEN AND TATE FIELDS 6 Proof. Assume there exists a central division algebra of dimension n 2 over K. There is a finite Galois extension L/K such that L = M n (L). If x,..., x n 2 is a K-basis for, then the reduced norm of any x, N(x) = det(x ) is a form in n 2 variables of degree n. Moreover, as x is invariant under the action of Gal(L/K) the coefficients of this form actually lie in K. By the main result, K is a T sen -field and N(x) has a non-trivial zero whenever n 2 > n. As the reduced norm is multiplicative, this contradicts N(x)N(x ) =. Hence, n = and the only central division algebra is K itself. If K is the functionfield of a surface, we also have an immediate application : Proposition 2.28 Let K be the functionfield of a surface defined over an algebraically closed field. If is a central simple K-algebra of dimension n 2, then the reduced norm map N : K is surjective. Proof. Let e,..., e n 2 be a K-basis of and k K, then N( x i e i ) kx n n 2 + is a form of degree n in n 2 + variables. Since K is a T sen 2 field, it has a non-trivial solution (x 0 i ), but then, δ = ( x 0 i e i)x n 2 + has reduced norm equal to k. From the cohomological description of the Brauer group it is clear that we need to have some control on the absolute Galois group G K = Gal(K/K). We will see that finite transcendence degree forces some cohomology groups to vanish. Definition 2.29 The cohomological dimension of a group G, cd(g) d if and only if H r (G, A) = 0 for all r > d and all torsion modules A G-mod. Definition 2.30 A field K is said to be a T ate d -field if the absolute Galois group G K = Gal(K/K) satisfies cd(g) d. First, we will reduce the condition cd(g) d to a more manageable one. To start, one can show that a profinite group G (that is, a projective limit of finite groups, see [30] for more details) has cd(g) d if and only if H d+ (G, A) = 0 for all torsion G-modules A Further, as all Galois cohomology groups of profinite groups are torsion, we can decompose the cohomology in its p-primary parts and relate their vanishing to the cohomological dimension of the p-sylow subgroups G p of G. This problem can then be verified by computing cohomology of finite simple G p -modules of p-power order, but for a profinite p-group there is just one such module namely Z/pZ with the trivial action. Combining these facts we have the following manageable criterium on cohomological dimension. Proposition 2.3 cd(g) d if H d+ (G, Z/pZ) = 0 for the simple G-modules with trivial action Z/pZ.

62 62 CHAPTER 2. BRAUER-SEVERI VARIETIES. We will need the following spectral sequence in Galois cohomology Proposition 2.32 (Hochschild-Serre spectral sequence) If N is a closed normal subgroup of a profinite group G, then holds for every continuous G-module A. E p,q 2 = H p (G/N, H q (N, A)) = H n (G, A) Now, we are in a position to state and prove Tate s theorem Theorem 2.33 Let K be of transcendence degree d over an algebraically closed field, then K is a T ate d -field. Proof. Let C denote the algebraically closed basefield, then K is algebraic over C(t,..., t d ) and therefore G K GC(t,...,t d ) Thus, K is T ate d if C(t,..., t d ) is T ate d. By induction it suffices to prove If cd(g L ) k then cd(g L(t) ) k + Let L be the algebraic closure of L and M the algebraic closure of L(t). As L(t) and L are linearly disjoint over L we have the following diagram of extensions and Galois groups L L(t) G L(t) M G L L G L L(t) G L(t) where G L(t) /G L(t) G L. We claim that cd(g L(t) ). Consider the exact sequence of G L(t) -modules 0 µp M ( )p M 0 where µ p is the subgroup (of C ) of p-roots of unity. As G L(t) acts trivially on µ p it is after a choice of primitive p-th root of one isomorphic to Z/pZ. Taking cohomology with respect to the subgroup G L(t) we obtain 0 = H (G L(t), M ) H 2 (G L(t), Z/pZ) H 2 (G L(t), M ) = Br(L(t)) But the last term vanishes by Tsen s theorem as L(t) is the functionfield of a curve defined over the algebraically closed field L. Therefore, H 2 (G L(t), Z/pZ) = 0 for all simple modules Z/pZ, whence cd(g L(t) ). By the inductive assumption we have cd(g L ) k and now we are going to use exactness of the sequence 0 G L G L(t) G L(t) 0 to prove that cd(g L(t) ) k +. For, let A be a torsion G L(t) -module and consider the Hochschild-Serre spectral sequence E p,q 2 = H p (G L, H q (G L(t), A)) = H n (G L(t), A)

63 2.7. CONIVEAU SPECTRAL SEQUENCE 63 By the restrictions on the cohomological dimensions of G L and G L(t) the level two term has following shape q E p,q 2 = p k k + k + 2 where the only non-zero groups are lying in the lower rectangular region. Therefore, all E p,q = 0 for p+q > k+. Now, all the composition factors of H k+2 (G L(t), A) are lying on the indicated diagonal line and hence are zero. Thus, H k+2 (G L(t), A) = 0 for all torsion G L(t) -modules A and hence cd(g L(t) ) k +. Theorem 2.34 If A is a constant sheaf of an Abelian torsion group A on K et, then H i et(k, A) = 0 whenever i > trdeg C (K). 2.7 Coniveau spectral sequence Consider the setting A i K π k where A is a discrete valuation ring in K with residue field A/m = k. As always, we will assume that A is a C-algebra. By now we have a grip on the Galois cohomology groups Het(K, i µ l n ) and Het(k, i µ l n ) and we will use this information to compute the étale cohomology groups Here, µ l n H i et(a, µ l n ) = µ n... µ }{{ n where the tensorproduct is as sheafs of invertible Z } n = l Z/nZ-modules. We will consider the Leray spectral sequence for i and hence have to compute the derived sheaves of the direct image Lemma R i µ l n. R 0 i µ l n µ l n on A et. µ l n concentrated in m. 3. R j i µ l n 0 whenever j 2.

64 64 CHAPTER 2. BRAUER-SEVERI VARIETIES. Proof. The strict Henselizations of A at the two primes {0, m} are resp. A sh 0 K and A sh m k{t} where K (resp. k) is the algebraic closure of K (resp. k). Therefore, which is zero for i and µ l n (R j i µ l n ) 0 = H j et(k, µ l n ) for j = 0. Further, A sh m A K is the field of fractions of k{t} and hence is of transcendence degree one over the algebraically closed field k, whence (R j i µ l n ) m = H j et(l, µ l n ) which is zero for j 2 because L is T ate. For the field-tower K L K we have that G L = Ẑ = lim µ m because the only Galois extensions of L are the Kummer extensions obtained by adjoining m t. But then, Het(L, µ l n ) = H (Ẑ, µ l n (K)) = Hom(Ẑ, µ l n (K)) = µ l n from which the claims follow. Theorem 2.36 We have a long exact sequence 0 H (A, µ l n ) H (K, µ l n ) H 0 (k, µ l n ) H 2 (A, µ l n ) H 2 (K, µ l n ) H (k, µ l n )... Proof. By the foregoing lemma, the second term of the Leray spectral sequence for i µ l n looks like H 0 (k, µ l n ) H (k, µ l n ) H 2 (k, µ l n )... H 0 (A, µ l n ) H (A, µ l n ) H 2 (A, µ l n )... with connecting morphisms Het i (k, µ l n ) α i H i+ et (A, µ l n ) The spectral sequences converges to its limiting term which looks like

65 2.7. CONIVEAU SPECTRAL SEQUENCE Ker α Ker α 2 Ker α 3... H 0 (A, µ l n ) H (A, µ l n ) Coker α... and the Leray sequence gives the short exact sequences 0 H et (A, µ l n ) H et (K, µ l n ) Ker α 0 0 Coker α H 2 et (K, µ l n ) Ker α2 0 0 Coker αi H i et (K, µ l n ) Ker αi 0 and gluing these sequences gives us the required result. In particular, if A is a discrete valuation ring of K with residue field k we have for each i a connecting morphism Het(K, i µ l n ) i,a Het i (k, µ l n ) Like any other topology, the étale topology can be defined locally on any scheme X. That is, we call a morphism of schemes Y f X an étale extension (resp. cover) if locally f has the form f a U i : A i = Γ(U i, O X ) Bi = Γ(f (U i ), O Y ) with A i B i an étale extension (resp. cover) of algebras. Again, we can construct the étale site of X locally and denote it with X et. Presheaves and sheaves of groups on X et are defined similarly and the right derived functors of the left exact global sections functor Γ : S ab (X et ) Ab will be called the cohomology functors and we denote R i Γ(G) = H i et(x, G) From now on we restrict to the case when X is a smooth, irreducible projective variety of dimension d over C. In this case, we can initiate the computation of the cohomology groups Het(X, i µ l n ) via Galois cohomology of functionfields of subvarieties using the coniveau spectral sequence Theorem 2.37 Let X be a smooth irreducible variety over C. Let X (p) denote the set of irreducible subvarieties x of X of codimension p with functionfield C(x), then there exists a coniveau spectral sequence E p.q = H q p et x X (p) (C(x), µ l p n ) = H p+q et (X, µ l n )

66 66 CHAPTER 2. BRAUER-SEVERI VARIETIES. In contrast to the spectral sequences used before, the existence of the coniveau spectral sequence by no means follows from general principles. In it, a lot of heavy machinery on étale cohomology of schemes is encoded. In particular, cohomology groups with support of a closed subscheme, see for example [20, p. 9-94], and cohomological purity and duality, see [20, p ] a detailed exposition of which would take us too far afield. For more details we refer the reader to [5]. Using the results on cohomological dimension and vanishing of Galois cohomology of µ k n when the index is larger than the transcendence degree, we see that the coniveau spectral sequence has the following shape q... d E p,q =... p where the only non-zero terms are in the indicated region. Let us understand the connecting morphisms at the first level, a typical instance of which is H i (C(x), µ n l p ) H i (C(y), µ n l p ) x X (p) y X (p+) and consider one of the closed irreducible subvarieties x of X of codimension p and one of those y of codimension p +. Then, either y is not contained in x in which case the component map H i (C(x), µ l p n ) H i (C(y), µ l p n ) is the zero map. Or, y is contained in x and hence defines a codimension one subvariety of x. That is, y defines a discrete valuation on C(x) with residue field C(y). In this case, the above component map is the connecting morphism defined above. In particular, let K be the functionfield of X. Then we can define the unramified cohomology groups F i,l n (K/C) = Ker H i (K, µ l n ) i,a H i (k A, µ l n ) where the sum is taken over all discrete valuation rings A of K (or equivalently, the irreducible codimension one subvarieties of X) with residue field k A. By definition, this is a (stable) birational invariant of X. In particular, if X is (stably) rational over C, then Fn i,l (K/C) = 0 for all i, l 0

67 2.8. THE ARTIN-MUMFORD EXACT SEQUENCE The Artin-Mumford exact sequence In this section S will be a smooth irreducible projective surface. Definition 2.38 S is called simply connected if every étale cover Y S is trivial, that is, Y is isomorphic to a finite disjoint union of copies of S. The first term of the coniveau spectral sequence of S has following shape H 2 (C(S), µn) C H (C(S), Zn) P µ n 0... H (C(S), µn) C Zn µn where C runs over all irreducible curves on S and P over all points of S. Lemma 2.39 For any smooth S we have H (C(S), µ n ) connected, Het(S, µ n ) = 0. C Z n. If S is simply Proof. Using the Kummer sequence µn Gm ( ) Gm and Hilbert 90 we obtain that H et(c(s), µ n ) = C(S) /C(S) n The first claim follows from the exact diagram describing divisors of rational functions µ n µ n 0 0 C C(S) div C Z 0 0 C ( ) n C(S) div C Z 0 n. 0 C Z n C Z n By the coniveau spectral sequence we have that Het(S, µ n ) is equal to the kernel of the morphism Het(C(S), µ n ) γ C Z n and in particular, H (S, µ n ) H (C(S), µ n ).

68 68 CHAPTER 2. BRAUER-SEVERI VARIETIES. As for the second claim, an element in H (S, µ n ) determines a cyclic extension L = C(S) n f with f C(S) /C(S) n such that in each fieldcomponent L i of L there is an étale cover T i S with C(Ti ) = L i. By assumption no non-trivial étale covers exist whence f = C(S) /C(S) n. If we invoke another major tool in étale cohomology of schemes, Poincaré duality, see for example [20, VI, ], we obtain the following information on the cohomology groups for S. Proposition 2.40 (Poincaré duality for S) If S is simply connected, then. H 0 et(s, µ n ) = µ n 2. H et(s, µ n ) = 0 3. H 3 et(s, µ n ) = 0 4. H 4 et(s, µ n ) = µ n Proof. The third claim follows from the second as both groups are dual to each other. The last claim follows from the fact that for any smooth irreducible projective variety X of dimension d one has that Het 2d (X, µ n ) µ n d We are now in a position to state and prove the important Theorem 2.4 (Artin-Mumford exact sequence) If S is a simply connected smooth projective surface, then the sequence 0 Brn (S) Brn (C(S)) C C(C) /C(C) n is exact. P µ n µ n 0 Proof. was The top complex in the first term of the coniveau spectral sequence for S H 2 (C(S), µ n ) α C H (C(C), Z n ) β P µ n The second term of the spectral sequence (which is also the limiting term) has the following form Ker α Ker β/im α Coker β 0... Ker γ Coker γ µn

69 2.8. THE ARTIN-MUMFORD EXACT SEQUENCE 69 By the foregoing lemma we know that Coker γ = 0. By Poincare duality we know that Ker β = Im α and Coker β = µ n. Hence, the top complex was exact in its middle term and can be extended to an exact sequence 0 H 2 (S, µ n ) H 2 (C(S), µ n ) C H (C(C), Z n ) P µ n µ n 0 As Z n µ n the third term is equal to C C(C) /C(C) n by the argument given before and the second term we remember to be Br n (C(S). The identification of Br n (S) with H 2 (S, µ n ) will be explained below. Some immediate consequences can be drawn from this : For a smooth simply connected surface S, Br n (S) is a birational invariant (it is the birational invariant Fn 2, (C(S)/C) of the foregoing section. In particular, if S = P 2 we have that Br n (P 2 ) = 0 and we obtain the description of Br n (C(x, y)) by Z n -wrinkles as 0 Brn C(x, y) C C(C) /C(C) n P µ n µn 0 Example 2.42 If S is not necessarily simply connected, show that any class in Br(C(S)) n determines a Z n -wrinkle. Example 2.43 If X is a smooth irreducible rational projective variety of dimension d, show that the obstruction to classifying Br(C(X)) n by Z n -wrinkles is given by H 3 et(x, µ n ). We will give a ringtheoretical interpretation of the maps in the Artin-Mumford sequence. Observe that nearly all maps are those of the top complex of the first term in the coniveau spectral sequence for S. We gave an explicit description of them using discrete valuation rings. The statements below follow from this description. Let us consider a discrete valuation ring A with field of fractions K and residue field k. Let be a central simple K-algebra of dimension n 2. Definition 2.44 An A-subalgebra Λ of will be called an A-order if it is a free A-module of rank n 2 with Λ.K =. An A-order is said to be maximal if it is not properly contained in any other order. In order to study maximal orders in (they will turn out to be all conjugated), we consider the completion  with respect to the m-adic filtration where m = At with t a uniformizing parameter of A. ˆK will denote the field of fractions of  and ˆ = K ˆK. Because ˆ is a central simple ˆK-algebra of dimension n 2 it is of the form ˆ = M t (D) where D is a division algebra with center ˆK of dimension s 2 and hence n = s.t. We call t the capacity of at A. In D we can construct a unique maximal Â-order Γ, namely the integral closure of  in D. We can view Γ as a discrete valuation ring extending the valuation v defined by A on K. If v : ˆK Z, then this extended valuation w : D n 2 Z is defined as w(a) = ( ˆK(a) : ˆK) v(n ˆK(a)/ ˆK(a))

70 70 CHAPTER 2. BRAUER-SEVERI VARIETIES. for every a D where ˆK(a) is the subfield generated by a and N is the norm map of fields. The image of w is a subgroup of the form e Z n 2.Z. The number e = e(d/ ˆK) is called the ramification index of D over ˆK. We can use it to normalize the valuation w to v D : D Z defined by v D (a) = e n 2 v(n D/ ˆK (a)) With these conventions we have that v D (t) = e. The maximal order Γ is then the subalgebra of all elements a D with v D (a) 0. It has a unique maximal ideal generated by a prime element T and we have that Γ T Γ Γ = is a division algebra finite dimensional over Â/t = k (but not necessarily having k as its center). The inertial degree of D over ˆK is defined to be the number f = f(d/ ˆK) = (Γ : k) and one shows that s 2 = e.f and e s whence s f After this detour, we can now take Λ = M t (Γ) as a maximal Â-order in ˆ. One shows that all other maximal Â-orders are conjugated to Λ. Λ has a unique maximal ideal M with Λ = M t (Γ). Definition 2.45 With notations as above, we call the numbers e = e(d/ ˆK), f = f(d/ ˆK) and t resp. the ramification, inertia and capacity of the central simple algebra at A. If e = we call Λ an Azumaya algebra over A, or equivalently, if Λ/tΛ is a central simple k-algebra of dimension n 2. Now let us consider the case of a discrete valuation ring A in K such that the residue field k is T sen. The center of the division algebra Γ is a finite dimensional field extension of k and hence is also T sen whence has trivial Brauer group and therefore must coincide with Γ. Hence, Γ = k(a) a commutative field, for some a Γ. But then, f s and we have e = f = s and k(a) is a cyclic degree s field extension of k. Because s n, the cyclic extension k(a) determines an element of H et(k, Z n ). Definition 2.46 Let Z be a normal domain with field of fractions K and let be a central simple K-algebra of dimension n 2. A Z-order B is a subalgebra which is a finitely generated Z-module. It is called maximal if it is not properly contained in any other order. One can show that B is a maximal Z-order if and only if Λ = B p is a maximal order over the discrete valuation ring A = Z p for every height one prime ideal p of Z. Return to the situation of an irreducible smooth projective surface S. If is a central simple C(S)-algebra of dimension n 2, we define a maximal order as a sheaf A of O S -orders in which for an open affine cover U i S is such that A i = Γ(U i, A) is a maximal Z i = Γ(U i, O S ) order in Any irreducible curve C on S defines a discrete valuation ring on C(S) with residue field C(C) which is T sen. Hence, the above argument can be applied to obtain from A a cyclic extension of C(C), that is, an element of C(C) /C(C) n. Definition 2.47 We call the union of the curves C such that A determines a nontrivial cyclic extension of C(C) the ramification divisor of (or of A).

71 2.9. BRAUER-SEVERI SCHEMES 7 The map in the Artin-Mumford exact sequence Br n (C(S)) C H et(c(c), µ n ) assigns to the class of the cyclic extensions introduced above. Definition 2.48 An S-Azumaya algebra (of index n) is a sheaf of maximal orders in a central simple C(S)-algebra of dimension n 2 such that it is Azumaya at each curve C, that is, such that [ ] lies in the kernel of the above map. Observe that this definition of Azumaya algebra coincides with the one given in the discussion of twisted forms of matrices. One can show that if A and A are S-Azumaya algebras of index n resp. n, then A OS A is an Azumaya algebra of index n.n. We call an Azumaya algebra trivial if it is of the form End(P) where P is a vectorbundle over S. The equivalence classes of S-Azumaya algebras can be given a group-structure called the Brauer-group Br(S) of the surface S. 2.9 Brauer-Severi schemes Now that we have some control over the central simple algebras over functionfields, we will generalize the classical notion of Brauer-Severi variety of a central simple algebra to the setting of (maximal) orders. Fix a projective normal variety X with function field C(X) and let be a central simple C(X)-algebra of dimension n 2. Let A be a sheaf of O X -algebras. We call A an O X -order in if and only if for every affine open subset U X we have that the sections A(U) = Γ(U, A) is a finite module over the integrally closed domain R(U) = Γ(U, O X ) such that A(U) R(U) C(X) We will define the Brauer-Severi scheme BS(A) of A locally so we fix an affine open set U and denote A = A(U) and R = R(U). Let K be the algebraic closure of C(X), then we have the natural inclusions A K M n (K) R C(X) K By Galois descent we can define a linear trace map δ t(δ) = tr(δ ) t C(X) such that for all with tr the usual trace map on M n (K). That is, A t A is a trace map on the order A satisfying the Cayley-Hamilton identities of n n matrices such that the image of the trace map is the center R.. One of the major results we will prove in chapter 4 is that this allows to reconstruct both the order A and the center R from geometrical data. Consider the affine scheme of all n-dimensional trace preserving representations of A, rep t n A = {A φ Mn (C) tr φ = φ t } where tr is the ordinary trace map on M n (C). Conjugation by GL n (C) in the target space M n (C) induces a GL n (C)-action on rep t n A. We will show in chapter 4 that

72 72 CHAPTER 2. BRAUER-SEVERI VARIETIES. A is the ring of GL n (C)-equivariant maps from rep t n A to M n (C), and R is the ring of polynomial GL n (C)-invariants on rep t n A. We use these representation spaces to define the Brauer-Severi scheme BS(A) in a fashion very similar to the Hilbert scheme construction in the previous chapter. In fact, historically these varieties were introduced and studied by M. Nori [22] who called them noncommutative Hilbert schemes. We follow here the account of M. Van den Bergh in [3]. Consider the GL n (C) action on the product scheme rep t n A C n given by g.(φ, v) = (g φ g, gv) In this product we consider the set of Brauer stable points which are Brauer s (A) = {(φ, v) φ(a)v = C n } which is also the subset of points with trivial stabilizer subgroup. Hence, every GL n (C)-orbit in Brauer s (A) is closed and we can form the orbit space which we call the Brauer-Severi scheme of the order A BS(A) = Brauer s (A)/GL n (C). This is shown to be a projective space bundle over the quotient variety rep t n A/GL n (C) which by the above is the variety corresponding to R, that is, the chosen affine open subset of the projective normal variety X. For arbitrary orders not much can be said about these Brauer-Severi schemes. We will now restrict to smooth orders A that is such that their representation space rep t n A is a smooth GL n (C)-variety. In chapter 5 we will prove that this geometric condition is equivalent to the algebraic characterization of A via the lifting property modulo nilpotent ideals in the category of algebras equipped with a trace map satisfying the Cayley-Hamilton identities of n n matrices. Lemma 2.49 Let A be a smooth order, that is, rep t n A is a smooth variety. Then, the Brauer-Severi scheme BS(A) is a smooth variety. Proof. As the action of GL n (C) on Brauer s (A) is free, it suffices to prove that Brauer s (A) is a smooth variety. As Brauer s (A) is a Zariski open subset of the variety rep t n A C n which is smooth by assumption, the result follows. Remains to classify the smooth orders A. The strategy we will follow is : first compute the étale local structure of these orders, that is, if m R is a maximal ideal of R we describe  m = A R ˆRm These structures will follow by combining the étale slice results in invariant theory with the geometric reconstruction of an order A from its representation space rep t n A. The local structures can be classified combinatorial by quiver-data. In the special case of orders over smooth surfaces we will show that the relevant data is given a Z n -loop. That is,

73 2.9. BRAUER-SEVERI SCHEMES 73 A two loop quiver A klm with dimension vector α = (,..., ) 2 k+l+m k+ k+l+ k+l k- k and the vertices numbered as indicated. In this picture we make the natural changes whenever k or l is zero. An unordered partition of n in k + l + m nonzero parts p = (p,..., p k+l+m ) associated to the vertices of the quiver We will prove in chapter 6 that a Z n -loop encodes the following algebraic data. Let m be a maximal ideal of R, then there is a closed GL n (C)-orbit in rep t n A corresponding to m. This closed orbit determines a semi-simple n-dimensional representation of A. the fact that all dimension components are equal to one asserts that all the simple components of this representation occur with multiplicity one and the components of the partition p give the dimensions of these simple components. We will prove in chapter 6 a local characterization of smooth orders in arbitrary dimension. In the special case of surfaces we have the following result Theorem 2.50 Let be a central simple algebra of dimension n 2 over C(S) where S is a smooth projective surface and let A be an O S -order in. Then A is a sheaf of smooth orders if and only if for every affine open subset U S and section algebras A = Γ(U, A) and R = Γ(U, O S ) we have for every maximal ideal m a Z n - loop (A klm, p) such that in local coordinates x, y of S near the point corresponding to m () (x) (y) () Â m (x) () (y) () Mn (C[[x, y]]) ()

74 74 CHAPTER 2. BRAUER-SEVERI VARIETIES. where at spot (i, j) with i, j k + l + m there is a block of dimension p i p j with entries the indicated ideal of C[[x, y]]. Using such an explicit local description of the order, it is also possible to determine the étale local structure of the Brauer-Severi variety BS(A) in a neighborhood of the closed fiber corresponding to m as well as the structure of the fiber π (m). Assume A is locally of type (A klm, p) and construct the extended quiver p p p p p 0 p p p p p 2 k+l+m k+ k+l+ k+l k- k That is, we add on extra vertex labeled zero and connect it to vertex i by p i directed arrows where p i is the i-th component of the unordered partition p. In chapter 8 we will prove that the local structure of BS(A) is determined by the moduli space of θ-stable representations of this extended quiver for a certain character θ. The fiber of the structural morphism BS(B) π S over the point corresponding to m we will show in chapter 8 to be the moduli space of the θ-stable representations in the nullcone of the quiver. Rather than introducing all these concepts here we will illustrate these results in the case of the smooth orders in quaternion algebras considered by M. Artin and D. Mumford in [3]. Example 2.5 Smooth quaternion orders over surfaces. Let A be a maximal order in a quaternion division algebra over a smooth projective surface S such that the ramification divisor is a disjoint union of smooth curves. We restrict to affine sections on an affine open subset U and call them again A and R. If m R is a maximal ideal corresponding to a point on S not contained in the ramification divisor of A, then A m is an Azumaya algebra and as the Brauer group of every Henselian local ring is trivial, it follows that in these points the étale local structure of A must be  m M 2 (C[[x, y]]) for suitable local variables x and y. If however m corresponds to a point on the ramification divisor we have seen before that a local description of A m is the free

75 2.9. BRAUER-SEVERI SCHEMES 75 R m -module spanned by {, i, j, ij} such that i 2 = a j 2 = bt ji = ij where a, b are units in R m and t is a local equation of the divisor in the point. That is, after splitting with say the quadratic extension by adding a we can view the up-tensored A m to be the subalgebra over the center by the matrices [ ] [ ] a t a b 0 Hence, if we take our local variables to be such that x = t we obtain for the étale local structure [ ] C[[x, y]] (x) Â m. C[[x, y]] C[[x, y]] In our quiver-approach there are just two possibilities for Z 2 -loops. They are type : A 00 = and p = 2, type 2 : A 0 = and p =. We observe that type is precisely the Azumaya case and type 2 corresponds to a point on the ramification divisor. To compute the fiber of BS(A) over a type point (an Azumaya point) we have to consider the quiver and we need to classify orbits of θ-stable representations where θ = (2, 2) in the nullcone containing the vertex space in v 0. Being in the nullcone means that all evaluations around oriented cycles in the quiver should be zero, so the two loop-matrices must be zero. Being θ-stable means that the representation has no proper subrepresentation, say with dimension vector β = (, n ) such that θ, β >= 2 2n > 0. In this case this means that either of the two matrices corresponding to the two extra arrows must be nonzero. That is, the relevant representation space is C 2. Considering the C C -action on these representations (λ, µ).(a, b) = µ λ (a, b) we see that the classifying space is P (C) as it should be over an Azumaya point. The fiber over a type 2 point (a ramified point) is determined by the quiver

76 76 CHAPTER 2. BRAUER-SEVERI VARIETIES. Again, we have to consider representations in the nullcone, meaning that the loopmatrix is zero and that at least one of the two arrows in the A 0 -quiver must be zero. This time we have to consider θ-stable representations where θ = (2,, ) which means that the representation is not allowed to have a proper subrepresentation of dimension vector β = (, n, n 2 ) such that θ, β = 2 n n 2 > 0. If one of the arrows a in A 0 is non-zero this means that the extra arrow ending in the source of a must be nonzero and if both arrows in A 0 are zero the two extra arrows must be non-zero. That is, we have to classify the orbits of the quiver-representations giving us the required P P as classifying space. In this book we will give combinatorial tools to extend these descriptions of Brauer-Severi schemes of smooth orders both the higher n and to higher dimensional base varieties.

77 2.9. BRAUER-SEVERI SCHEMES 77 Classical Structures. Among the plenitude of Quillen-smooth algebras there is a small subclass of well understood examples : the path algebras of quivers. The main result we will prove in this part asserts that for an arbitrary Quillen-smooth algebra A, the local study of the approximation at level n, A@ n, is controlled by a quiver setting. In this introduction, we will first give an example of this theory and then we will briefly indicate the relationship between this reduction result and the theory of A -algebras. Consider Artin s braid group B 3 on three strings. B 3 has the presentation B 3 L, R LR L = R LR where L and R are the fundamental 3-braids L R If we let S = LR L and T = R L, an algebraic manipulation shows that B 3 = S, T T 3 = S 2 is an equivalent presentation for B 3. The center of B 3 is the infinite cyclic group generated by the braid Z = S 2 = (LR L) 2 = (R L) 3 = T 3 It follows from the second presentation of B 3 that the quotient group modulo the center is isomorphic to B 3 Z s, t s2 = = t 3 Z 2 Z 3 the free product of the cyclic group of order 2 (with generator s) and the cyclic group of order 3 (with generator t). This group is isomorphic to the modular group P SL 2 (Z) via L [ ] 0 [ ] and R 0 It is well known that the modular group P SL 2 (Z) acts on the upper half-plane H 2 by left multiplication in the usual way, that is [ ] a b c d : H 2 H 2 given by z az + b cz + d

78 78 CHAPTER 2. BRAUER-SEVERI VARIETIES. The fundamental domain H 2 /P SL 2 (Z) for this action is the hyperbolic triangle and the action defines a quilt-tiling [?] on the hyperbolic plane, indexed by elements of P SL 2 (Z) = Z 2 Z 3 By the above, the representation theory of the 3-string braid group B 3 is essentially reduced to that of the modular group P SL 2 (Z). The latter can be studied using noncommutative geometry as the group algebra CP SL 2 (Z) is a Quillen-smooth algebra. Indeed, CP SL 2 (Z) CZ 2 CZ 3 (C C) (C C C) and the free product of Quillen-smooth algebras is again Quillen-smooth as follows immediately from the universal property of free products. Phrased differently, the group algebra CP SL 2 (Z) is the coordinate ring of the noncommutative product of two commutative points with three commutative points. For a fixed integer n we want to determine the isomorphism classes of all n- dimensional representations of P SL 2 (Z), or equivalently, of the Quillen-smooth algebra CP SL 2 (Z). We give a geometric reformulation of this problem. Let rep n CP SL 2 (Z) be the representation variety of CP SL 2 (Z), that is, its geometric points are algebra morphisms CP SL 2 (Z) φ Mn (C) From the presentation of P SL 2 (Z) we see that we can identify it with the closed subvariety of the affine space M n (C) M n (C) rep n CP SL 2 (Z) = {(A, B) M n (C) M n (C) A 2 = n = B 3 } It follows from Quillen-smoothness that rep n CP SL 2 (Z) is a smooth affine variety (though not necessarily connected) for every n. On this representation space, the group of invertible matrices GL n (C) acts by simultaneous conjugation, that is g.(a, B) = (gag, gbg )

79 2.9. BRAUER-SEVERI SCHEMES 79 and isomorphism classes of representations correspond to GL n (C)-orbits. That is, we have to classify the orbits in rep n CP SL 2 (Z). For n > there is no Hausdorff orbit-space due to the existence of non-closed orbits. For this reason, the classification of isomorphism classes of n-dimensional representations of P SL 2 (Z) is split into two sub-problems : The construction of an algebraic quotient map rep n CP SL 2 (Z) π issn CP SL 2 (Z) classifying closed GL n (C)-orbits, which we will show to be the same as isomorphism classes of semi-simple n-dimensional representations. For x iss n CP SL 2 (Z), the classification of all orbits in the fiber π (ξ). That is, if M ξ is the corresponding semi-simple n-dimensional representation, we want to classify all representations having M ξ as the direct sum of its Jordan-Hölder components. To solve the first, we introduce the approximation at level n, denoted by CP SL 2 (Z)@ n. To define it we first adjoin formal traces to the groupalgebra. That is, we consider the commutative polynomial algebra in the variables t w where w runs through all necklaces w of length l 0 w where each of beads is either s or t subject to the conditions that no two (resp. three) consecutive beads are labeled s (resp. t). With CP SL 2 (Z) t we denote the tensor product C[t w w necklace] C CP SL 2 (Z) and there is a natural C-linear trace map CP SL 2 (Z) t tr C[t w w necklace] defined by sending each monomial m = s i... t i l in the noncommuting variables s and t to t w where w is the cyclic word constructed from m. As we are interested in n-dimensional representations we would like to interpret t w as the character tr(m) = T r(a i... B i l ). For this reason we consider the quotient CP SL 2 (Z) = CP SL 2(Z) t (t n, χ (n) m (m)) where for each monomial m we define the formal Cayley-Hamilton polynomial χ (n) m (t) of m of degree n to be the polynomial in C[t w w][t] obtained after expressing the coefficients of the polynomial f(t) = n i= (t λ i) which are symmetric functions in the λ i as polynomials in the Newton functions η i = λ i j and replacing η i by tr(m i ). By construction it follows that there is a one-to-one correspondence between n-dimensional representations of the group algebra CP SL 2 (Z) and n-dimensional trace preserving representations of CP SL 2 (Z)@ n. We will prove in

80 80 CHAPTER 2. BRAUER-SEVERI VARIETIES. chapter 4 that CP SL 2 (Z)@ n is a finitely generated module over the commutative central subalgebra tr(cp SL 2 (Z)@ n ) which is also tr(cp SL 2 (Z)@ n ) = C[iss n P SL 2 (Z)] the coordinate ring of the quotient variety as semi-simple representations are determined by their characters. The algebraic quotient map rep n P SL 2 (Z) issn P SL 2 (Z) is given by sending an n-dimensional representation to its set of characters tr(a i... B i l ). Moreover, we will prove in chapter 5 that the approximation at level n can be geometrically reconstructed CP SL 2 (Z)@ n = { rep n P SL 2 (Z) equiv Mn (C) } as the algebra of all GL n (C)-equivariant maps from the representation space to n n matrices. Both results are valid for any affine C-algebra A and follow from invariant theory and the generic case of m-tuples of n n matrices under simultaneous conjugation, which we will prove in chapter 3. Now, let ξ iss n P SL 2 (Z) be the point corresponding to the semi-simple n- dimensional representation M ξ = S e... S e k k where the S i are distinct irreducible d i -dimensional representations which occur in M ξ with multiplicity e i (hence, n = d i e i ). Using the theory of étale slices we will prove in chapter 5 that the étale local structure of the quotient variety iss n P SL 2 (Z) in a neighborhood of ξ is fully determined by combinatorial data consisting of a quiver Q ξ on k vertices corresponding to the distinct simple components of M ξ, and a dimension vector α ξ = (e,..., e k ) corresponding to the multiplicities of these simple components in M ξ. The local quiver Q ξ is constructed as follows (a proof will be given in chapter 7). If S is a simple P SL 2 (Z)-representation, we can decompose its restrictions to the cyclic subgroups Z 2 and Z 3 into one-dimensional eigenspaces { S Z2 S Z3 E a E a2 F b F b2 ζ F b3 ζ 2 where E λ resp. F λ are the one-dimensional simple representations on which s resp. t acts via multiplication with λ. We will call this 5-tuple a τ(s) = a2 to be the type of S. If the dimension of S, d(s) = d, we will show in chapter 7 that these numbers must satisfy the relations (or see [?] for another proof) b b 2 b 3 { d = a + a 2 = b + b 2 + b 3 a i b j for all i, j

81 2.9. BRAUER-SEVERI SCHEMES 8 With these notations, the local quiver Q ξ has the following local shape for every two of its vertices v i and v j a ii e i a ij a ji e j a jj where the numbers of multiple arrows and loops are determined by the formulas { a ij = d(s i )d(s j ) τ(s i ), τ(s j ) when i j a ii = + d(s i ) 2 τ(s i ), τ(s i ) where, is the usual inproduct on 5-tuples. For example, iss 4 P SL 2 (Z) has several components of dimension 3 and 2. For one of the three 3-dimensional components, the different types of semi-simples M ξ and corresponding local quivers Q ξ of the 3-dimensional component are listed below In chapter 5 we prove that the étale local structure of iss n P SL 2 (Z) near ξ is isomorphic to that of iss αξ Q ξ near the trivial representation. The local algebra of the latter is generated by traces along oriented cycles in Q ξ. That is, for every arrow a e j we e i take an e j e i matrix M a of indeterminates. Multiplying these matrices along an oriented cycle in Q ξ and taking the trace of the square matrix obtained gives an invariant function. Such invariants generate the local algebra of iss αξ Q ξ in the trivial semi-simple representation. Therefore, to verify whether

82 82 CHAPTER 2. BRAUER-SEVERI VARIETIES. iss n P SL 2 (Z) is smooth in ξ it suffices to prove that the traces along oriented cycle for the quiver-setting (Q ξ, α ξ ) generate a polynomial algebra. For example, consider a point ξ iss 4 P SL 2 (Z) of type » a b 2 Then, the traces along oriented cycles in Q ξ are generated by the following three algebraic independent polynomials x = ac + bd y = eg + fh z = (cg + dh)(ea + fb) and hence iss 4 P SL 2 (Z) is smooth in ξ. The other cases being easier, we see that this component of iss 4 P SL 2 (Z) is a smooth manifold. Another application of this local quiver-setting (Q ξ, α ξ ) is that one can construct families of irreducible representations of P SL 2 (Z) starting from known ones. For example consider the point ξ of type Then, M ξ is determined by the following matrices ( , 0 ζ ζ 0 ) ˆc a d ˆe» g h The quiver-setting (Q ξ, α ξ ) implies that any nearby orbit is determined by a matrixcouple b 0 0 b ( a d 0 0 c f, a 2 ζ 2 d c 2 ζ f 2 ) 0 0 e 0 0 e 2 and as there is just one arrow in each direction these entries must satisfy 0 = a a 2 = b b 2 = c c 2 = d d 2 = e e 2 = f f 2 As the square of the first matrix must be the identity matrix 4, we have in addition that 0 = a b = c d = e f Hence, we get several sheets of 3-dimensional families of representations (possibly, matrix-couples lying on different sheets give isomorphic P SL 2 (Z)-representations, as the isomorphism holds in the étale topology and not necessarily in the Zariski topology). One of the sheets has representatives b 0 0 ( a d , 0 ζ c ζ f ) 0 0 e b c d f e f

83 2.9. BRAUER-SEVERI SCHEMES 83 From the description of dimension vectors of semi-simple quiver representations we will give in chapter 6 it follows that such a representation is simple if and only if ab 0 cd 0 and ef 0 Moreover, these simples are not-isomorphic unless their traces ab, cd and ef evaluate to the same numbers. A final application of the local quiver-setting is that it solves the second subproblem. That is, assume that ξ iss n P SL 2 (Z) has local quiver-setting (Q ξ, α ξ ), then the isomorphism classes of P SL 2 (Z)-representations having as direct sum of its Jordan-Hölder components the semi-simple representation M ξ are in one-to-one correspondence with the GL(α) = GL e (C)... GL ek (C)-orbits in the nullcone of the quiver representation space rep αξ Q ξ. In chapter 8 we will see how we can stratify these nullcones to get a handle on this problem. In the above example, this nullcone problem is quite trivial. A representation has M ξ as Jordan-Hölder sum if and only if all traces vanish, that is, ab = cd = ef = 0 Under the action of the group GL(α ξ ) = C C C C, these orbits are easily seen to be classified by the arrays a c e b d f filled with zeroes and ones subject to the rule that no column can have two s, giving 27 = 3 3 -orbits. In chapter 3 we will give more applications to the representation theory of B 3, P SL 2 (Z) and, more generally, knot groups. Although we will arrive at the local quiver-setting (Q ξ, α ξ ) by invariant theory we will indicate an alternative approach using the theory of A -algebras. More details can be found in the excellent notes of B. Keller [?]. In recent years some families of multi-linear objects satisfying certain extended associativity constraints have been studied which are naturally associated to topological operads. For our purposes, the relevant operad is the tiny interval operad. That is, let D (n) be the collection of all configurations i i 2 in consisting of the unit interval with n closed intervals cut out, each gap given a label i j where (i, i 2,..., i n ) is a permutation of (, 2,..., n). Clearly, D (n) is a 2n-dimensional C -manifold having n! connected components, each of which is a contractible space. the operadic structure comes from the collection of composition maps D (n) (D (m )... D (m n )) D (m m n ) defined by resizing the configuration in the D (m i )-component such that it fits

84 84 CHAPTER 2. BRAUER-SEVERI VARIETIES. precisely in the i-th gap of the configuration of the D (n)-component. That is, i i in k k 2 km i j j 2 jm i l l 2 lm in We then obtain a unit interval having m m n gaps which are labeled in the natural way, that is the first m labels are for the gaps in the D (m )-configuration fitted in gap, the next m 2 labels are for the gaps in the D (m 2 )-configuration fitted in gap 2 and so on. The tiny interval operator consists of a collection of topological spaces D (n) for n 0, a continuous action of S n on D (n) by relabeling, for every n, an identity element id D (), continuous composition maps m (n,m,...,m n) satisfying a list of axioms. For every topological operad, we can take its homology operad and define a class of algebras over it, see for example [?] or [?] for details. Rather than introducing all these concepts here we will list the set of axioms defining the algebra-objects associated to the tiny interval operad : the A -algebras. Definition 2.52 An A -algebra is a Z-graded complex vectorspace B = p Z B p endowed with homogeneous C-linear maps m n : B n B of degree 2 n for all n, satisfying the following relations We have m m = 0, that is (B, m ) is a differential complex... m Bi m Bi m Bi+ m... We have the equality of maps B B B m m 2 = m 2 (m + m ) where is the identity map on the vectorspace B. That is, m is a derivation with respect to the multiplication B B m2 B.

85 2.9. BRAUER-SEVERI SCHEMES 85 We have the equality of maps B B B B m 2 ( m2 m 2 ) = m m 3 + m 3 (m + m + m ) where the right second expression is the associator for the multiplication m 2 and the first is a boundary of m 3, implying that m 2 is associative up to homology. More generally, for n we have the relations ( ) i+j+k m l ( i m j k ) = 0 where the sum runs over all decompositions n = i+j+k and where l = i++k. These identities can be pictorially represented by b i+ m j ± = 0 b m l Observe that an A -algebra B is in general not associative for the multiplication m 2, but its homology H B = H (B, m 2 ) is an associative graded algebra for the multiplication induced by m 2. Further, if m n = 0 for all n 3, then B is an associative differentially graded algebra and conversely every differentially graded algebra yields an A -algebra with m n = 0 for all n 3. Let A be an associative C-algebra and M a left A-module. Choose an injective resolution of M 0 M I 0 I... with the I k injective left A-modules and denote by I the complex I : 0 I 0 d I d... Let B = HOM A (I, I ) be the morphism complex. That is, its n-th component are the graded A-linear maps I I of degree n. This space can be equipped with a differential d(f) = d f ( ) n f d for f in the n-th part Then, B is a differentially graded algebra where the multiplication is the natural composition of graded maps. The homology algebra H B = Ext A(M, M)

86 86 CHAPTER 2. BRAUER-SEVERI VARIETIES. is the extension algebra of M. This extension algebra has a canonical structure of A -algebra with m = 0 and m 2 he usual multiplication. Now, let M,..., M k be A-modules (for example, finite dimensional representations) and with filt(m,..., M k ) we denote the full subcategory of all A-modules whose objects admit finite filtrations with subquotients among the M i. We have the following result, see for example [?, 6]. Theorem 2.53 Let M = M... M k. The canonical A -structure on the extension algebra Ext A (M, M) contains enough information to reconstruct the category filt(m,..., M k ). Finally, let us elucidate the connection between this result and the local quiversetting (Q ξ, α ξ ) associated to a semi-simple n-dimensional representation M ξ = S e... S e k k of a Quillen-smooth algebra A. In chapter 9 we will prove that for a Quillen-smooth algebra Ext i A (M, M) = 0 whenever i 2. That is, the extension algebra contains only two terms B ξ = Ext A(M ξ, M ξ ) Ext 0 A (M ξ, M ξ ) = Hom A (M ξ, M ξ ) and using the above decomposition this space is equal to M e (C)... M ek (C) and hence determines the dimension vector α ξ = (e,..., e k ). Ext A (M ξ, M ξ ) which by the decomposition is equal to k i,j= M ej e i (Ext A(S i, S j )) and we will prove that dim C Ext A (S i, S j ) determines the number of arrows (or loops) in Q ξ between the vertices v i and v j. By the theorem above, this quiver-setting must contain enough information to describe filt(s,..., S k ) and hence in particular all n-dimensional representations having as their Jordan-Hölder components the simples S i.

87 Chapter 3 Generic Matrices. The results of this section are essential in the geometric study of noncommutative smooth algebras. If A is an affine C-algebra, say with generators {a,..., a m }, we will study in chapter 4 the variety (actually, a scheme) of all n-dimensional representations rep n A of A. The crucial result we will prove in chapter 4 is that the canonical Cayley-Hamilton algebra of degree n, n, associated to A can be recovered from the natural GL n (C)-structure on rep n A as the ring of all GL n (C)- equivariant polynomial maps from rep n A to M n (C). In this chapter we will study the generic case, that is, when A is the free associative C-algebra C x,..., x m on m noncommuting generators. In this case, rep n C x,..., x m = M n (C)... M n (C) }{{} m as every C-algebra map to M n (C) is determined by the images of the generators x i. The GL n (C)-action on the representation variety, determining isomorphisms of representations, is given by simultaneous conjugation, that is, g.(a,..., A m ) = (ga g,..., ga m g ). In the special case when m =, the Jordan-normal form of an n n matrix provides us with a set-theoretical description of the GL n (C)-orbits. However, as we will see in section, one cannot define a Hausdorff topology on this set or orbits due to the existence of non-closed orbits. Invariant theory provides us with the best continuous approximation to such an orbit-space. We will see that all functions on M n (C) which are constant on conjugacy classes are actually functions in the coefficients of the characteristic polynomial of the matrix. That is, we have an algebraic quotient map M n (C) C n A (σ (A),..., σ n (A)) where σ i (A) is the i-th elementary symmetric function in the eigenvalues of A. A characteristic property of this quotient map is that every fiber contains a unique closed orbit. In trying to extend this to arbitrary m we are faced with the problem that there are no known canonical forms for m-tuples of n n matrices, except for small values of m and n such as (m, n) = (2, 2), in which case a complete description of the orbits is given in section 2. The combinatorial tools which will be developed in part 3 will allow us later to extend such a complete classification (at least in principle) of all orbits for moderate values of n. In this chapter we will prove the important results, due to C. Procesi [24] on the invariant theory of m-tuples of n n matrices under simultaneous conjugation. 87

88 88 CHAPTER 3. GENERIC MATRICES. In section 3 we will determine the ring of invariant polynomials for simultaneous conjugation. The approach is classical in invariant theory. First we determine the multilinear invariant polynomials and then we will use polarization and restitution to find all invariants. It turns out that all invariants are generated by traces of necklaces. Consider a noncommutative word in the variables x i w = x i x i2... x ik determined only up to cyclic permutation of the terms, that is w should really be viewed as a necklace having k beads w Replace each of the beads x i by an n n matrix X i having all its coefficients being indeterminates. That is X i is a generic n n matrix x (i)... x n (i) i = X i =.. x n (i)... x nn (i) Then multiplying these generic matrices along the necklace and taking the trace of the n n matrix obtained, we get an invariant polynomial. We will then use some results on the representation theory of the symmetric groups to bound the length of necklaces necessary to generate the whole algebra of invariants N m n = C[rep n C x,..., x m ] GLn(C). The best bound on this length we will obtain is n 2 +. Further, we will study in section 4 the C-algebra of all GL n (C)-equivariant polynomial maps M n (C)... M n (C) M n (C) which form the trace algebra T m n. We will prove that T m n is generated by the ring of invariants N m n and the generic matrices X i introduced above. In sections 6 and 7 we will then determine all relations holding among the necklace invariants and prove that they are all formal consequences of the Cayley-Hamilton equation holding for n n matrices. In fact, we will show in the next chapter that the trace algebra T m n is the generic object in the category of all Cayley-Hamilton algebras of degree n. We have tried to keep this chapter as self-contained as possible. More details on symmetric groups can be found for example in [?]. 3. Conjugacy classes of matrices From now on we will denote by M n the space of all n n matrices M n (C) and by GL n the general linear group GL n (C). A matrix A M n determines by left multiplication a linear operator on the n-dimensional vectorspace V n = C n of column vectors. If g GL n is the matrix describing the base change from the canonical

89 3.. CONJUGACY CLASSES OF MATRICES 89 basis of V n to a new basis, then the linear operator expressed in this new basis is represented by the matrix gag. For a given matrix A we want to find an suitable basis such that the conjugated matrix gag has a simple form. That is, we consider the linear action of GL n on the n 2 -dimensional vectorspace M n of n n matrices determined by GL n M n M n (g, A) g.a = gag. The orbit O(A) = {gag g GL n } of A under this action is called the conjugacy class of A. We look for a particularly nice representant in a given conjugacy class. The answer to this problem is, of course, given by the Jordan normal form of the matrix. With e ij we denote the matrix whose unique non-zero entry is at entry (i, j). Recall that the group GL n is generated by the following three classes of matrices : the permutation matrices p ij = n + e ij + e ji e ii e jj for all i j, the addition matrices a ij (λ) = n + λe ij for all i j and 0 λ, and the multiplication matrices m i (λ) = n + (λ )e ii for all i and 0 λ. Conjugation by these matrices determine the three types of Jordan moves on n n matrices, where the altered rows and columns are dashed : λ. λ. i j i j +λ. i λ. i j type p i j type a i type m Therefore, it suffices to consider sequences of these moves on a given n n matrix A M n. The characteristic polynomial of A is defined to be the polynomial of degree n in the variable t χ A (t) = det(t n A) C[t]. As C is algebraically closed, χ A (t) decomposes as a product of linear terms e (t λ i ) di i= where the {λ,..., λ e } are called the eigenvalues of the matrix A. Observe that λ i is an eigenvalue of A if and only if there is a non-zero eigenvector v V n = C n with eigenvalue λ i, that is, A.v = λ i v. In particular, the rank r i of the matrix A i = λ i n A satisfies n d i r i < n. A nice inductive procedure using Jordan moves given in [?] gives a proof of the following Jordan-Weierstrass theorem. Theorem 3. Let A M n with characteristic polynomial χ A (t) = e i= (t λ i) di. Then, A determines unique partitions p i = (a i, a i2,..., a imi ) of d i

90 90 CHAPTER 3. GENERIC MATRICES. associated to the eigenvalues λ i of A such that A is conjugated to a unique (up to permutation of the blocks) block-diagonal matrix J (p,...,p e) = B B 2... with m = m +...+m e and exactly one block B l of the form J aij (λ i ) for all i e and j m i where λ i. J aij (λ i ) = λ.. i... M a ij (C) λi B m For example, let us prove uniqueness of the partitions p i of d i corresponding to the eigenvalue λ i of A. Assume A is conjugated to another Jordan block matrix J (q,...,q e), necessarily with partitions q i = (b i,..., b im i ) of d i. To begin, observe that for a Jordan block of size k we have that rk J k (0) l = k l for all l k and if µ 0 then rk J k (µ) l = k for all l. As J (p,...,p e) is conjugated to J (q,...,q e) we have for all λ C and all l rk (λ n J (p,...,p e)) l = rk (λ n J (q,...,q e)) l Now, take λ = λ i then only the Jordan blocks with eigenvalue λ i are important in the calculation and one obtains for the ranks l l n #{j a ij h} respectively n #{j b ij h}. h= Now, for any partition p = (c,..., c u ) and any natural number h we see that the number z = #{j c j h} h= c c 2 c z c z+ c u h is the number of blocks in the h-th row of the dual partition p which is defined to be the partition obtained by interchanging rows and columns in the Young diagram of p. Therefore, the above rank equality implies that p i = qi and hence that p i = q i. As we can repeat this argument for the other eigenvalues we have the required uniqueness. Hence, the Jordan normal form shows that the classification of GL n -orbits in M n consists of two parts : a discrete part choosing

91 3.. CONJUGACY CLASSES OF MATRICES 9 a partition p = (d, d 2,..., d e ) of n, and for each d i, a partition p i = (a i, a i2,..., a imi ) of d i, determining the sizes of the Jordan blocks and a continuous part choosing an e-tuple of distinct complex numbers (λ, λ 2,..., λ e ). fixing the eigenvalues. Moreover, this e-tuple (λ,..., λ e ) is determined only up to permutations of the subgroup of all permutations π in the symmetric group S e such that p i = p π(i) for all i e. Whereas this gives a satisfactory set-theoretical description of the orbits we cannot put an Hausdorff topology on this set due to the existence of non-closed orbits in M n. For example, if n = 2, consider the matrices A = [ ] λ 0 λ and B = [ ] λ 0 0 λ which are in different normal form so correspond to distinct orbits. For any ɛ 0 we have that [ ] [ ] [ ] [ ] ɛ 0 λ ɛ 0 λ ɛ.. = 0 0 λ 0 0 λ belongs to the orbit of A. Hence if ɛ 0, we see that B lies in the closure of O(A). As any matrix in O(A) has trace 2λ, the orbit is contained in the 3-dimensional subspace [ ] λ + x y M2 z λ x In this space, the orbit-closure O(A) is the set of points satisfying x 2 + yz = 0 (the determinant has to be λ 2 ), which is a cone having the origin as its top : The orbit O(B) is the top of the cone and the orbit O(A) is the complement. Still, for general n we can try to find the best separated topological quotient space for the action of GL n on M n. We will prove that this space coincide with the quotient variety determined by the invariant polynomial functions. If two matrices are conjugated A B, then A and B have the same unordered n-tuple of eigenvalues {λ,..., λ n } (occurring with multiplicities). Hence any symmetric function in the λ i will have the same values in A as in B. In particular this is the case for the elementary symmetric functions σ l σ l (λ,..., λ l ) = λ i λ i2... λ il. i <i 2<...<i l Observe that for every A M n with eigenvalues {λ,..., λ n } we have n (t λ j ) = χ A (t) = det(t n A) = t n + j= n ( ) i σ i (A)t n i i=

92 92 CHAPTER 3. GENERIC MATRICES. Developing the determinant det(t n A) we see that each of the coefficients σ i (A) is in fact a polynomial function in the entries of A. A fortiori, σ i (A) is a complex valued continuous function on M n. The above equality also implies that the functions σ i : M n C are constant along orbits. We now construct the continuous map M n π C n sending a matrix A M n to the point (σ (A),..., σ n (A)) in C n. Clearly, if A B then they map to the same point in C n. We claim that π is surjective. Take any point (a,..., a n ) C n and consider the matrix A M n 0 a n 0 a n A = (3.) 0 a 2 a then we will show that π(a) = (a,..., a n ), that is, det(t n A) = t n a t n + a 2 t n ( ) n a n. Indeed, developing the determinant of t n A along the first column we obtain t a n t a n t 0 0 a n- t 0 0 a n t t a n-2. a t t a n-2. a t a 0 0 t a Here, the second determinant is equal to ( ) n a n and by induction on n the first determinant is equal to t.(t n a t n ( ) n a n ), proving the claim. Next, we will determine which n n matrices can be conjugated to a matrix in the canonical form A as above. We call a matrix B M n cyclic if there is a (column) vector v C n such that C n is spanned by the vectors {v, B.v, B 2.v,..., B n.v}. Let g GL n be the basechange transforming the standard basis to the ordered basis (v, B.v, B 2.v, B 3.v,..., ( ) n B n.v). In this new basis, the linear map determined by B (or equivalently, g.b.g ) is equal to the matrix in canonical form 0 b n 0 b n b 2 b where B n.v has coordinates (b n,..., b 2, b ) in the new basis. Conversely, any matrix in this form is a cyclic matrix.

93 3.. CONJUGACY CLASSES OF MATRICES 93 We claim that the set of all cyclic matrices in M n is a dense open subset. To see this take v = (x,..., x n ) τ C n and compute the determinant of the n n matrix v Bv... B n- v This gives a polynomial of total degree n in the x i with all its coefficients polynomial functions c j in the entries b kl of B. Now, B is a cyclic matrix if and only if at least one of these coefficients is non-zero. That is, the set of non-cyclic matrices is exactly the intersection of the finitely many hypersurfaces in the vectorspace M n. V j = {B = (b kl ) k,l M n c j (b, b 2,..., b nn ) = 0} Theorem 3.2 The best continuous approximation to the orbit space is given by the surjection π C n M n mapping a matrix A M n (C) to the n-tuple (σ (A),..., σ n (A)). Let f : M n C be a continuous function which is constant along conjugacy classes. We will show that f factors through π, that is, f is really a continuous function in the σ i (A). Consider the diagram s M n f C C n π... f =f s where s is the section of π (that is, π s = id C n) determined by sending a point (a,..., a n ) to the cyclic matrix in canonical form A as in equation (3.). Clearly, s is continuous, hence so is f = f s. The approximation property follows if we prove that f = f π. By continuity, it suffices to check equality on the dense open set of cyclic matrices in M n. There it is a consequence of the following three facts we have proved before : () : any cyclic matrix lies in the same orbit as one in standard form, (2) : s is a section of π and (3) : f is constant along orbits. Example 3.3 Orbits in M 2. A 2 2 matrix A can be conjugated to an upper triangular matrix with diagonal entries the eigenvalues λ, λ 2 of A. As the trace and determinant of both matrices are equal we have σ (A) = tr(a) and σ 2 (A) = det(a). The best approximation to the orbitspace is therefore given by the surjective map» π M 2 C 2 a b (a + d, ad bc) c d

94 94 CHAPTER 3. GENERIC MATRICES. The matrix A has two equal eigenvalues if and only if the discriminant of the characteristic polynomial t 2 σ (A)t + σ 2 (A) is zero, that is when σ (A) 2 4σ 2 (A) = 0. This condition determines a closed curve C in C 2 where C = {(x, y) C 2 x 2 4y = 0}. C Observe that C is a smooth -dimensional submanifold of C 2. We will describe the fibers (that is, the inverse images of points) of the surjective map π. If p = (x, y) C 2 C, then π (p) consists of precisely one orbit (which is then necessarily closed in M 2 ) namely that of the diagonal matrix» λ 0 where λ 0 λ,2 = x ± p x 2 4y 2 2 If p = (x, y) C then π (p) consists of two orbits, O " # λ and O " # λ 0 0 λ 0 λ where λ = x. We have seen that the second orbit lies in the closure of the first. Observe that 2 the second orbit reduces to one point in M 2 and hence is closed. Hence, also π (p) contains a unique closed orbit. To describe the fibers of π as closed subsets of M 2 it is convenient to write any matrix A as a linear combination» 0 A = u(a) 2 0 2» 0 + v(a) w(a) 2» 0 + z(a) 0 0» Expressed in the coordinate functions u, v, w and z the fibers π (p) of a point p = (x, y) C 2 are the common zeroes of ( u = x v 2 + 4wz = x 2 4y The first equation determines a three dimensional affine subspace of M 2 in which the second equation determines a quadric If p / C this quadric is non-degenerate and thus π (p) is a smooth 2-dimensional submanifold of M 2. If p C, the quadric is a cone with top lying in the point x 2 2. Under the GL 2 -action, the unique singular point of the cone must be clearly fixed giving us the closed orbit of dimension 0 corresponding to the diagonal matrix. The other orbit is the complement of the top and hence is a smooth 2-dimensional (non-closed) submanifold of M 2. The graphs represent the orbit-closures and the dimensions of the orbits.

95 3.. CONJUGACY CLASSES OF MATRICES 95 Example 3.4 Orbits in M 3. π We will describe the fibers of the surjective map M 3 C 3. If a 3 3 matrix has multiple eigenvalues then the discriminant d = (λ λ 2 ) 2 (λ 2 λ 3 ) 2 (λ 3 λ ) 2 is zero. Clearly, d is a symmetric polynomial and hence can be expressed in terms of σ, σ 2 and σ 3. More precisely, d = 4σσ σ σ3 2 σσ σ σ 2 σ 3 The set of points in C 3 where d vanishes is a surface S with singularities. These singularities are the common zeroes of the d for i 3. One computes that these σ i singularities form a twisted cubic curve C in C 3, that is, C = {(3c, 3c 2, c 3 ) c C}. The description of the fibers π (p) for p = (x, y, z) C 3 is as follows. When p / S, then π (p) consists of a unique orbit (which is therefore closed in M 3 ), the conjugacy class of a matrix with paired distinct eigenvalues. If p S C, then π (p) consists of the orbits of 2 A = 4 λ λ 05 and A 2 = 4 λ λ µ 0 0 µ Finally, if p C, then the matrices in the fiber π (p) have a single eigenvalue λ = x and the 3 fiber consists of the orbits of the matrices 2 B = 4 λ λ 5 B 2 = 4 λ λ 05 B 3 = 4 λ λ λ 0 0 λ 0 0 λ We observe that the strata with distinct fiber behavior (that is, C 3 S, S C and C) are all submanifolds of C 3. The dimension of an orbit O(A) in M n is computed as follows. Let C A be the subspace of all matrices in M n commuting with A. Then, the stabilizer subgroup of A is a dense open subset of C A whence the dimension of O(A) is equal to n 2 dim C A. Performing these calculations for the matrices given above, we obtain the following graphs representing orbit-closures and the dimensions of orbits 6 O A 6 O B 6 O A2 4 O B2 4 O B3 0 C 3 S S C Returning to M n, the set of cyclic matrices is a Zariski open subset of M n. For, consider the generic matrix of coordinate functions and generic column vector x... x n v X =.. and v =. x n... x nn v n C

96 96 CHAPTER 3. GENERIC MATRICES. and form the square matrix [ v X.v X 2.v... X n.v ] M n (C[x, x 2,..., x nn, v,..., v n ]) Then its determinant can be written as z l= p l(x ij )q l (v k ) where the q l are polynomials in the v k and the p l polynomials in the x ij. Let A M n be such that at least one of the p l (A) 0, then the polynomial d = l p l(a)q l (v k ) C[v,..., v k ] is non-zero. But then there is a c = (c,..., c n ) C n such that d(c) 0 and hence c τ is a cyclic vector for A. The converse implication is obvious. Theorem 3.5 Let f : M n C is a regular (that is, polynomial) function on M n which is constant along conjugacy classes, then f C[σ (X),..., σ n (X)] Proof. Consider again the diagram s M n f C C n π... f =f s The function f = f s is a regular function on C n whence is a polynomial in the coordinate functions of C m (which are the σ i (X)), so f C[σ (X),..., σ n (X)] C[M n ]. Moreover, f and f are equal on a Zariski open (dense) subset of M n whence they are equal as polynomials in C[M n ]. The ring of polynomial functions on M n which are constant along conjugacy classes can also be viewed as a ring of invariants. The group GL n acts as algebra automorphisms on the polynomial ring C[M n ]. The automorphism φ g determined by g GL n sends the variable x ij to the (i, j)-entry of the matrix g.x.g which is a linear form in C[M n ]. This action is determined by the property that for all g GL n, A A and f C[M n ] we have that φ g (f)(a) = f(g.a.g ) The ring of polynomial invariants is the algebra of polynomials left invariant under this action C[M n ] GLn = {f C[M n ] φ g (f) = f for all g GL n } and hence is the ring of polynomial functions on M n which are constant along orbits. The foregoing theorem determines the ring of polynomials invariants C[M n ] GLn = C[σ (X),..., σ n (X)] We will give an equivalent description of this ring below. Consider the variables λ,..., λ n and consider the polynomial f n (t) = n (t λ i ) = t n + i= n ( ) i σ i t n i i=

97 3.. CONJUGACY CLASSES OF MATRICES 97 then σ i is the i-th elementary symmetric polynomial in the λ j. We know that these polynomials are algebraically independent and generate the ring of symmetric polynomials in the λ j, that is, C[σ,..., σ n ] = C[λ,..., λ n ] Sn where S n is the symmetric group on n letters acting by automorphisms on the polynomial ring C[λ,..., λ n ] via π(λ i ) = λ π(i) and the algebra of polynomials which are fixed under these automorphisms are precisely the symmetric polynomials in the λ j. Consider the symmetric Newton functions s i = λ i +...+λ i n, then we claim that this is another generating set of symmetric polynomials, that is, C[σ,..., σ n ] = C[s,..., s n ]. To prove this it suffices to express each σ i as a polynomial in the s j. More precisely, we claim that the following identities hold for all j n s j σ s j + σ 2 s j ( ) j σ j s + ( ) j σ j.j = 0 (3.2) For j = n this identity holds because we have 0 = n f n (λ i ) = s n + i= n ( ) i σ i s n i if we take s 0 = n. Assume now j < n then the left hand side of equation 3.2 is a symmetric function in the λ i of degree j and is therefore a polynomial p(σ,..., σ j ) in the first j elementary symmetric polynomials. Let φ be the algebra epimorphism C[λ,..., λ n ] φ C[λ,..., λ j ] defined by mapping λ j+,..., λ j to zero. Clearly, φ(σ i ) is the i-th elementary symmetric polynomial in {λ,..., λ j } and φ(s i ) = λ i λ i j. Repeating the above j = n argument (replacing n by j) we have 0 = j f j (λ i ) = φ(s j ) + i= i= j ( ) i φ(σ i )φ(s n i ) (this time with s 0 = j). But then, p(φ(σ ),..., φ(σ j )) = 0 and as the φ(σ k ) for k j are algebraically independent we must have that p is the zero polynomial finishing the proof of the claimed identity. If λ,..., λ n are the eigenvalues of an n n matrix A, then A can be conjugated to an upper triangular matrix B with diagonal entries (λ,..., λ ). Hence, the trace tr(a) = tr(b) = λ +...+λ n = s. In general, A i can be conjugated to B i which is an upper triangular matrix with diagonal entries (λ i,..., λ i n) and hence the traces of A i and B i are equal to λ i λ i n = s i. Concluding, we have Theorem 3.6 Consider the action of conjugation by GL n on M n. Let X be the generic matrix of coordinate functions on M n x... x nn X =.. x n... x nn Then, the ring of polynomial invariants is generated by the traces of powers of X, that is, C[M n ] GLn = C[tr(X), tr(x 2 ),..., tr(x n )] i=

98 98 CHAPTER 3. GENERIC MATRICES. Proof. The result follows from theorem 3.5 and the fact that C[σ (X),..., σ n (X)] = C[tr(X),..., tr(x n )] 3.2 Simultaneous conjugacy classes. For applications to noncommutative algebras it is crucial to extend what we have done for conjugacy classes of matrices to simultaneous conjugacy classes of m-tuples of matrices. Consider the mn 2 -dimensional complex vectorspace M m n = M n... M n }{{} m of m-tuples (A,..., A m ) of n n-matrices A i M n. On this space we let the group GL n act by simultaneous conjugation, that is g.(a,..., A m ) = (g.a.g,..., g.a m.g ) for all g GL n and all m-tuples (A,..., A m ). Unfortunately, there is no substitute for the Jordan normalform result in this more general setting. Still, for small m and n one can work out the GL n -orbits by ad hoc methods. Example 3.7 Orbits in M2 2 = M 2 M 2. We can try to mimic the geometric approach to the conjugacy class problem, that is, we will try to approximate the orbitspace via polynomial functions on M2 2 which are constant along orbits. For (A, B) M2 2 = M 2 M 2 clearly the polynomial functions we have encountered before tr(a), det(a) and tr(b), det(b) are constant along orbits. However, there are more : for example tr(ab). Later, we will show that these five functions generate all polynomials functions which are constant along orbits. Here, we will show that the map M2 2 = M π 2 M 2 C 5 defined by (A, B) (tr(a), det(a), tr(b), det(b), tr(ab)) is surjective such that each fiber contains precisely one closed orbit. In the next chapter, we will see that this property characterizes the best polynomial approximation to the (non-existent) orbit space. First, we will show surjectivity of π, that is, for every (x,..., x 5 ) C 5 we will construct a couple of 2 2 matrices (A, B) (or rather its orbit) such that π(a, B) = (x,..., x 5 ). Consider the open set where x 2 4x 2. We have seen that this property characterizes those A M 2 such that A has distinct eigenvalues and hence diagonalizable. Hence, we can take a representative of the orbit O(A, B) to be a couple (» λ 0 0 µ,» c c 2 c 3 c 4 with λ µ. We need a solution to the set of equations 8 >< x 3 = c + c 4 x 4 = c c 4 c 2 c 3 >: x 5 = λc + µc 4 Because λ µ the first and last equation uniquely determine c, c 4 and substitution in the second gives us c 2 c 3. Analogously, points of C 5 lying in the open set x 2 3 x 4 lie in the image of π. Finally, for a point in the complement of these open sets, that is when x 2 = x 2 and x 2 3 = 4x 4 we can consider a couple (A, B) (» λ 0 λ,» µ 0 c µ where λ = 2 x and µ = 2 x 3. Observe that the remaining equation x 5 = tr(ab) = 2λµ + c has a solution in c. Now, we will describe the fibers of π. Assume (A, B) is such that A and B have a common eigenvector v. Simultaneous conjugation with a g GL n expressing a basechange from the ) )

99 3.2. SIMULTANEOUS CONJUGACY CLASSES. 99 standard basis to {v, w} for some w shows that the orbit O(A, B) contains a couple of uppertriangular matrices. We want to describe the image of these matrices under π. Take an upper triangular representative in O(A, B) (» a a 2 0 a 3,» b b 2 0 b 3 with π-image (x,..., x 5 ). The coordinates x, x 2 determine the eigenvalues a, a 3 of A only as an unordered set (similarly, x 3, x 4 only determine the set of eigenvalues {b, b 3 } of B). Hence, tr(ab) is one of the following two expressions and therefore satisfies the equation a b + a 3 b 3 or a b 3 + a 3 b (tr(ab) a b a 3 b 3 )(tr(ab) a b 3 a 3 b ) = 0. Recall that x = a + a 3, x 2 = a a 3, x 3 = b + b 3, x 4 = b b 3 and x 5 = tr(ab) we can express this equation as x 2 5 x x 3 x 5 + x 2 x 4 + x 2 3 x 2 4x 2 x 4 = 0. This determines an hypersurface H C 5. If we view the left-hand side as a polynomial f in the coordinate functions of C 5 we see that H is a four dimensional subset of C 5 with singularities the common zeroes of the partial derivatives f x i for i 5 These singularities for the 2-dimensional submanifold S of points of the form (2a, a 2, 2b, b 2, 2ab). We now claim that the smooth submanifolds C 5 H, H S and S of C 5 describe the different types of fiber behavior. In chapter 6 we will see that the subsets of points with different fiber behavior (actually, of different representation type) are manifolds for m-tuples of n n matrices. If p / H we claim that π (p) is a unique orbit, which is therefore closed in M2 2. Let (A, B) π and assume first that x 2 4x 2 then there is a representative in O(A, B) of the form»» λ 0 c c (, 2 ) 0 µ c 3 c 4 with λ µ. Moreover, c 2 c 3 0 (for otherwise A and B would have a common eigenvector whence p H) hence we may assume that c 2 = (eventually after simultaneous conjugation with a suitable diagonal matrix diag(t, t )). The value of λ, µ is determined by x, x 2. Moreover, c, c 3, c 4 are also completely determined by the system of equations 8 >< x 3 = c + c 4 x 4 = c c 4 c 3 >: x 5 = λc + µc 4 and hence the point p = (x,..., x 5 ) completely determines the orbit O(A, B). Remains to consider the case when x 2 = 4x 2 (that is, when A has a single eigenvalue). Consider the couple (ua+vb, B) for u, v C. To begin, ua+vb and B do not have a common eigenvalue. Moreover, p = π(a, B) determines π(ua + vb, B) as 8 >< tr(ua + vb) = utr(a) + vtr(b) det(ua + vb) = u >: 2 det(a) + v 2 det(b) + uv(tr(a)tr(b) tr(ab)) tr((ua + vb)b) = utr(ab) + v(tr(b) 2 2det(B)) Assume that for all u, v C we have the equality tr(ua + vb) 2 = 4det(uA + vb) then comparing coefficients of this equation expressed as a polynomial in u and v we obtain the conditions x 2 = 4x 2, x 2 3 = 4x 4 and 2x 5 = x x 3 whence p S H, a contradiction. So, fix u, v such that ua + vb has distinct eigenvalues. By the above argument O(uA + vb, B) is the unique orbit lying over π(ua + vb, B), but then O(A, B) must be the unique orbit lying over p. Let p H S and (A, B) π (p), then A and B are simultaneous upper triangularizable, with eigenvalues a, a 2 respectively b, b 2. Either a a 2 or b b 2 for otherwise p S. Assume a a 2, then there is a representative in the orbit O(A, B) of the form (» ai 0 0 a j,» bk b ) 0 b l for {i, j} = {, 2} = {k, l}. If b 0 we can conjugate with a suitable diagonal matrix to get b = hence we get at most 9 possible orbits. Checking all possibilities we see that only three of them are distinct, those corresponding to the couples» a 0 ( 0 a 2,» b ) ( 0 b 2» a 0, 0 a 2 ).»»» b 0 a2 0 b ) (, ) 0 b 2 0 a 0 b 2

100 00 CHAPTER 3. GENERIC MATRICES. Clearly, the first and last orbit have the middle one lying in its closure. Observe that the case assuming that b b 2 is handled similarly. Hence, if p H S then π (p) consists of three orbits, two of dimension three whose closures intersect in a (closed) orbit of dimension two. Finally, consider the case when p S and (A, B) π (p). Then, both A and B have a single eigenvalue and the orbit O(A, B) has a representative of the form»» a x b y (, ) 0 a 0 b for certain x, y C. If either x or y are non-zero, then the subgroup of GL 2 fixing this matrix consists of the matrices of the form»» c u v Stab = { u C 0 c 0 u, v C} but these matrices also fix the second component. Therefore, if either x or y is nonzero, the orbit is fully determined by [x : y] P. That is, for p S, the fiber π (p) consists of an infinite family of orbits of dimension 2 parameterized by the points of the projective line P together with the orbit of (» a 0 0 a,» b 0 0 b which consists of one point (hence is closed in M2 2 ) and lies in the closure of each of the 2- dimensional orbits. Concluding, we see that each fiber π (p) contains a unique closed orbit (that of minimal dimension). The orbitclosure and dimension diagrams have the following shapes ) P C 5 H H S S 3.3 Matrix invariants and necklaces In this section we will determine the ring of all polynomial maps M m n = M n... M n }{{} m f C which are constant along orbits under the action of GL n on Mn m by simultaneous conjugation. The strategy we will use is classical in invariant theory. First, we will determine the multilinear maps which are constant along orbits, equivalently, the linear maps M m n = M n... M n }{{} m C which are constant along GL n -orbits where GL n acts by the diagonal action, that is, g.(a... A m ) = ga g... ga m g.

101 3.3. MATRIX INVARIANTS AND NECKLACES 0 Afterwards, we will be able to obtain from them all polynomial invariant maps by using polarization and restitution operations. First, we will translate our problem into one studied in classical invariant theory of GL n. Let V n C n be the n-dimensional vectorspace of column vectors on which GL n acts naturally by left multiplication C ν C ν 2 V n =. C with action g. In order to define an action on the dual space Vn = Hom(V n, C) C n of covectors (or, row vectors) we have to use the contragradient action. ν n V n = [ C C... C ] with action [ φ φ 2... φ n ].g Observe, that we have an evaluation map Vn V n scalar product f(v) for all f Vn and v V n ν [ ] ν 2 φ φ 2... φ n.. = φ ν + φ 2 ν φ n ν n ν n C which is given by the which is invariant under the diagonal action of GL n on Vn V n. Further, we have the natural identification C M n = V n Vn C =. [ C C... C ]. C Under this identification, a pure tensor v f corresponds to the rank one matrix or rank one endomorphism of V n defined by v f : V n V n with w f(w)v and observe that the rank one matrices span M n. The diagonal action of GL n on V n Vn is then determined by its action on the pure tensors where it is equal to ν g. ν 2 [ ] φ φ 2... φ n.g ν n and therefore coincides with the action of conjugation on M n. Now, let us consider the identification (Vn m Vn m ) End(Vn m ) obtained from the nondegenerate pairing given by the formula End(Vn m ) (Vn m Vn m ) C λ, f... f m v... v m = f... f m (λ(v... v m ))

102 02 CHAPTER 3. GENERIC MATRICES. GL n acts diagonally on Vn m and hence again by conjugation on End(Vn m ) after embedding GL n GL(V m n ) = GL mn. Thus, the above identifications are isomorphism as vectorspaces with GL n -action. But then, the space of GL n -invariant linear maps Vn m Vn m C can be identified with the space End GLn (Vn m ) of GL n -linear endomorphisms of. We will now give a different presentation of this vectorspace relating it to V m n the symmetric group. Apart from the diagonal action of GL n on Vn m given by g.(v... v m ) = g.v... g.v m we have an action of the symmetric group S m on m letters on V m n σ.(v... v m ) = v σ()... v σ(m) given by These two actions commute with each other and give embeddings of GL n and S m in End(V m n ). GL n GL(V m n ) End(V m n ) S m The subspace of V m n spanned by the image of GL n will be denoted by GL n. Similarly, with S m we denote the subspace spanned by the image of S m. Theorem 3.8 With notations as above we have :. GL n = End Sm (V m n ) 2. S m = End GLn (V m n ) Proof. () : Under the identification End(Vn m ) = End(V n ) m an element g GL n is mapped to the symmetric tensor g... g. On the other hand, the image of End Sm (Vn m ) in End(V n ) m is the subspace of all symmetric tensors in End(V ) m. We can give a basis of this subspace as follows. Let {e,..., e n 2} be a basis of End(V n ), then the vectors e i... e im form a basis of End(V n ) m which is stable under the S m -action. Further, any S m -orbit contains a unique representative of the form e h... e h n 2 n 2 with h h n 2 = m. If we denote by r(h,..., h n 2) the sum of all elements in the corresponding S m -orbit then these vectors are a basis of the symmetric tensors in End(V n ) m. The claim follows if we can show that every linear map λ on the symmetric tensors which is zero on all g... g with g GL n is the zero map. Write e = x i e i, then λ(e... e) = x h... xh n 2 n 2 λ(r(h,..., h n 2))

103 3.3. MATRIX INVARIANTS AND NECKLACES 03 is a polynomial function on End(V n ). As GL n is a Zariski open subset of End(V ) on which by assumption this polynomial vanishes, it must be the zero polynomial. Therefore, λ(r(h,..., h n 2)) = 0 for all (h,..., h n 2) finishing the proof. (2) : Recall that the groupalgebra CS m of S m is a semisimple algebra. Any epimorphic image of a semisimple algebra is semisimple. Therefore, S m is a semisimple subalgebra of the matrixalgebra End(Vn m ) M nm. By the double centralizer theorem (see for example [23]), it is therefore equal to the centralizer of End Sm (Vm m ). By the first part, it is the centralizer of GL n in End(Vn m ) and therefore equal to End GLn (Vn m ). Because End GLn (Vn m ) = S m, every GL n -endomorphism of Vn m can be written as a linear combination of the morphisms λ σ describing the action of σ S m on Vn m. Our next job is to trace back these morphisms λ σ through the canonical identifications until we can express them in terms of matrices. To start let us compute the linear invariant µ σ : V m n V m n C corresponding to λ σ under the identification (Vn m Vn m ) End(Vn m ). By the identification we know that µ σ (f... f m v... v m ) is equal to λ σ, f... f m v... v m = f... f m (v σ()... v σ(m) ) = i f i(v σ i)) ( That is, we have proved Proposition 3.9 Any multilinear GL n -invariant map γ : V m n is a linear combination of the invariants V m n C µ σ (f... f m v... v m ) = i f i (v σ(i) ) for σ S m. Using the identification M n (C) = V n V n (V n V ) m n = V m n corresponds to a multilinear GL n -invariant map a multilinear GL n -invariant map V m n C M n (C)... M n (C) C We will now give a description of the generating maps µ σ in terms of matrices. Under the identification, matrix multiplication is induced by composition on rank one endomorphisms and here the rule is given by v f.v f = f(v )v f ν.. [ ν ] φ... φ n... [ ν ] φ... φ n =.. f(v ) [ φ... φ n]. ν n ν n Moreover, the trace map on M n is induced by that on rank one endomorphisms where it is given by the rule tr(v f) = f(v) ν n

104 04 CHAPTER 3. GENERIC MATRICES. ν tr(. [ ν φ... ν φ n ] φ... φ n ) = tr(..... ) = ν i φ i = f(v) ν n ν n φ... ν n φ i n With these rules we can now give a matrix-interpretation of the GL n -invariant maps µ σ. Proposition 3.0 Let σ = (i i 2... i α )(j j 2... j β )... (z z 2... z ζ ) be a decomposition of σ S m into cycles (including those of length one). Then, under the above identification we have µ σ (A... A m ) = tr(a i A i2... A iα )tr(a j A j2... A jβ )... tr(a z A z2... A zζ ) Proof. Both sides are multilinear hence it suffices to verify the equality for rank one matrices. Write A i = v i f i, then we have that µ σ (A... A m ) = µ σ (v... v m f... f m ) = i f i(v σ(i) ) Consider the subproduct Now, look at the matrixproduct f i (v i2 )f i2 (v i3 )... f iα (v iα ) = S which is by the product rule equal to v i f i.v i2 f i2.....v iα f iα f i (v i2 )f i2 (v i3 )... f iα (v iα )v i f iα Hence, by the trace rule we have that α tr(a i A i2... A iα ) = f ij (v σ(ij)) = S j= Having found a description of the multilinear invariant polynomial maps M m n = M n... M n }{{} m C we will now describe all polynomial maps which are constant along orbits by polarization. The coordinate algebra C[Mn m ] is the polynomial ring in mn 2 variables x ij (k) where k m and i, j n. Consider the m generic n n matrices k = X k = x (k)... x n (k).. x n (k)... x nn (k) M n (C[M m n ]). The action of GL n on polynomial maps f C[M m n ] is fully determined by the action on the coordinate functions x ij (k). As in the case of one n n matrix we see that this action is given by g.x ij (k) = (g.x k.g) ij.

105 3.3. MATRIX INVARIANTS AND NECKLACES 05 We see that this action preserves the subspaces spanned by the entries of any of the generic matrices. Hence, we can define a gradation on C[M m n ] by deg(x ij (k)) = (0,..., 0,, 0,..., 0) (with at place k) and decompose C[M m n ] = (d,...,d m) N m C[M m n ] (d,...,d m) where C[M m n ] (d,...,d m) is the subspace of all multihomogeneous forms f in the x ij (k) of degree (d,..., d m ), that is, in each monomial term of f there are exactly d k factors coming from the entries of the generic matrix X k for all k m. The action of GL n stabilizes each of these subspaces, that is, if f C[M m n ] (d,...,d m) then g.f C[M m n ] (d,...,d m) for all g GL n. In particular, if f determines a polynomial map on Mn m which is constant along orbits, that is, if f belongs to the ring of invariants C[Mn m ] GLn then each of its multihomogeneous components is also an invariant and therefore it suffices to determine all multihomogeneous invariants. Let f C[Mn m ] (d,...,d m) and take for each k m d k new variables t (k),..., t dk (k). Expand f(t ()A () t d A d (),..., t (m)a (m) t dm (m)a dm (m)) as a polynomial in the variables t i (k), then we get an expression t () s()... t s d () d... t (m) s(m)... t dm (m) s dm (m). f (s(),...,s d (),...,s (m),...,s dm (m))(a (),..., A d (),..., A (m),..., A dm (m)) such that for all k m we have d k i= s i(k) = d k. Moreover, each of the f (s(),...,s d (),...,s (m),...,s dm (m)) is a multi-homogeneous polynomial function on M n... M }{{ n M } n... M n... M }{{} n... M }{{ n } d d 2 d m of multi-degree (s (),..., s d (),..., s (m),..., s dm (m)). Observe that if f is an invariant polynomial function on M m n, then each of these multi homogeneous functions is an invariant polynomial function on M D n where D = d d m. In particular, we consider the multi-linear function f,..., : M D n = M d n... M dm n C which we call the polarization of the polynomial f and denote with P ol(f). Observe that P ol(f) in symmetric in each of the entries belonging to a block M d k n for every k m. If f is invariant under GL n, then so is the multilinear function P ol(f) and we know the form of all such functions by the results given before (replacing Mn m by Mn D ). Finally, we want to recover f back from its polarization. We claim to have the equality P ol(f)(a,..., A }{{},..., A m,..., A m ) = d }{{}!... d m!f(a,..., A m ) d d m and hence we recover f. This process is called restitution. The claim follows from the observation that f(t ()A t d ()A,..., t (m)a m t dm (m)a m ) = f((t () t d ())A,..., (t (m) t dm (m))a m ) = (t () t d ()) d... (t (m) t dm (m)) dm f(a,..., A m )

106 06 CHAPTER 3. GENERIC MATRICES. and the definition of P ol(f). Hence we have proved that any multi-homogeneous invariant polynomial function f on Mn m of multidegree (d,..., d m ) can be obtained by restitution of a multilinear invariant function P ol(f) : M D n = M d n... M dm n C If we combine this fact with our description of all multilinear invariant functions on M n... M n we finally obtain : Theorem 3. Any polynomial function Mn m f C which is constant along orbits under the action of GL n by simultaneous conjugation is a polynomial in the invariants tr(x i... X il ) where X i... X il run over all possible noncommutative polynomials in the generic matrices {X,..., X m }. We will call the algebra C[Mn m ] generated by these invariants the necklace algebra N m n = C[Mn m ] GLn. The terminology is justified by the observation that the generators tr(x i X i2... X il ) are only determined up to cyclic permutation of the generic matrices X j. That is, the generators are determined by necklace words w such as i 5 i 6 i 4 i 7 i 3 w i 2 i 8 i 9 i where each bead corresponds to a generic matrix i = X i. They are multiplied cyclicly to obtain an n n matrix with coefficients in M n (C[M m n ]). The trace of this matrix is called tr(w) and the result asserts that these elements generate the ring of polynomial invariants. 3.4 The trace algebra. In this section we will prove a bound on the length of the necklace words w necessary for the tr(w) to generate N m n. In the last section, after we have determined the relations between these necklaces tr(w), we will be able to improve this bound. First, we will characterize all GL n -equivariant maps from Mn m to M n, that is all polynomial maps Mn m f M n such that for all g GL n the diagram below is commutative M m n f M n g.g M m n f Mn g.g

107 3.4. THE TRACE ALGEBRA. 07 With pointwise addition and multiplication in the target algebra M n, these polynomial maps form a noncommutative algebra T m n called the trace algebra. Obviously, the trace algebra is a subalgebra of the algebra of all polynomial maps from Mn m to M n, that is, T m n M n (C[Mn m ]) Clearly, using the diagonal embedding of C in M n any invariant polynomial on M m n determines a GL n -equivariant map. Equivalently, using the diagonal embedding of C[M m n ] in M n (C[M m n ]) we can embed the necklace algebra N m n = C[M m n ] GLn T m n Another source of GL n -equivariant maps are the coordinate maps X i : M m n = M n... M m n Mn (A,..., A m ) A i Observe that the coordinate map X i is represented by the generic matrix i = X i in M n (C[M m n ]). Proposition 3.2 As an algebra over the necklace algebra N m n, the trace algebra T m n is generated by the elements {X,..., X m }. Proof. Consider a GL n -equivariant map Mn m polynomial map f M n and associate to it the M m+ n = M m n M n tr(fx m+) C defined by sending (A,..., A m, A m+ ) to tr(f(a,..., A m ).A m+ ). For all g GL n we have that f(g.a.g,..., g.a m.g ) is equal to g.f(a,..., A m ).g and hence tr(f(g.a.g,..., g.a m.g ).g.a m+.g ) = tr(g.f(a,..., A m ).g.g.a m+.g ) = tr(g.f(a,..., A m ).A m+.g ) = tr(f(a,..., A m ).A m+ ) so tr(fx m+ ) is an invariant polynomial function on M m+ n which is linear in X m+. By theorem 3. we can therefore write tr(fx m+ ) = g i...i }{{} l tr(x i... X il X m+ ) N m n Here, we used the necklace property allowing to permute cyclicly the trace terms in which X m+ occurs such that X m+ occurs as the last factor. But then, tr(fx m+ ) = tr(gx m+ ) where g = g i...i l X i... X il. Finally, using the nondegeneracy of the trace map on M n (that is, if A, B M n such that tr(ac) = tr(bc) for all C M n, then A = B) it follows that f = g. If we give each of the generic matrices X i degree one, we see that the trace algebra T m n is a connected positively graded algebra T m n = T 0 T T 2... with T 0 = C. Our aim is to bound the length of the monomials in the X i necessary to generate T m n as a module over the necklace algebra N m n. Before we can do this we need to make a small detour in one of the more exotic realms of noncommutative algebra : the Nagata-Higman problem.

108 08 CHAPTER 3. GENERIC MATRICES. Theorem 3.3 (Nagata-Higman) Let R be an associative algebra without a unit element. Assume there is a fixed natural number n such that x n = 0 for all x R. Then, R 2n = 0, that is x.x x 2 n = 0 for all x j R. Proof. We use induction on n, the case n = being obvious. Consider for all x, y R f(x, y) = yx n + xyx n 2 + x 2 yx n x n 2 yx + x n y. Because for all c C we must have that 0 = (y + cx) n = x n c n + f(x, y)c n y n it follows that all the coefficients of the c i with i < n must be zero, in particular f(x, y) = 0. But then we have for all x, y, z R that 0 = f(x, z)y n + f(x, zy)y n 2 + f(x, zy 2 )y n f(x, zy n ) = nx n zy n + zf(y, x n ) + xzf(y, x n 2 ) + x 2 zf(y, x n 3 ) x n 2 zf(y, x) and therefore x n zy n = 0. Let I R be the twosided ideal of R generated by all elements x n, then we have that I.R.I = 0. In the quotient algebra R = R/I every element x satisfies x n = 0. By induction we may assume that R 2n = 0, or equivalently that R 2 n is contained in I. But then, R 2n = R 2(2n )+ = R 2n.R.R 2n I.R.I = 0 finishing the proof. Proposition 3.4 The trace algebra T m n is spanned as a module over the necklace algebra N m n by all monomials in the generic matrices of degree l 2 n. X i X i2... X il Proof. By the diagonal embedding of N m n in M n (C[Mn m ]) it is clear that N m n commutes with any of the X i. Let T + and N + be the strict positive degrees of T m n and N m n and form the graded associative algebra (without unit element) R = T + /N +.T + Observe that any element t T + satisfies an equation of the form t n + c t n + c 2 t n c n = 0 with all of the c i N +. Indeed we have seen that all the coefficients of the characteristic polynomial of a matrix can be expressed as polynomials in the traces of powers of the matrix. But then, for any x R we have that x n = 0. By the Nagata-Higman theorem we know that R 2n = (R ) 2n = 0. Let T be the graded N m n -submodule of T m n spanned by all monomials in the generic matrices X i of degree at most 2 n, then the above can be reformulated as T m n = T + N +.T m n.

109 3.4. THE TRACE ALGEBRA. 09 We claim that T n m = T. Assume not, then there is a homogeneous t T m n of minimal degree d not contained in T but still we have a description t = t + c.t c s.t s with t and all c i, t i homogeneous elements. As deg(t i ) < d, t i T for all i but then is t T a contradiction. Finally we are in a position to bound the length of the necklaces generating N m n as an algebra. Theorem 3.5 The necklace algebra N m n is generated by all necklaces tr(w) where w is a necklace word w of length l 2 n where each of the beads is a generic matrix i = X i. Proof. Let T be the C-subalgebra of T m n generated by the generic matrices X i. Then, tr(t +) generates the ideal N +. Let S be the set of all monomials in the X i of degree at most 2 n. By the foregoing proposition we know that T N m n.s. The trace map tr : T m n N m n is N m n -linear and therefore, because T + T.(CX CX m ) we have tr(t +) tr(n m n.s.(cx CX m )) N m n.tr(s ) where S is the set of monomials in the X i of degree at most 2 n. If N is the C- subalgebra of N m n generated by all tr(s ), then we have tr(t +) N m n.n +. But then, we have N + = N m n tr(t + ) N m n N + and thus N m n = N + N m n N + from which it follows that N m n = N by a similar argument as in the foregoing proof. Example 3.6 The algebras N 2 2 and T 2 2. When working with 2 2 matrices, the following identities are often helpful 0 = A 2 tr(a)a + det(a) A.B + B.A = tr(ab) tr(a)tr(b) + tr(a)b + tr(b)a for all A, B M 2. Let N be the subalgebra of N 2 2 generated by tr(x ), tr(x 2 ), det(x ), det(x 2 ) and tr(x X 2 ). Using the two formulas above and N 2 2 -linearity of the trace on T2 2 we see that the trace of any monomial in X and X 2 of degree d 3 can be expressed in elements of N and traces of monomials of degree d. Hence, we have N 2 2 = C[tr(X ), tr(x 2 ), det(x ), det(x 2 ), tr(x X 2 )].

110 0 CHAPTER 3. GENERIC MATRICES. Observe that there can be no algebraic relations between these generators as we have seen that the induced map π : M2 2 C 5 is surjective. Another consequence of the above identities is that over N 2 2 any monomial in the X, X 2 of degree d 3 can be expressed as a linear combination of, X, X 2 and X X 2 and so these elements generate T 2 2 as a N2 2-module. In fact, they are a basis of T 2 2 over N2 2. Assume otherwise, there would be a relation say X X 2 = αi 2 + βx + γx 2 with α, β, γ C(tr(X ), tr(x 2 ), det(x ), det(x 2 ), tr(x X 2 )). Then this relation has to hold for all matrix couples (A, B) M2 2 and we obtain a contradiction if we take the couple»»» A = B = whence AB = Concluding, we have the following description of N 2 2 and T2 2 as a subalgebra of C[M 2 2] respectively M 2 (C[M2 2]) ( N 2 2 = C[tr(X ), tr(x 2 ), det(x ), det(x 2 ), tr(x X 2 )] T 2 2 = N2 2.I 2 N 2 2.X N 2 2.X 2 N 2 2.X X 2 Observe that we might have taken the generators tr(x 2 i ) rather than det(x i) because det(x i ) = 2 (tr(x i) 2 tr(x i ) 2 ) as follows from taking the trace of characteristic polynomial of X i. 3.5 The symmetric group. Let S d be the symmetric group of all permutations on d letters. The group algebra C S d is a semisimple algebra. In particular, any simple S d -representation is isomorphic to a minimal left ideal of C S d which is generated by an idempotent. We will now determine these idempotents. To start, conjugacy classes in S d correspond naturally to partitions λ = (λ,..., λ k ) of d, that is, decompositions in natural numbers d = λ λ k with λ λ 2... λ k The correspondence associates to a partition λ = (λ,..., λ k ) the conjugacy class of a permutation consisting of disjoint cycles of lengths λ,..., λ k. It is traditional to assign to a partition λ = (λ,..., λ k ) a Young diagram with λ i boxes in the i-th row, the rows of boxes lined up to the left. The dual partition λ = (λ,..., λ r) to λ is defined by interchanging rows and columns in the Young diagram of λ. For example, to the partition λ = (3, 2,, ) of 7 we assign the Young diagram λ = λ = with dual partition λ = (4, 2, ). A Young tableau is a numbering of the boxes of a Young diagram by the integers {, 2,..., d}. For example, two distinct Young tableaux of type λ are Now, fix a Young tableau T of type λ and define subgroups of S d by P λ = {σ S d σ preserves each row }

111 3.5. THE SYMMETRIC GROUP. Q λ = {σ S d σ preserves each column } For example, for the second Young tableaux given above we have that { P λ = S {,3,5} S {2,4} {(6)} {(7)} Q λ = S {,2,6,7} S {3,4} {(5)} Observe that different Young tableaux for the same λ define different subgroups and different elements to be defined below. Still, the simple representations we will construct from them turn out to be isomorphic. Using these subgroups, we define the following elements in the groupalgebra CS d a λ = e σ, b λ = sgn(σ)e σ and c λ = a λ.b σ σ P λ σ Q λ The element c λ is called a Young symmetrizer. The next result gives an explicit oneto-one correspondence between the simple representations of CS d and the conjugacy classes in S d (or, equivalently, Young diagrams). Theorem 3.7 For every partition λ of d the left ideal CS d.c λ = V λ is a simple S d -representations and, conversely, any simple S d -representation is isomorphic to V λ for a unique partition λ. Proof. Observe that P λ Q λ = {e} (any permutation preserving rows as well as columns preserves all boxes) and so any element of S d can be written in at most one way as a product p.q with p P λ and q Q λ. In particular, the Young symmetrizer can be written as c λ = ±e σ with σ = p.q for unique p and q and the coefficient ± = sgn(q). From this it follows that for all p P λ and q Q λ we have p.a λ = a λ.p = a λ, sgn(q)q.b λ = b λ.sgn(q)q = b λ, p.c λ.sgn(q)q = c λ Moreover, we claim that c λ is the unique element in CS d (up to a scalar factor) satisfying the last property. This requires a few preparations. Assume σ / P λ.q λ and consider the tableaux T = σt, that is, replacing the label i of each box in T by σ(i). We claim that there are two distinct numbers which belong to the same row in T and to the same column in T. If this were not the case, then all the distinct numbers in the first row of T appear in different columns of T. But then we can find an element q in the subgroup σ.q λ.σ preserving the columns of T to take all these elements to the first row of T. But then, there is an element p T λ such that p T and q T have the same first row. We can proceed to the second row and so on and obtain elements p P λ and q σ.q λ, σ such that the tableaux pt and q T are equal. Hence, pt = q σt entailing that p = q σ. Further, q = σ.q.σ but then p = q σ = σq whence σ = p.q P λ.q λ, a contradiction. Therefore, to σ / P λ.q λ we can assign a transposition τ = (ij) (replacing the two distinct numbers belonging to the same row in T and to the same column in T ) for which p = τ P λ and q = σ.τ.σ Q λ. After these preliminaries, assume that c = a σ e σ is an element such that p.c.sgn(q)q = c for all p P λ, q Q λ We claim that a σ = 0 whenever σ / P λ.q λ. For take the transposition τ found above and p = τ, q = σ.τ.σ, then p.σ.q = τ.σ.σ.τ.σ = σ. However, the coefficient of σ in c is a σ and that of p.c.q is a σ proving the claim. That is, c = a pq e p.q p,q

112 2 CHAPTER 3. GENERIC MATRICES. but then by the property of c we must have that a pq = sgn(q)a e whence c = a e c λ finishing the proof of the claimed uniqueness of the element c λ. As a consequence we have for all elements x CS d that c λ.x, c λ = α x c λ for some scalar α x C and in particular that c 2 λ = n λc λ, for, p.(c λ.x.c λ ).sgn(q)q = p.a λ.b λ.x.a λ.b λ.sgn(q)q = a λ.b λ.x.a λ.b λ = c λ.x.c λ and the statement follows from the uniqueness result for c λ. Define V λ = CS d.c λ then we have c λ.v λ Cc λ. We claim that V λ is a simple S d - representation. Let W V λ be a simple subrepresentation, then being a left ideal of CS d we can write W = CS d.x with x 2 = x (note that W is a direct summand). Assume that c λ.w = 0, then W.W CS d.c λ.w = 0 implying that x = 0 whence W = 0, a contradiction. Hence, c λ.w = Cc λ W, but then V λ = CS d.c λ W whencev λ = W is simple. Remains to show that for different partitions, the corresponding simple representations cannot be isomorphic. We put a lexicographic ordering on the partitions by the rule that λ > µ if the first nonvanishing λ i µ i is positive We claim that if λ > µ then a λ.cs d.b µ = 0. It suffices to check that a λ.σ.b µ = 0 for σ S d. As σ.b µ.σ is the b-element constructed from the tableau b.t where T is the tableaux fixed for µ, it is sufficient to check that a λ.b µ = 0. As λ > µ there are distinct numbers i and j belonging to the same row in T and to the same column in T. If not, the distinct numbers in any fixed row of T must belong to different columns of T, but this can only happen for all rows if µ λ. So consider τ = (ij) which belongs to P λ and to Q µ, whence a λ.τ = a λ and τ.b µ = b µ. But then, a λ.b µ = a λ.τ, τ, b µ = a λ.b µ proving the claim. If λ µ we claim that V λ is not isomorphic to V µ. Assume that λ > µ and φ a CS d -isomorphism with φ(v λ ) = V µ, then φ(c λ V λ ) = c λ φ(v λ ) = c λ V µ = c λ CS d c µ = 0 Hence, c λ V λ = Cc λ 0 lies in the kernel of an isomorphism which is clearly absurd. Summarizing, we have constructed to distinct partitions of d, λ and µ nonisomorphic simple CS d -representations V λ and V µ. As we know that there are as many isomorphism classes of simples as there are conjugacy classes in S d (or partitions), the V λ form a complete set of isomorphism classes of simple S d -representations, finishing the proof of the theorem. 3.6 Necklace relations. In this section we will prove that all the relations holding among the elements of the necklace algebra N m n are formal consequences of the Cayley-Hamilton equation. First, we will have to set up some notation to clarify what we mean by this. For technical reasons it is sometimes convenient to have an infinite supply of noncommutative variables {x, x 2,..., x i,...}. Two monomials of the same degree d in these variables M = x i x i2... x id and M = x j x j2... x jd

113 3.6. NECKLACE RELATIONS. 3 are said to be equivalent if M is obtained from M by a cyclic permutation, that is, there is a k such that i = j k and all i a = j b with b = k + a mod d. That is, if they determine the same necklace word w with each of the beads one of the noncommuting variables i = x i. To each equivalence class we assign a formal variable that we denote by t(x i x i2... x id ). The formal necklace algebra N is then the polynomial algebra on all these (infinitely many) letters. Similarly, we define the formal trace algebra T to be the algebra T = N C C x, x 2,..., x i,... that is, the free associative algebra on the noncommuting variables x i with coefficients in the polynomial algebra N. Crucial for our purposes is the existence of an N -linear formal trace map t : T N defined by the formula t( a i...i k x i... x ik ) = a i...i k t(x i... x ik ) where a i...i k N. In an analogous manner we will define infinite versions of the necklace and trace algebras. Let Mn be the space of all ordered sequences (A, A 2,..., A i,...) with A i M n and all but finitely many of the A i are the zero matrix. Again, GL n acts on Mn by simultaneous conjugation and we denote the infinite necklace algebra to be the algebra of polynomial functions f N n M n f C which are constant along orbits. Clearly, N n is generated as C-algebra by the invariants tr(m) where M runs over all monomials in the coordinate generic matrices X k = (x ij (k)) i,j belonging to the k-th factor of Mn. Similarly, the infinite trace algebra T n is the algebra of GL n -equivariant polynomial maps M n Mn. Clearly, T n is the C-algebra generated by N n and the generic matrices X k for k <. Observe that T n is a subalgebra of the matrixring T n M n (C[M n ])

114 4 CHAPTER 3. GENERIC MATRICES. and as such has a trace map tr defined on it and from our knowledge of the generators of N n we know that tr(t n ) = N n. Now, there are natural algebra epimorphisms T τ T n and N ν N n defined by τ(t(x i... x ik )) = ν(t(x i... x ik )) = tr(x i... X ik ) and τ(x i ) = X i. That is, ν and τ are compatible with the trace maps T τ T n t N ν N n tr We are interested in describing the necklace relations, that is, the kernel of ν. In the next section we will describe the trace relations which is the kernel of τ. Note that we obtain the relations holding among the necklaces in N m n by setting all x i = 0 with i > m and all t(x i... x ik ) = 0 containing a variable with i j > m. In the description a map T : CS d N will be important. Let S d be the symmetric group of permutations on {,..., d} and let σ = (i i... i α )(j j 2... j β )... (z z 2... z ζ ) be a decomposition of σ S d into cycles including those of length one. The map T assigns to σ a formal necklace T σ (x,..., x d ) defined by T σ (x,..., x d ) = t(x i x i2... x iα )t(x j x j2... x jβ )... t(x z x z2... x zζ ) Let V = V n be again the n-dimensional vectorspace of column vectors, then S d acts naturally on V d via σ.(v... v d ) = v σ()... v σ(d) hence determines a linear map λ σ End(V d ). Recall from section 3 that under the natural identifications the map λ σ defines the multilinear map (M d n ) (V d V d ) End(V d ) µ σ : M n... M n }{{} d C defined by (using the cycle decomposition of σ as before) µ σ (A... A d ) = tr(a i A i2... A iα )tr(a j A j2... A jβ )... tr(a z A z2... A zζ ). Therefore, a linear combination a σ T σ (x,..., x d ) is a necklace relation (that is, belongs to Ker ν) if and only if the multilinear map a σ µ σ : Mn d C is zero. This, in turn, is equivalent to the endomorphism a σ λ σ End(V m ), induced by the action of the element a σ e σ CS d on V d, being zero. In order to answer the latter problem we have to understand the action of a Young symmetrizer c λ CS d on V d.

115 3.6. NECKLACE RELATIONS. 5 Let λ = (λ, λ 2,..., λ k ) be a partition of d and equip the corresponding Young diagram with the standard tableau (that is, order first the boxes in the first row from left to right, then the second row from left to right and so on). d The subgroup P λ of S d which preserves each row then becomes P λ = S λ S λ2... S λk Sd. As a λ = p P λ e p we see that the image of the action of a λ on V d is the subspace Im(a λ ) = Sym λ V Sym λ2 V... Sym λ k V V d. Here, Sym i V denotes the subspace of symmetric tensors in V i. Similarly, equip the Young diagram of λ with the tableau by ordering first the boxes in the first column from top to bottom, then those of the second column from top to bottom and so on. d Equivalently, give the Young diagram corresponding to the dual partition of λ λ = (µ, µ 2,..., µ l ) the standard tableau. Then, the subgroup Q λ of S d which preserves each row of λ (or equivalently, each column of λ ) is Q λ = S µ S µ2... S µl Sd As b λ = q Q λ sgn(q)e q we see that the image of b λ on V d is the subspace Im(b λ ) = µ µ 2 µ l V V... V V d. Here, i V is the subspace of all anti-symmetric tensors in V i. Note that i V = 0 whenever i is greater than the dimension dim V = n. That is, the image of the action of b λ on V d is zero whenever the dual partition λ contains a row of length n +, or equivalently, whenever λ has n + rows. Because the Young symmetrizer c λ = a λ.b λ C S d we have proved the first result on necklace relations. Proposition 3.8 A formal necklace σ S d a σ T σ (x,..., x d )

116 6 CHAPTER 3. GENERIC MATRICES. is a necklace relation (for n n matrices) if and only if the element aσ e σ CS d belongs to the ideal of CS d spanned by the Young symmetrizers c λ relative to partitions λ = (λ,..., λ k ) n with a least n + rows, that is, k n +. Example 3.9 (Fundamental necklace and trace relation.) Consider the partition λ = (,,..., ) of n +, with corresponding Young tableau 2. n+ Then, P λ = {e}, Q λ = S n+ and we have the Young symmetrizer a λ = b λ = c λ = X σ S n+ sgn(σ)e σ. The corresponding element is called the fundamental necklace relation F(x,..., x n+ ) = X sgn(σ)t σ(x,..., x n+ ). σ S n+ Clearly, F(x,..., x n+ ) is multilinear of degree n + in the variables {x,..., x n+ }. Conversely, any multilinear necklace relation of degree n+ must be a scalar multiple of F(x,..., x n+ ). This follows from the proposition as the ideal described there is for d = n + just the scalar multiples of P σ S n+ sgn(σ)e σ. Because F(x,..., x n+ ) is multilinear in the variables x i we can use the cyclic permutation property of the formal trace t to write it in the form F(x,..., x n+ ) = t(ch(x,..., x n)x n+ ) with CH(x,..., x n) T Observe that CH(x,..., x n) is multilinear in the variables x i. Moreover, by the nondegeneracy of the trace map tr and the fact that F(x,..., x n+ ) is a necklace relation, it follows that CH(x,..., x n) is a trace relation. Again, any multilinear trace relation of degree n in the variables {x,..., x n} is a scalar multiple of CH(x,..., x n). This follows from the corresponding uniqueness result for F(x,..., x n+ ). We can give an explicit expression of this fundamental trace relation nx X X CH(x,..., x n) = ( ) k x i x i2... x ik sgn(σ)t σ(x j,..., x jn k ) k=0 i i 2... i k σ S J where J = {,..., n} {i,..., i k }. In a moment we will see that CH(x,..., x n) and hence also F(x,..., x n+ ) is obtained by polarization of the Cayley-Hamilton identity for n n matrices. We will explain what we mean by the Cayley-Hamilton polynomial for an element of T. Recall that when X M n (A) is a matrix with coefficients in a commutative C-algebra A its characteristic polynomial is defined to be χ X (t) = det(t n X) A[t] and by the Cayley-Hamilton theorem we have the basic relation that χ X (X) = 0. We have seen that the coefficients of the characteristic polynomial can be expressed as polynomial functions in the tr(x i ) for i n.

117 3.6. NECKLACE RELATIONS. 7 For example if n = 2, then the characteristic polynomial can we written as χ X (t) = t 2 tr(x)t + 2 (tr(x)2 tr(x 2 )). For general n the method for finding these polynomial functions is based on the formal recursive algorithm expressing elementary symmetric functions in term of Newton functions, usually expressed by the formulae f(t) = f (t) f(t) n (t λ i ), i= = d log f(t) dt = n i= t λ i = n t k+ ( λ k i ) Note, if λ i are the eigenvalues of X M n, then f(t) = χ X (t) and n i= λk i = tr(xk ). Therefore, one can use the formulae to express f(t) in terms of the elements n i= λk i. To get the required expression for the characteristic polynomial of X one only has to substitute n i= λk i with tr(xk ). This allows us to construct a formal Cayley-Hamilton polynomial χ x (x) T of an element x T by replacing in the above characteristic polynomial the term tr(x k ) with t(x k ) and t l with x l. If x is one of the variables x i then χ x (x) is an element of T homogeneous of degree n. Moreover, by the Cayley-Hamilton theorem it follows immediately that χ x (x) is a trace relation. Hence, if we fully polarize χ x (x) (say, using the variables {x,..., x n }) we obtain a multilinear trace relation of degree n. By the argument given in the example above we know that this element must be a scalar multiple of CH(x,..., x n ). In fact, one can see that this scale factor must be ( ) n as the leading term of the multilinearization is σ S n x σ()... x σ(n) and compare this with the explicit form of CH(x,..., x n ). Example 3.20 Consider the case n = 2. The formal Cayley-Hamilton polynomial of an element x T is k=0 χ x(x) = x 2 t(x)x + 2 (t(x)2 t(x 2 )). Polarization with respect to the variables x and x 2 gives the expression i= x x 2 + x 2 x t(x )x 2 t(x 2 )x + t(x )t(x 2 ) t(x x 2 ) which is CH(x, x 2 ). Indeed, multiplying it on the right with x 3 and applying the formal trace t to it we obtain t(x x 2 x 3 ) + t(x 2 x x 3 ) t(x )t(x 2 x 3 ) t(x 2 )t(x x 3 ) + t(x )t(x 2 )t(x 3 ) t(x x 2 )t(x 3 ) as required. = T (23) (x, x 2, x 3 ) + T (23) (x, x 2, x 3 ) T ()(23) (x, x 2, x 3 ) T (2)(3) (x, x 2, x 3 ) + T ()(2)(3) (x, x 2, x 3 ) T (2)(3) (x, x 2, x 3 ) = X σ S 3 T σ(x, x 2, x 3 ) = F(x, x 2, x 3 ) Theorem 3.2 The necklace relations Ker ν is the ideal of N generated by all the elements F(m,..., m n+ ) where the m i run over all monomials in the variables {x, x 2,..., x i,...}. Proof. Take a homogeneous necklace relation f Ker ν of degree d and polarize it to get a multilinear element f N. Clearly, f is also a necklace relation and if we can show that f belongs to the described ideal, then so does f as the process of restitution maps this ideal into itself.

118 8 CHAPTER 3. GENERIC MATRICES. Therefore, we may assume that f is multilinear of degree d. A priori f may depend on more than d variables x k, but we can separate f as a sum of multilinear polynomials f i each depending on precisely d variables such that for i j f i and f j do not depend on the same variables. Setting some of the variables equal to zero, we see that each of the f i is again a necklace relation. Thus, we may assume that f is a multilinear necklace identity of degree d depending on the variables {x,..., x d }. But then we know from proposition 3.8 that we can write f = τ S d a τ T τ (x,..., x d ) where a τ e τ CS d belongs to the ideal spanned by the Young symmetrizers of Young diagrams λ having at least n + rows. We claim that this ideal is generated by the Young symmetrizer of the partition (,..., ) of n + under the natural embedding of S n+ into S d. Let λ be a Young diagram having k n + boxes and let c λ be a Young symmetrizer with respect to a tableau where the boxes in the first column are labeled by the numbers I = {i,..., i k } and let S I be the obvious subgroup of S d. As Q λ = S I Q we see that b λ = ( σ S I sgn(σ)e σ ).b with b CQ. Hence, c λ belongs to the twosided ideal generated by c I = σ S I sgn(σ)e σ but this is also the twosided ideal generated by c k = σ S k sgn(σ)e σ as one verifies by conjugation with a partition sending I to {,..., k}. Moreover, by induction one shows that the twosided ideal generated by c k belongs to the twosided ideal generated by c d = σ S d sgn(σ)e σ, finishing the proof of the claim. From this claim, we can write τ S d a τ e τ = τ i,τ j S d a ij e τi.( σ S n+ sgn(σ)e σ ).e τj and therefore it suffices to analyze the form of the necklace identity associated to an element of the form e τ.( σ S n+ sgn(σ)e σ ).e τ with τ, τ S d Now, if a groupelement µ S d b µ e µ corresponds to the formal necklace polynomial G(x,..., x d ), then the element e τ.( µ S d b µ e µ ).e τ corresponds to the formal necklace polynomial G(x τ(),..., x τ(d) ). Therefore, we may replace the element e τ.( σ S n+ sgn(σ)e σ ).e τ by the element ( sgn(σ)e σ ).e η with η = τ.τ S d σ S n+ We claim that we can write η = σ.θ with σ S n+ and θ S d such that each cycle of θ contains at most one of the elements from {, 2,..., n+}. Indeed assume that η contains a cycle containing more than one element from {,..., n + }, say and 2, that is η = (i i 2... i r 2j j 2... j s )(k... k α )... (z... z ζ ) then we can express the product (2).η in cycles as (i i 2... i r )(2j j 2... j s )(k... k α )... (z... z ζ ) Continuing in this manner we reduce the number of elements from {...., n + } in every cycle to at most one.

119 3.7. TRACE RELATIONS. 9 But then as σ S n+ we have seen that ( sgn(σ)e σ ).e σ = sgn(σ )( sgn(σ)e σ ) and consequently ( sgn(σ)e σ ).e η = ±( sgn(σ)e σ ).e θ σ S n+ σ S n+ where each cycle of θ contains at most one of {,..., n + }. Let us write θ = (i... i α )(2j... j β )... (n + s... s κ )(t... t λ )... (z... z ζ ) Now, let σ S n+ then the cycle decomposition of σ.θ is obtained as follows : substitute in each cycle of σ the element formally by the string i... i α, the element 2 by the string 2j... j β, and so on until the element n + by the string n+s... s κ and finally adjoin the cycles of θ in which no elements from {,..., n+ } appear. Finally, we can write out the formal necklace element corresponding to the element ( σ S n+ sgn(σ)e σ ).e θ as F(x x i... x iα, x 2 x j... x jβ,..., x n+ x s... x sκ )t(x t... x tλ )... t(x z... x zζ ) finishing the proof of the theorem. 3.7 Trace relations. We will again use the non-degeneracy of the trace map to deduce the trace relations, that is, Ker τ from the description of the necklace relations. Theorem 3.22 The trace relations Ker τ is the twosided ideal of the formal trace algebra T generated by all elements F(m,..., m n+ ) and CH(m,..., m n ) where the m i run over all monomials in the variables {x, x 2,..., x i,...}. Proof. Consider a trace relation H(x,..., x d ) Ker τ. Then, we have a necklace relation of the form t(h(x,..., x d )x d+ ) Ker ν By theorem 3.2 we know that this element must be of the form ni...i n+ F(m i,..., m in+ ) where the m i are monomials, the n i...i n+ N and the expression must be linear in the variable x d+. That is, x d+ appears linearly in each of the terms nf(m,..., m n+ ) so appears linearly in n or in precisely one of the monomials m i. If x d+ appears linearly in n we can write n = t(n.x d+ ) where n T. If x d+ appears linearly in one of the monomials m i we may assume that it does so in m n+, permuting the monomials if necessary. That is, we may assume m n+ = m n+.x d+.m n+ with m, m monomials. But then, we can write nf(m,..., m n+ ) = nt(ch(m,..., m n ).m n+.x d+.m n+ ) = t(n.m n+.ch(m,..., m n ).m n+.x d+ )

120 20 CHAPTER 3. GENERIC MATRICES. using N -linearity and the cyclic permutation property of the formal trace t. But then, separating the two cases, one can write the total expression t(h(x,..., x d )x d+ ) = t([ i n i...i n+ F(m i,..., m in+ ) + j n j...j n+.m jn+.ch(m j,..., m jn ).m j n+ ] x d+ ) Finally, observe that two formal trace elements H(x,..., x d ) and K(x,..., x d ) are equal if the formal necklaces t(h(x,..., x d )x d+ ) = t(k(x,..., x d )x d+ ) are equal, finishing the proof. We will give another description of the necklace relations Ker τ which is better suited for the categorical interpretation of T n to be given in the next chapter. Consider formal trace elements m, m 2,..., m i,... with m j T. The formal substitution f f(m, m 2,..., m i,...) is the uniquely determined algebra endomorphism of T which maps the variable x i to m i and is compatible with the formal trace t. That is, the substitution sends a monomial x i x i... x ik to the element g i g i2... g ik and an element t(x i x i2... x ik ) to the element t(g i g i2... g ik ). A substitution invariant ideal of T is a twosided ideal of T that is closed under all possible substitutions as well as under the formal trace t. For any subset of elements E T there is a minimal substitution invariant ideal containing E. This is the ideal generated by all elements obtained from E by making all possible substitutions and taking all their formal traces. We will refer to this ideal as the substitution invariant ideal generated by E. Recall the definition of the formal Cayley-Hamilton polynomial χ x (x) of an element x T given in the previous section. Theorem 3.23 The trace relations Ker τ is the substitution invariant ideal of T generated by the formal Cayley-Hamilton polynomials χ x (x) for all x T Proof. The result follows from theorem 3.22 and the definition of a substitution invariant ideal once we can show that the full polarization of χ x (x), which we have seen is CH(x,..., x n ), lies in the substitution invariant ideal generated by the χ x (x). This is true since we may replace the process of polarization with the process of multilinearization, whose first step is to replace, for instance χ x (x) by χ x+y (x + y) χ x (x) χ y (y). The final result of multilinearization is the same as of full polarization and the claim follows as multilinearizing a polynomial in a substitution invariant ideal, we remain in the same ideal. We will use our knowledge on the necklace and trace relations to improve the bound of 2 n in the Nagata-Higman problem to n 2. Recall that this problem asks

121 3.7. TRACE RELATIONS. 2 for a number N(n) with the property that if R is an associative C-algebra without unit such that r n = 0 for all r R, then we must have for all r i R the identity r r 2... r N(n) = 0 in R. We start by reformulating the problem. Consider the positive part F + of the free C-algebra generated by the variables {x, x 2,..., x i,...} F + = C x, x 2,..., x i,... + which is an associative C-algebra without unit. Let T (n) be the twosided ideal of F + generated by all n-powers f n with f F +. Note that the ideal T (n) is invariant under all substitutions of F +. The Nagata-Higman problem then asks for a number N(n) such that the product x x 2... x N(n) T (n). We will now give an alternative description of the quotient algebra F + /T (n). Let N + be the positive part of the infinite necklace algebra N n and T + the positive part of the infinite trace algebra T n. Consider the quotient associative C-algebra without unit T + = T + /(N + T n ). Observe the following facts about T + : as a C-algebra it is generated by the variables X, X 2,... as all the other algebra generators of the form t(x i... x ir ) of T are mapped to zero in T +. Further, from the form of the Cayley-Hamilton polynomial it follows that every t T + satisfies t n = 0. That is, we have an algebra epimorphism F + /T (n) T + and we claim that it is also injective. To see this, observe that the quotient T /N + T is just the free C-algebra on the variables {x, x 2,...}. To obtain T + we have to factor out the ideal of trace relations. Now, a formal Cayley-Hamilton polynomial χ x (x) is mapped to x n in T /N + T. That is, to obtain T + we factor out the substitution invariant ideal (observe that t is zero here) generated by the elements x n, but this is just the definition of F + /T (n). Therefore, a reformulation of the Nagata-Higman problem is to find a number N = N(n) such that the product of the first N generic matrices X X 2... X N N + T n or, equivalently that tr(x X 2... X N X N+ ) can be expressed as a linear combination of products of traces of lower degree. Using the description of the necklace relations given in proposition 3.8 we can reformulate this conditions in terms of the group algebra CS N+. Let us introduce the following subspaces of the groupalgebra : A will be the subspace spanned by all N + cycles in S N+, B will be the subspace spanned by all elements except N + cycles, L(n) will be the ideal of CS N+ spanned by the Young symmetrizers associated to partitions L(n) with n rows, and n

122 22 CHAPTER 3. GENERIC MATRICES. M(n) will be the ideal of CS N+ spanned by the Young symmetrizers associated to partitions M(n) n having more than n rows. With these notations, we can reformulate the above condition as (2... NN + ) B + M(n) and consequently CS N+ = B + M(n) Define an inner product on the groupalgebra CS N+ such that the groupelements form an orthonormal basis, then A and B are orthogonal complements and also L(n) and M(n) are orthogonal complements. But then, taking orthogonal complements the condition can be rephrased as (B + M(n)) = A L(n) = 0. Finally, let us define an automorphism τ on CS N+ induced by sending e σ to sgn(σ)e σ. Clearly, τ is just multiplication by ( ) N on A and therefore the above condition is equivalent to A L(n) τl(n) = 0. Moreover, for any Young tableau λ we have that τ(a λ ) = b λ and τ(b λ ) = a λ. Hence, the automorphism τ sends the Young symmetrizer associated to a partition to the Young symmetrizer of the dual partition. This gives the following characterization τl(n) is the ideal of CS N+ spanned by the Young symmetrizers associated to partitions τl(n) n with n columns. Now, specialize to the case N = n 2. Clearly, any Young diagram having n 2 + boxes must have either more than n columns or more than n rows n and consequently we indeed have for N = n 2 that A L(n) τl(n) = 0 finishing the proof of the promised refinement of the Nagata-Higman bound

123 3.7. TRACE RELATIONS. 23 Theorem 3.24 Let R be an associative C-algebra without unit element. Assume that r n = 0 for all r R. Then, for all r i R we have r r 2... r n 2 = 0 Theorem 3.25 The necklace algebra N m n is generated as a C-algebra by all elements of the form tr(x i X i2... X il ) with l n 2 +. The trace algebra T m n is spanned as a module over the necklace algebra N m n by all monomials in the generic matrices of degree l n 2. X i X i2... X il

124 24 CHAPTER 3. GENERIC MATRICES.

125 Chapter 4 Reconstructing Algebras. Let A be an affine C-algebra, generated by {a,..., a m }. A running theme of this book is to study A by investigating its level n approximations A@ n for all natural numbers n. These algebras are defined in two stages. First, we define the category alg t of C-algebras equipped with a linear trace map A t A. This map has to satisfy for all a, b A t(a)b = bt(a) t(ab) = t(ba) and t(t(a)b) = t(a)t(b) Morphisms in alg t are C-algebra morphisms compatible with the trace structure. The forgetful functor alg t alg has a left adjoint alg (.)t alg t A A t where A t is constructed (as in the special case of T in the foregoing chapter) by adding formal traces to necklaces with beads running through a C-basis of A. The algebra A t is trace-generated by m elements, that is, we have a commutative diagram C x,..., x m A T m A t where T m is the subalgebra with trace of T generated by {x,..., x m }. The vertical maps are the natural ones and the lower map is trace preserving. For any a A and natural number n we can define a formal Cayley-Hamilton polynomial χ (n) a (t) of degree n by expressing f(t) = n (t λ i ) with the λ i indeterminates i= as a polynomial in t with coefficients which are polynomials in the Newton functions n i= λk i (as they are symmetric functions in the λ i). Replacing each occurrence of n i= λk i by t(a k ) gives χ (n) a (t) A[t]. The approximation of A at level n is then defined to be A@ n = A t (t() n, χ (n) a (a) a A) 25

126 26 CHAPTER 4. RECONSTRUCTING ALGEBRAS. with the induced trace map, that is we can complete the above diagram T m A t T m n A@n The main result of this chapter gives a geometric interpretation of the algebra A@ n. The representation variety rep n A has as its geometric points the n- dimensional representations of A, that is, C-algebra morphisms A ρ Mn (C) It will be shown that this variety comes equipped with a GL n -structure, the orbits of which correspond to isomorphism classes of n-dimensional representations. We will prove that A@ n is the algebra of GL n -equivariant polynomial maps rep n A M n with the algebra structure coming from those of the target space M n (C). Further, we will prove that the the commutative central subalgebra t(a@ n ) of A@ n classifies the isomorphism classes of n-dimensional semi-simple representations of A. The geometric interpretation of t(a@ n ) proves it to be the coordinate ring of the quotient variety rep n A/GL n classifying the closed orbits in rep n A which correspond by the Artin-Voigt result to semi-simple representations. These two main results follow from the description of necklace and trace algebras given in the foregoing chapter and invariant theory, the basics of which we will recall in this chapter. At an intermediate stage, we will introduce also trace preserving representation varieties rep tr A when the algebra A is equipped with a trace map. n The above results then follow from the natural GL n -isomorphisms rep n A rep tr n At rep tr n A@ n coming from the left adjointness. The level n approximation A@ n is a special case of a Cayley-Hamilton algebra of degree n, other natural examples are orders over normal affine varieties in central simple algebras of dimension n 2 over the functionfield. The results in this chapter prove that there is a functor from the category CH(n) of Cayley-Hamilton algebras of degree n to the category of affine GL n -varieties (actually schemes) which assigns to an algebra A with trace map t the trace preserving representation variety rep tr A and that this functor has a left n inverse assigning to an affine GL n -variety the algebra of GL n -equivariant maps from the variety to M n. This left inverse is not an equivalence of categories, however, and the characterization of those affine GL n -varieties which are trace preserving representation varieties is a difficult problem. In chapter we will show that the formal structure defined on them may be a first step in solving this riddle. 4. Representation varieties. When A is a noncommutative affine algebra with generating set {a,..., a m }, there is an epimorphism C x,..., x m φ A defined by φ(x i ) = a i. Hence, we have a presentation of A as A C x,..., x m /I A

127 4.. REPRESENTATION VARIETIES. 27 where I A is the twosided ideal of relations holding among the a i. For example, if A = C[x,..., x m ], then I A is the twosided ideal of C x,..., x m generated by the elements x i x j x j x i for all i, j m. Observe that there is no analog of the Hilbert basis theorem for C x,..., x m, that is, I A is not necessarily finitely generated as a twosided ideal. If it is, we say that A is finitely presented. An n-dimensional representation of A is an algebra morphism A ψ Mn from A to n n matrices over C. If A is generated by {a,..., a m }, then ψ is fully determined by the point (ψ(a ), ψ(a 2 ),..., ψ(a m )) M m n = M n... M }{{ n. } m We claim that mod n (A), the set of all n-dimensional representations of A, forms a Zariski closed subset of M m n. To begin, observe that rep n (C x,..., x m ) = M m n as any m-tuple of n n matrices (A,..., A m ) Mn m determines an algebra morphism C x,..., x m ψ M n by taking ψ(x i ) = A i. Now, given a presentation A = C x,..., x m /I A an m-tuple (A,..., A m ) Mn m determines an n-dimensional representation of A if (and only if) for every noncommutative polynomial r(x,..., x m ) I A C x,..., x m we have that r(a,..., A m ) =.. M n Hence, consider the ideal I A (n) of C[M m n ] = C[x ij (k) i, j n, k m] generated by all the entries of the matrices in M n (C[M m n ]) of the form r(x,..., X m ) for all r(x,..., x m ) I A. By the above observation we see that the reduced representation variety rep n A is the set of simultaneous zeroes of the ideal I A (n), that is, rep n A = V(I A (n)) M m n proving the claim, where V denotes the closed set in the Zariski topology determined by an ideal, the complement of which we will denotye with X). Observe that even when A is not finitely presented, the ideal I A (n) is finitely generated as an ideal of the commutative polynomial algebra C[Mn m ]. Often, the ideal I A (n) contains more information than the closed subset rep n (A) = V(I A (n)) which only determines the radical ideal of I A (n). This forces us to consider also the representation variety (or module scheme) rep n A which we will introduce in a moment. Example 4. It may happen that rep n A =. For example, consider the Weyl algebra A (C) = C x, y /(xy yx ) If a couple of n n-matrices (A, B) rep n A (C) then we must have A.B B.A = n M n However, taking traces on both sides gives a contradiction as tr(ab) = tr(ba) and tr( n ) = n 0.

128 28 CHAPTER 4. RECONSTRUCTING ALGEBRAS. In the foregoing chapter we studied the action of GL n by simultaneous conjugation on M m n. We claim that rep n A M m n is stable under this action, that is, if (A,..., A m ) rep n A, then also (ga g,..., ga m g ) rep n A. This is clear by composing the n-dimensional representation ψ of A determined by (A,..., A m ) with the algebra automorphism of M n given by conjugation with g GL n, A ψ M n... g.ψ M n g.g That is, rep n A is a GL n -variety. We will give an interpretation of the orbits under this action. Recall that a left A-module M is a vectorspace on which elements of A act on the left as linear operators satisfying the conditions.m = m and a.(b.m) = (ab).m for all a, b A and all m M. An A-module morphism M f N between two left A-modules is a linear map such that f(a.m) = a.f(m) for all a A and all m M. An A-module automorphism is an A-module morphism M f N such that there is an A-module morphism N g M such that f g = idm and g f = id N. Assume the A-module M has complex dimension n, then after fixing a basis we can identify M with C n (column vectors). For any a A we can represent the linear action of a on M by an n n matrix ψ(a) M n. The condition that a.(b.m) = (ab).m for all m M asserts that ψ(ab) = ψ(a)ψ(b) for all a, b A, that is, ψ is an algebra morphism A ψ Mn and hence M determines an n-dimensional representation of A. Conversely, an n-dimensional representation A ψ Mn determines an A-module structure on C n by the rule a.v = ψ(a)v for all v C n. Hence, there is a one-to-one correspondence between the n-dimensional representations of A and the A-module structures on C n. For this reason we call the reduced variety rep n A the reduced representation variety of A. If two n-dimensional A- module structures M and N on C n are isomorphic (determined by a linear invertible map g GL n ) then for all a A we have the commutative diagram M g N a. a. g M N Hence, if the action of a on M is represented by the matrix A, then the action of a on M is represented by the matrix g.a.g. Therefore, two A-module structures on C n are isomorphic if and only if the points of rep n A corresponding to them lie in the same GL n -orbit. Concluding, studying n-dimensional A-modules up to isomorphism is the same as studying the GL n -orbits in the reduced representation variety rep n A. If the defining ideal I A (n) is a radical ideal (as we will see, this is the case when A is a Quillen-smooth algebra) the above suffices. In general, the scheme structure

129 4.2. SOME ALGEBRAIC GEOMETRY. 29 of the representation variety rep n A will be important. By definition, the module scheme rep n A is the functor assigning to any (affine) commutative C-algebra R, the set rep n A(R) = Alg C (C[M m n ]/I A (n), R) of C-algebra morphisms C[M m n ] ψ I A (n) R. Such a map ψ is determined by the image ψ(x ij (k)) = r ij (k) R. That is, ψ rep n A(R) determines an m-tuple of n n matrices with coefficients in R (r,..., r m ) M n (R)... M n (R) }{{} m where r k = r (k)... r n (k)... r n (k)... r nn (k) Clearly, for any r(x,..., x m ) I A we must have that r(r,..., r m ) is the zero matrix in M n (R). That is, ψ determines uniquely an R-algebra morphism ψ : R C A Mn (R) by mapping x k r k. Alternatively, we can identify the set rep n (R) with the set of left R A-module structures on the free R-module R n of rank n. In section 8, we will introduce the representation variety rep t n A and teh reduced representation variety rept n A of trace preserving n-dimensional representations. 4.2 Some algebraic geometry. Throughout this book we assume that the reader has some familiarity with algebraic geometry, such as the first two chapters of the textbook [9]. In this section we restrict to the dimension formulas and the relation between Zariski and analytic closure, illustrating them with examples from module varieties. We will work only with reduced varieties in this section. A morphism X φ Y between two affine irreducible varieties is said to be dominant if the image φ(x) is Zariski dense in Y. On the level of the coordinate algebras dominance is equivalent to φ : C[Y ] C[X] being injective and hence inducing a fieldextension φ : C(Y ) C(X) between the functionfields. Indeed, for f C[Y ] the function φ (f) is by definition the composition X φ Y f C and therefore φ (f) = 0 iff f(φ(x)) = 0 iff f(φ(x)) = 0. A morphism X φ Y between two affine varieties is said to be finite if under the algebra morphism φ the coordinate algebra C[X] is a finite C[Y ]-module. An important property of finite morphisms is that they are closed, that is the image of a closed subset is closed. Indeed, we can replace without loss of generality Y by the closed subset φ(x) = V Y (Ker φ ) and hence assume that φ is an inclusion C[Y ] C[X]. The claim then follows from the fact that in a finite extension there exists for any maximal ideal N C[Y ] a maximal ideal M C[X] such that M C[Y ] = C[X]. Example 4.2 Let X be an irreducible affine variety of dimension d. By the Noether normalization result C[X] is a finite module over a polynomial subalgebra C[f,..., f d ]. But then, the finite inclusion C[f,..., f d ] C[X] determines a finite surjective morphism X φ C d

130 30 CHAPTER 4. RECONSTRUCTING ALGEBRAS. An important source of finite morphisms is given by integral extensions. Recall that, if R S is an inclusion of domains we call S integral over R if every s S satisfies an equation n s n = r i s i with r i R. i=0 A normal domain R has the property that any element of its field of fractions which is integral over R belongs already to R. If X φ Y is a dominant morphism between two irreducible affine varieties, then φ is finite if and only if C[X] in integral over C[Y ] for the embedding coming from φ. Proposition 4.3 Let X φ Y be a dominant morphism between irreducible affine varieties. Then, for any x X and any irreducible component C of the fiber φ (φ(z)) we have dim C dim X dim Y. Moreover, there is a nonempty open subset U of Y contained in the image φ(x) such that for all u U we have dim φ (u) = dim X dim Y. Proof. Let d = dim X dim Y and apply the Noether normalization result to the affine C(Y )-algebra C(Y )C[X]. Then, we can find a function g C[Y ] and algebraic independent functions f,..., f d C[X] g (g clears away any denominators that occur after applying the normalization result) such that C[X] g is integral over C[Y ] g [f,..., f d ]. That is, we have the commutative diagram X X (g) ρ X Y (g) C d pr X φ Y X Y (g) where we know that ρ is finite and surjective. But then we have that the open subset X Y (g) lies in the image of φ and in X Y (g) all fibers of φ have dimension d. For the first part of the statement we have to recall the statement of Krull s Hauptideal result : if X is an irreducible affine variety and g,..., g r C[X] with (g,..., g r ) C[X], then any component C of V X (g,..., g r ) satisfies the inequality dim C dim X r. If dim Y = r apply this result to the g i determining the morphism X φ Y C r where the latter morphism is the one from example 4.2. In fact, a stronger result holds. Chevalley s theorem asserts the following. Theorem 4.4 Let X φ Y be a morphism between affine varieties, the function X N defined by x dim x φ (φ(x)) is upper-semicontinuous. That is, for all n N, the set {x X dim x φ (φ(x)) n} is Zariski open in X.

131 4.2. SOME ALGEBRAIC GEOMETRY. 3 Proof. Let Z(φ, n) be the set {x X dim x φ (φ(x)) n}. We will prove that Z(φ, n) is closed by induction on the dimension of X. We first make some reductions. We may assume that X is irreducible. For, let X = i X i be the decomposition of X into irreducible components, then Z(φ, n) = Z(φ X i, n). Next, we may assume that Y = φ(x) whence Y is also irreducible and φ is a dominant map. Now, we are in the setting of proposition 4.3. Therefore, if n dim X dim Y we have Z(φ, n) = X by that proposition, so it is closed. If n > dim X dim Y consider the open set U in Y of proposition 4.3. Then, Z(φ, n) = Z(φ (X φ (U)), n). the dimension of the closed subvariety X φ (U) is strictly smaller that dim X hence by induction we may assume that Z(φ (X φ (U)), n) is closed in X φ (U) whence closed in X. An immediate consequence of the foregoing proposition is that for any morphism X φ Y between affine varieties, the image φ(x) contains an open dense subset of φ(z) (reduce to irreducible components and apply the proposition). Example 4.5 Let A be an affine C-algebra and M rep n A. We claim that the orbit O(M) = GL n.m is Zariski open in its closure O(M). Consider the orbit-map GL n φ repn A defined by g g.m. then, by the above remark O(M) = φ(gl n) contains a Zariski open subset U of O(M) contained in the image of φ which is O(M). But then, O(M) = GL n.m = g GLn g.u is also open in O(M). Next, we claim that O(M) contains a closed orbit. Indeed, assume O(M) is not closed, then the complement C M = O(M) O(M) is a proper Zariski closed subset whence dim C < dim O(M). But, C is the union of GL n-orbits O(M i ) with dim O(M i ) < dim O(M). Repeating the argument with the M i and induction on the dimension we will obtain a closed orbit in O(M). Next, we want to relate the Zariski closure with the C-closure. Whereas they are usually not equal (for example, the unit circle in C ), we will show that they coincide for the important class of constructible subsets. A subset Z of an affine variety X is said to be locally closed if Z is open in its Zariski closure Z. A subset Z is said to be constructible if Z is the union of finitely many locally closed subsets. Clearly, finite unions, finite intersections and complements of constructible subsets are again constructible. The importance of constructible sets for algebraic geometry is clear from the following result. Proposition 4.6 Let X φ Y be a morphism between affine varieties. If Z is a constructible subset of X, then φ(z) is a constructible subset of Y. Proof. Because every open subset of X is a finite union of special open sets which are themselves affine varieties, it suffices to show that φ(x) is constructible. We will use induction on dim φ(x). There exists an open subset U φ(x) which is contained in φ(x). Consider the closed complement W = φ(x) U and its inverse image X = φ (W ). Then, X is an affine variety and by induction we may assume that φ(x ) is constructible. But then, φ(x) = U φ(x ) is also constructible. Example 4.7 Let A be an affine C-algebra. The subset ind n A rep n A of the indecomposable n-dimensional A-modules is constructible. Indeed, define for any pair k, l such that k + l = n the morphism GL n rep k A rep l A repn A by sending a triple (g, M, N) to g.(m N). By the foregoing result the image of this map is constructible. The decomposable n-dimensional A-modules belong to one of these finitely many sets whence are constructible, but then so is its complement which in ind n A.

132 32 CHAPTER 4. RECONSTRUCTING ALGEBRAS. Apart from being closed, finite morphisms often satisfy the going-down property. That is, consider a finite and surjective morphism X φ Y where X is irreducible and Y is normal (that is, C[Y ] is a normal domain). Let Y Y an irreducible Zariski closed subvariety and x X with image φ(x) = y Y. Then, the going-down property asserts the existence of an irreducible Zariski closed subvariety X X such that x X and φ(x ) = Y. In particular, the morphism X φ Y is again finite and surjective and in particular dim X = dim Y. An important application of this property is that any two points of an irreducible affine variety can be connected through an irreducible curve. Lemma 4.8 Let x X an irreducible affine variety and U a Zariski open subset. Then, there is an irreducible curve C X through x and intersecting U. Proof. If d = dim X consider the finite surjective morphism X φ C d of example 4.2. Let y C d φ(x U) and consider the line L through y and φ(x). By the going-down property there is an irreducible curve C X containing x such that φ(c) = L and by construction C U. Proposition 4.9 Let X φ Y be a dominant morphism between irreducible affine varieties any y Y. Then, there is an irreducible curve C X such that y φ(c). Proof. Consider an open dense subset U Y contained in the image φ(x). By the lemma there is a curve C Y containing y and such that C U. Then, again applying the lemma to an irreducible component of φ (C ) not contained in a fiber, we obtain an irreducible curve C X with φ(c) = C. Any affine variety X C k can also be equipped with the induced C-topology from C k which is much finer than the Zariski topology. Usually there is no relation between the closure Z C of a subset Z X in the C-topology and the Zariski closure Z. Lemma 4.0 Let U C k containing a subset V which is Zariski open and dense in U. Then, U C = U Proof. By reducing to irreducible components, we may assume that U is irreducible. Assume first that dim U =, that is, U is an irreducible curve in C k. Let U s be the subset of points where U is a complex manifold, then U U s is finite and by the implicit function theorem in analysis every u U s has a C-open neighborhood which is C-homeomorphic to the complex line C, whence the result holds in this case. If U is general and x U we can take by the lemma above an irreducible curve C U containing z and such that C V. Then, C V is Zariski open and dense in C and by the curve argument above x (C V ) C U C. We can do this for any x U finishing the proof. Consider the embedding of an affine variety X C k, proposition 4.6 and the fact that any constructible set Z contains a subset U which is open and dense in Z we deduce from the lemma at one the next result.

133 4.3. THE GERSTENHABER-HESSELINK THEOREM. 33 Proposition 4. If Z is a constructible subset of an affine variety X, then Z C = Z Example 4.2 Let A be an affine C-algebra and M rep n A. We have proved in example 4.5 that the orbit O(M) = GL n.m is Zariski open in its closure O(M). Therefore, the orbit O(M) is a constructible subset of rep n A. By the proposition above, the Zariski closure O(M) of the orbit coincides with the closure of O(M) in the C-topology. 4.3 The Gerstenhaber-Hesselink theorem. In the next sections we will study orbit-closure and closed orbits in rep n A. In this section we give one of the rare instances (but which is very important in applications) where everything can be fully determined : the orbits in rep n C[x] or, equivalent, conjugacy classes of n n matrices. It is sometimes convenient to relax our definition of partitions to include zeroes at its tail. That is, a partition p of n is an integral n-tuple (a, a 2,..., a n ) with a a 2... a n 0 with n i= a i = n. As before, we represent a partition by a Young diagram by omitting rows corresponding to zeroes. If q = (b,..., b n ) is another partition of n we say that p dominates q and write p > q if and only if r r a i b i for all r n. i= i= For example, the partitions of 4 are ordered as indicated below > > > > Note however that the dominance relation is not a total ordering whenever n 6. For example, the following two partition of 6 are not comparable. The dominance order is induced by the Young move of throwing a row-ending box down the diagram. Indeed, let p and q be partitions of n such that p > q and assume there is no partition r such that p > r and r > q. Let i be the minimal number such that a i > b i. Then by the assumption a i = b i +. Let j > i be minimal such that a j b j, then we have b j = a j + because p dominates q. But then, the remaining rows of p and q must be equal. That is, a Young move can be depicted as i i j j p = q =

134 34 CHAPTER 4. RECONSTRUCTING ALGEBRAS. For example, the Young moves between the partitions of 4 given above are as indicated.... A Young p-tableau is the Young diagram of p with the boxes labeled by integers from {, 2,..., s} for some s such that each label appears at least ones. A Young p- tableau is said to be of type q for some partition q = (b,..., b n ) of n if the following conditions are met : the labels are non-decreasing along rows, the labels are strictly increasing along columns, and the label i appears exactly b i times. For example, if p = (3, 2,, ) and q = (2, 2, 2, ) then the p-tableau below is of type q (observe that p > q and even p q). In general, let p = (a,..., a n ) and q = (b,..., b n ) be partitions of n and assume that p q. Then, there is a Young p-tableau of type q. For, fill the Young diagram of q by putting s in the first row, 2 s in the second and so on. Then, upgrade the fallen box together with its label to get a Young p-tableau of type q. In the example above = Conversely, assume there is a Young p-tableau of type q. The number of boxes labeled with a number i is equal to b b i. Further, any box with label i must lie in the first i rows (because the labels strictly increase along a column). There are a a i boxes available in the first i rows whence i b i j= i a i for all i n j= and therefore p > q. After these preliminaries on partitions, let us return to nilpotent matrices. Let A be a nilpotent matrix of type p = (a,..., a n ), that is, conjugated to a matrix with Jordan blocks (all with eigenvalue zero) of sizes a i. We have seen before that the subspace V l of column vectors v C n such that A l.v = 0 has dimension l #{j a j h} = h= l h= a h

135 4.3. THE GERSTENHABER-HESSELINK THEOREM. 35 where p = (a,..., a n) is the dual partition of p. Choose a basis {v,..., v n } of C n such that for all l the first a a l base vectors span the subspace V l. For example, if A is in Jordan normal form of type p = (3, 2,, ) then p = (4, 2, ) and we can choose the standard base vectors ordered as follows 0 0 {e, e 4, e 6, e }{{} 7, e 2, e 5, e }{{} 3 }. }{{} 4 2 Take a partition q = (b,..., b n ) with p q (in particular, p > q), then for the dual partitions we have q p (and thus q > p ). But then there is a Young q -tableau of type p. In the example with q = (2, 2, 2, ) we have q = (4, 3) and p = (4, 2, ) and we can take the Young q -tableau of type p Now label the boxes of this tableau by the base vectors {v,..., v n } such that the boxes labeled i in the Young q -tableau of type p are filled with the base vectors from V i V i. Call this tableau T. In the example, we can take e e 4 e 6 e 7 T = e 2 e 5 e 3 Define a linear operator F on C n by the rule that F (v i ) = v j if v j is the label of the box in T just above the box labeled v i. In case v i is a label of a box in the first row of T we take F (v i ) = 0. Obviously, F is a nilpotent n n matrix and by construction we have that rk F l = n (b b l ) That is, F is nilpotent of type q = (b,..., b n ). Moreover, F satisfies F (V i ) V i for all i by the way we have labeled the tableau T and defined F. In the example above, we have F (e 2 ) = e, F (e 5 ) = e 4, F (e 3 ) = e 6 and all other F (e i ) = 0. That is, F is the matrix which is seen to be of type (2, 2, 2, ) after performing a few Jordan moves. 0

136 36 CHAPTER 4. RECONSTRUCTING ALGEBRAS. Returning to the general case, consider for all ɛ C the n n matrix : F ɛ = ( ɛ)f + ɛa. We claim that for all but finitely many values of ɛ we have F ɛ O(A). Indeed, we have seen that F (V i ) V i where V i is defined as the subspace such that A i (V i ) = 0. Hence, F (V ) = 0 and therefore F ɛ (V ) = ( ɛ)f + ɛa(v ) = 0. Assume by induction that F i ɛ (V i ) = 0 holds for all i < l, then we have that Fɛ(V l l ) = Fɛ l (( ɛ)f + ɛa)(v l ) Fɛ l (V l ) = 0 because A(V l ) V l and F (V l ) V l. But then we have for all l that rk F l ɛ dim V l = n (a a l ) = rk A l def = r l. Then for at least one r l r l submatrix of Fɛ l its determinant considered it as a polynomial of degree r l in ɛ is not identically zero (as it is nonzero for ɛ = ). But then this determinant is non-zero for all but finitely many ɛ. Hence, rk Fɛ l = rk A l for all l for all but finitely many ɛ. As these numbers determine the dual partition p of the type of A, F ɛ is a nilpotent n n matrix of type p for all but finitely many values of ɛ, proving the claim. But then, F 0 = F which we have proved to be a nilpotent matrix of type q belongs to the closure of the orbit O(A). That is, we have proved the difficult part of the Gerstenhaber-Hesselink theorem. Theorem 4.3 Let A be a nilpotent n n matrix of type p = (a,..., a n ) and B nilpotent of type q = (b,..., b n ). Then, B belongs to the closure of the orbit O(A), that is, B O(A) if and only if p > q in the domination order on partitions of n. To prove the theorem we only have to observe that if B is contained in the closure of A, then B l is contained in the closure of A l and hence rk A l rk B l (because rk A l < k is equivalent to vanishing of all determinants of k k minors which is a closed condition). But then, n l a i n i= for all l, that is, q > p and hence p > q. The other implication was proved above if we remember that the domination order was induced by the Young moves and clearly we have that if B O(C) and C O(A) then also B O(A). Example 4.4 Nilpotent matrices for small n. We will apply theorem 4.3 to describe the orbit-closures of nilpotent matrices of 8 8 matrices. The following table lists all partitions (and their dual in the other column) l i= The partitions of 8. b i a (8) v (,,,,,,,) b (7,) u (2,,,,,,) c (6,2) t (2,2,,,,) d (6,,) s (3,,,,,) e (5,3) r (2,2,2,,)

137 4.3. THE GERSTENHABER-HESSELINK THEOREM. 37 f (5,2,) q (3,2,,,) g (5,,,) p (4,,,,) h (4,4) o (2,2,2,2) i (4,3,) n (3,2,2,) j (4,2,2) m (3,3,,) k (3,3,2) k (3,3,2) l (4,2,,) l (4,2,,) The domination order between these partitions can be depicted as follows where all the Young moves are from left to right a b c e h k o r t d f g i j l m n p q Of course, from this graph we can read off the dominance order graphs for partitions of n 8. The trick is to identify a partition of n with that of 8 by throwing in a tail of ones and to look at the relative position of both partitions in the above picture. Using these conventions we get the following graph for partitions of 7 g m n s b d f l q p i and for partitions of 6 the dominance order is depicted as follows c g l m s t p q r r u t v u s v u v The dominance order on partitions of n 5 is a total ordering. The Gerstenhaber-Hesselink theorem can be applied to describe the module varieties of the algebras C[x] (x r ). C[x] Example 4.5 The representation variety rep n (x r ). Any algebra morphism C[x] Mn is determined by the image of x, whence rep n(c[x]) = M n. We have seen that conjugacy classes in M n are classified by the Jordan normalform. Let A is conjugated to a matrix in normalform 2 3 J 6 4 J 2... J s 7 5 where J i is a Jordan block of size d i, hence n = d + d d s. C[x]-module M determined by A can be decomposed uniquely as Then, the n-dimensional M = M M 2... M s where M i is a C[x]-module of dimension d i which is indecomposable, that is, cannot be decomposed as a direct sum of proper submodules. Now, consider the quotient algebra R = C[x]/(x r ), then the ideal I R (n) of C[x, x 2,..., x nn] is generated by the n 2 entries of the matrix 2 3r 6 4 x... x n.. x n... x nn For example if r = m = 2, then the ideal is generated by the entries of the matrix» 2» x x 2 x 2 = + x 2 x 3 x 2 (x + x 4 ) x 3 x 4 x 3 (x + x 4 ) x x 2x

138 38 CHAPTER 4. RECONSTRUCTING ALGEBRAS. That is, the ideal with generators I R = (x 2 + x 2 x 3, x 2 (x + x 4 ), x 3 (x + x 4 ), (x x 4 )(x + x 4 )) The variety V(I R ) M 2 consists of all matrices A such that A 2 = 0. Conjugating A to an upper triangular form we see that the eigenvalues of A must be zero, hence»» rep 2 C[x]/(x ) = O( ) O( ) and we have seen that this variety is a cone with top the zero matrix and defining equations V(x + x 4, x 2 + x 2 x 3 ) and we see that I R is properly contained in this ideal. Still, we have that rad(i R ) = (x + x 4, x 2 + x 3x 4 ) 3 for an easy computation shows that x + x 4 = 0 C[x, x 2, x 3, x 4 ]/I R. Therefore, even in the easiest of examples, the representation variety does not have to be reduced. For the general case, observe that when J is a Jordan block of size d with eigenvalue zero an easy calculation shows that d J d 0 = and J d.. = Therefore, we see that the representation variety rep n C[x]/(x r ) is the union of all conjugacy classes of matrices having 0 as only eigenvalue and all of which Jordan blocks have size r. Expressed in module theoretic terms, any n-dimensional R = C[x]/(x r )-module M is isomorphic to a direct sum of indecomposables M = I e I e Ir er where I j is the unique indecomposable j-dimensional R-module (corresponding to the Jordan block of size j). Of course, the multiplicities e i of the factors must satisfy the equation e + 2e 2 + 3e re r = n In M we can consider the subspaces for all i r M i = {m M x i.m = 0} the dimension of which can be computed knowing the powers of Jordan blocks (observe that the dimension of M i is equal to n rank(a i )) t i = dim C M i = e + 2e (i )e i + i(e i + e i e r) Observe that giving n and the r -tuple (t, t 2,..., t n ) is the same as giving the multiplicities e i because 2t = t 2 + e 2t 2 = t 3 + t + e 2 2t 3 = t 4 + t 2 + e 3 8><. 2t n 2 = t n + t n 3 + e n 2 2t n = n + t n 2 + e n >: n = t n + e n Let n-dimensional C[x]/(x r )-modules M and M (or associated matrices A and A ) be determined by the r -tuples (t,..., t r ) respectively (t,..., t r ) then we have that O(A ) O(A) if and only if t t, t 2 t 2,..., t r t r Therefore, we have an inverse order isomorphism between the orbits in rep n(c[x]/(x r )) and the r -tuples of natural numbers (t,..., t r ) satisfying the following linear inequalities (which follow from the above system) 2t t 2, 2t 2 t 3 + t, 2t 3 t 4 + t 2,..., 2t n n + t n 2, n t n 2. Let us apply this general result in a few easy cases. First, consider r = 2, then the orbits in rep n C[x]/(x 2 ) are parameterized by a natural number t satisfying the inequalities n t and 2t n, the multiplicities are given by e = 2t n and e 2 = n t. Moreover, the orbit of the module M(t ) lies in the closure of the orbit of M(t ) whenever t t.

139 4.4. THE HILBERT CRITERIUM. 39 That is, if n = 2k + δ with δ = 0 or, then rep n C[x]/(x 2 ) is the union of k + orbits and the orbitclosures form a linear order as follows (from big to small) I δ I k 2 I δ+2 I k 2... I n If r = 3, orbits in rep n C[x]/(x 3 ) are determined by couples of natural numbers (t, t 2 ) satisfying the following three linear inequalities 8 >< 2t t 2 2t 2 n + t >: n t 2 For example, for n = 8 we obtain the following situation 2t = t 2 2t 2 = 8 + t t 2 = 8 Therefore, rep 8 C[x]/(x 3 ) consists of 0 orbits with orbitclosure diagram as below (the nodes represent the multiplicities [e e 2 e 3 ]). [02] [202] [2] [040] [3] [50] [230] [420] [60] [800] Here we used the equalities e = 2t t 2, e 2 = 2t 2 n t and e 3 = n t 2. For general n and r this result shows that rep n C[x]/(x r ) is the closure of the orbit of the module with decomposition M gen = I e r I s if n = er + s 4.4 The Hilbert criterium. A one parameter subgroup of a linear algebraic group G is a morphism λ : C G

140 40 CHAPTER 4. RECONSTRUCTING ALGEBRAS. of affine algebraic groups. That is, λ is both a groupmorphism and a morphism of affine varieties. The set of all one parameter subgroup of G will be denoted by Y (G). If G is commutative algebraic group, then Y (G) is an Abelian group with additive notation λ + λ 2 : C G with (λ + λ 2 )(t) = λ (t).λ 2 (t) Recall that an n-dimensional torus is an affine algebraic group isomorphic to C... C }{{ = T } n n the closed subgroup of invertible diagonal matrices in GL n. Lemma 4.6 Y (T n ) Z n. The correspondence is given by assigning to (r,..., r n ) Z n the one-parameter subgroup λ : C Tn given by t (t r,..., t rn ) Proof. For any two affine algebraic groups G and H there is a canonical bijection Y (G H) = Y (G) Y (H) so it suffices to verify that Y (C ) Z with any λ : C C given by t t r for some r Z. This is obvious as λ induces the algebra morphism C[C ] = C[x, x ] λ C[x, x ] = C[C ] which is fully determined by the image of x which must be an invertible element. Now, any invertible element in C[x, x ] is homogeneous of the form cx r for some r Z and c C. The corresponding morphism maps t to ct r which is only a groupmorphism if it maps the identity element to so c =, finishing the proof. Proposition 4.7 Any one-parameter subgroup λ : C t r 0 t g.....g 0 t rn GLn is of the form for some g GL n and some n-tuple (r,..., r n ) Z n. Proof. Let H be the image under λ of the subgroup µ of roots of unity in C. We claim that there is a basechange matrix g GL n such that C 0 g.h.g... 0 C Assume h H not a scalar matrix, then h has a proper eigenspace decomposition V W = C n. We use that h l = n and hence all its Jordan blocks must have size one as for any λ 0 we have l λ 0 λ l lλ l =..... lλ l n λ λ l

141 4.4. THE HILBERT CRITERIUM. 4 Because H is commutative, both V and W are stable under H. By induction on n we may assume that the images of H in GL(V ) and GL(W ) are diagonalizable, but then the same holds in GL n. As µ is infinite, it is Zariski dense in C and because the diagonal matrices are Zariski closed in GL n we have g.λ(c ).g = g.h.g Tn and the result follows from the lemma above Let V be a general GL n -representation considered as an affine space with GL n - action, let X be a GL n -stable closed subvariety and consider a point x X. A one-parameter subgroup C λ GLn determines a morphism C λ x X defined by t λ(t).x Observe that the image λ x (C ) lies in the orbit GL n.x of x. Assume there is a continuous extension of this map to the whole of C. We claim that this extension must then be a morphism. If not, the induced algebra morphism C[X] λ x C[t, t ] does not have its image in C[t], so for some f C[Z] we have that λ x(f) = a 0 + a t a z t z t s with a 0 0 and s > 0 But then λ x(f)(t) ± when t goes to zero, that is, λ x cannot have a continuous extension, a contradiction. So, if a continuous extension exists there is morphism λ x : C X. Then, λ x (0) = y and we denote this by limλ(t).x = y t 0 Clearly, the point y X must belong to the orbitclosure GL n.x in the Zariski topology (or in the C-topology as orbits are constructible). Conversely, one might ask whether if y GL n.x we can always approach y via a one-parameter subgroup. The Hilbert criterium gives situations when this is indeed possible. The only ideals of the formal power series C[[t]] are principal and generated by t r for some r N +. With C((t)) we will denote the field of fractions of the domain C((t)). Lemma 4.8 Let V be a GL n -representation, v V and a point w V lying in the orbitclosure GL n.v. Then, there exists a matrix g with coefficients in the field C((t)) and det(g) 0 such that (g.v) t=0 is well defined and is equal to w Proof. Note that g.v is a vector with coordinates in the field C((t)). If all coordinates belong to C[[t]] we can set t = 0 in this vector and obtain a vector in V. It is this vector that we denote with (g.v) t=0. Consider the orbit map µ : GL n V defined by g g.v. As w GL n.v we have seen that there is an irreducible curve C GL n such that w µ(c).

142 42 CHAPTER 4. RECONSTRUCTING ALGEBRAS. We obtain a diagram of C-algebras C[GL n ] C[C] C(C) µ µ C[V ] C[µ(C)] C[C ] Here, C[C] is defined to be the integral closure of C[µ(C)] in the functionfield C(C) of C. Two things are important to note here : C µ(c) is finite, so surjective and take c C be a point lying over w µ(c). Further, C having an integrally closed coordinate ring is a complex manifold. Hence, by the implicit function theorem polynomial functions on C can be expressed in a neighborhood of c as power series in one variable, giving an embedding C[C ] C[[t]] with (t) C[C ] = M c. This inclusion extends to one on the level of their fields of fractions. That is, we have a diagram of C-algebra morphisms C[GL n ] C(C) = C(C ) C((t)) µ C[V ] C[µ(C)] C[C ] C[[t]] The upper composition defines an invertible matrix g(t) with coefficients in C((t)), its (i, j)-entry being the image of the coordinate function x ij C[GL n ]. Moreover, the inverse image of the maximal ideal (t) C[[t]] under the lower composition gives the maximal ideal M w C[V ]. This proves the claim. Lemma 4.9 Let g be an n n matrix with coefficients in C((t)) and det g 0. Then there exist u, u 2 GL n (C[[t]]) such that t r 0 g = u.....u 2 0 t rn with r i Z and r r 2... r n. Proof. By multiplying g with a suitable power of t we may assume that g = (g ij (t)) i,j M n (C[[t]]). If f(t) = i=0 f it i C[[t]] define v(f(t)) to be the minimal i such that a i 0. Let (i 0, j 0 ) be an entry where v(g ij (t)) attains a minimum, say r. That is, for all (i, j) we have g ij (t) = t r t r f(t) with r 0 and f(t) an invertible element of C[[t]]. By suitable row and column interchanges we can take the entry (i 0, j 0 ) to the (, )-position. Then, multiplying with a unit we can replace it by t r and by elementary row and column operations all the remaining entries in the first row and column can be made zero. That is, we have invertible matrices a, a 2 GL n (C[[t]]) such that [ t r 0 τ ] g = a..a 0 g 2 Repeating the same idea on the submatrix g and continuing gives the result. We can now state and prove the Hilbert criterium which allows us to study orbitclosures by one parameter subgroups.

143 4.4. THE HILBERT CRITERIUM. 43 Theorem 4.20 Let V be a GL n -representation and X V a closed GL n -stable subvariety. Let O(x) = GL n.x be the orbit of a point x X. Let Y O(x) be a closed GL n -stable subset. Then, there exists a one-parameter subgroup λ : C GL n such that limλ(t).x Y t 0 Proof. It suffices to prove the result for X = V. By lemma 4.8 there is an invertible matrix g M n (C((t))) such that (g.x) t=0 = y Y By lemma 4.9 we can find u, u 2 GL n (C[[t]]) such that t r 0 g = u.λ (t).u 2 with λ (t) =... 0 t rn a one-parameter subgroup. There exist x i V such that u 2.x = i=0 z it i in particular u 2 (0).x = x 0. But then, (λ (t).u 2.x) t=0 = (λ (t).x i t i ) t=0 i=0 = (λ (t).x 0 ) t=0 + (λ (t).x t) t= But one easily verifies (using a basis of eigenvectors of λ (t)) that lim (s).(λ (t)x s 0 λ i t i ) t=0 = { (λ (t).x 0 ) t=0 if i = 0, 0 if i 0 As (λ (t).u 2.x) t=0 Y and Y is a closed GL n -stable subset, we also have that lim s 0 λ (s).(λ (t).u 2.x) t=0 Y that is, (λ (t).x 0 ) t=0 Y But then, we have for the one-parameter subgroup λ(t) = u 2 (0).λ (t).u 2 (0) that limλ(t).x Y t 0 finishing the proof. An important special case occurs when x V belongs to the nullcone, that is, when the orbit closure O(x) contains the fixed point 0 V. The original Hilbert criterium asserts the following. Proposition 4.2 Let V be a GL n -representation and x V in the nullcone. Then, there is a one-parameter subgroup C λ GL n such that lim λ(t).x = 0 t 0 In the statement of theorem 4.20 it is important that Y is closed. In particular, it does not follow that any orbit O(y) O(x) can be reached via one-parameter subgroups. In the next section we will give an example of such a situation.

144 44 CHAPTER 4. RECONSTRUCTING ALGEBRAS. 4.5 Semisimple modules In this section we will characterize the closed GL n -orbits in the module variety rep n A for an affine C-algebra A. We have seen that any point ψ rep n A, that is any n-dimensional representation A ψ Mn determines an n-dimensional A- module which we will denote with M ψ. A finite filtration F on an n-dimensional module M is a sequence of A- submodules F : 0 = M t+ M t... M M 0 = M. The associated graded A-module is the n-dimensional module gr F M = t i=0m i /M i+. We have the following ringtheoretical interpretation of the action of one-parameter subgroups of GL n on the representation variety rep n A. Lemma 4.22 Let ψ, ρ rep n A. Equivalent are,. There is a one-parameter subgroup C λ GL n such that lim λ(t).ψ = ρ t 0 2. There is a finite filtration F on the A-module M ψ such that as A-modules. gr F M ψ M ρ Proof. () (2) : If V is any GL n -representation and C λ GLn a oneparameter subgroup, we have an induced weight space decomposition of V V = i V λ,i where V λ,i = {v V λ(t).v = t i v, t C }. In particular, we apply this to the underlying vectorspace of M ψ which is V = C n (column vectors) on which GL n acts by left multiplication. We define M j = i>j V λ,i and claim that this defines a finite filtration on M ψ with associated graded A-module M ρ. For any a A (it suffices to vary a over the generators of A) we can consider the linear maps φ ij (a) : V λ,i V = Mψ a. Mψ = V Vλ,j (that is, we express the action of a in a blockmatrix Φ a with respect to the decomposition of V ). Then, the action of a on the module corresponding to λ(t).ψ is given by the matrix Φ a = λ(t).φ a.λ(t) with corresponding blocks V λ,i φ ij(a) V λ,j λ(t) λ(t) V λ,i φ ij (a) V λ,j

145 4.5. SEMISIMPLE MODULES 45 that is φ ij (a) = tj i φ ij (a). Therefore, if lim t 0 λ(t).ψ exists we must have that φ ij (a) = 0 for all j < i. But then, the action by a sends any M k = i>k V λ,i to itself, that is, the M k are A-submodules of M ψ. Moreover, for j > i we have lim φ t 0 ij(a) = lim t j i φ ij (a) = 0 t 0 Consequently, the action of a on ρ is given by the diagonal blockmatrix with blocks φ ii (a), but this is precisely the action of a on V i = M i /M i, that is, ρ corresponds to the associated graded module. (2) () : Given a finite filtration on M ψ F : 0 = M t+... M M 0 = M ψ we have to find a one-parameter subgroup C λ GLn which induces the filtration F as in the first part of the proof. Clearly, there exist subspaces V i for 0 i t such that V = t i=0v i and M j = t j=iv i. Then we take λ to be defined by λ(t) = t i Id Vi for all i and verifies the claims. Example 4.23 Let M ψ we the 2-dimensional C[x]-module determined by the Jordan block and consider the canonical basevectors»»» λ 0 e 0 λ = e 0 2 = Then, Ce is a C[x]-submodule of M ψ and we have a filtration 0 = M 2 Ce = M Ce Ce 2 = M 0 = M ψ Using the conventions of the second part of the above proof we then have V = Ce V 2 = Ce 2 hence λ(t) =» t 0 0 Indeed, we then obtain that» t 0. 0» λ. 0 λ» t 0 = 0» λ t 0 λ and the limit t the diagonal matrix. 0 exists and is the associated graded module grf M ψ = M ρ determined by Consider two modules M ψ, M ψ rep n A. Assume that O(M ρ ) O(M ψ ) and that we can reach the orbit of M ρ via a one-parameter subgroup. Then, lemma 4.22 asserts that M ρ must be decomposable as it is the associated graded of a nontrivial filtration on M ψ. This gives us a criterium to construct examples showing that the closedness assumption in the formulation of Hilbert s criterium is essential. Example 4.24 (Nullcone of M 2 3 = M 3 M 3 ) In chapter 8 we will describe a method to work-out the nullcones of m-tuples of n n matrices. The special case of matrices has been worked out by H.P. Kraft in [4, p.202]. We depict

146 46 CHAPTER 4. RECONSTRUCTING ALGEBRAS. the orbits here and refer to chapter 8 for more details In this picture, each node corresponds to a torus. The right hand number is the dimension of the torus and the left hand number is the dimension of the orbit represented by a point in the torus. The solid or dashed lines indicate orbitclosures. For example, the dashed line corresponds to the following two points in M3 2 = M 3 M 3 2 ψ = ( , ) ρ = ( , ) We claim that M ρ is an indecomposable 3-dimensional module of C x, y. Indeed, the only subspace of the column vectors C 3 left invariant under both x and y is equal to C 05 0 hence M ρ cannot have a direct sum decomposition of two or more modules. Next, we claim that O(M ρ) O(M ψ ). Indeed, simultaneous conjugating ψ with the invertible matrix 2 4 ɛ we obtain the couple ( ɛ5, ) 0 0 ɛ and letting ɛ 0 we see that the limiting point is ρ. The Jordan-Hölder theorem, see for example [23, 2.6] asserts that any finite dimensional A-module M has a composition series, that is, M has a finite filtration F : 0 = M t+ M t... M M 0 = M such that the successive quotients S i = M i /M i+ are all simple A-modules for 0 i t. Moreover, these composition factors S and their multiplicities are independent of the chosen composition series, that is, the set {S 0,..., S t } is the same for every composition series. In particular, the associated graded module for a composition series is determined only up to isomorphism and is the semisimple n-dimensional module gr M = t i=0s i Theorem 4.25 Let A be an affine C-algebra and M rep n A.. The orbit O(M) is closed in rep n A if and only if M is an n-dimensional semisimple A-module. 2. The orbitclosure O(M) contains exactly one closed orbit, corresponding to the direct sum of the composition factors of M.

147 4.5. SEMISIMPLE MODULES The points of the quotient variety of rep n A under GL n parameterize the isomorphism classes of n-dimensional semisimple A-modules. We will denote the quotient variety by iss n A. Proof. () : Assume that the orbit O(M) is Zariski closed. Let gr M be the associated graded module for a composition series of M. From lemma 4.22 we know that O(gr M) is contained in O(M) = O(M). But then gr M M whence M is semisimple. Conversely, assume M is semisimple. We know that the orbitclosure O(M) contains a closed orbit, say O(N). By the Hilbert criterium we have a one-parameter subgroup C λ GL n such that lim λ(t).m = N N. t 0 By lemma 4.22 this means that there is a finite filtration F on M with associated graded module gr F M N. For the semisimple module M the only possible finite filtrations are such that each of the submodules is a direct sum of simple components, so gr F M M, whence M N and hence the orbit O(M) is closed. (2) : Remains only to prove uniqueness of the closed orbit in O(M). This either follows from the Jordan-Hölder theorem or, alternatively, from the separation property of the quotient map to be proved in the next section. (3) : We will prove in the next section that the points of a quotient variety parameterize the closed orbits. Example 4.26 Recall the description of the orbits in M2 2 = M 2 M 2 given in the previous chapter P C 5 H H S S and each fiber contains a unique closed orbit. The one over a point in H S corresponding to the matrix couple»» a 0 b 0 (, ) 0 a 2 0 b 2 which is indeed a semi-simple module of C x, y (the direct sum of teh two -dimensional simple representations determined by x a i and y b i. In case a = a 2 and b = b 2 then these two simples coincide and the semi-simple module having this factor with multiplicity two is the unique closed orbit in the fiber of a point in S. Example 4.27 Assume A is a finite dimensional C-algebra. Then, there are only a finite number, say k, of nonisomorphic n-dimensional semisimple A-modules. Hence iss n A is a finite number of k points, whence rep n A is the disjoint union of k connected components, each consisting of all n-dimensional A-modules with the same composition factors. Connectivity follows from the fact that the orbit of the sum of the composition factors lies in the closure of each orbit. Example 4.28 Let A be an affine commutative algebra with presentation A = C[x,..., x m]/i A and let X be the affine variety V(I A ). Observe that any simple A-module is onedimensional hence corresponds to a point in X. (Indeed, for any algebra A a simple k-dimensional module determines an epimorphism A Mk and M k is only commutative if k = ). Applying the Jordan-Hölder theorem we see that iss n A X (n) = X... X /S {z } n n

148 48 CHAPTER 4. RECONSTRUCTING ALGEBRAS. the n-th symmetric product of X. 4.6 Some invariant theory. The results in this section hold for any reductive algebraic group. As we will need them primarily for GL n (or products of GL ni ) we will prove them only in that case. Our first aim is to prove that GL n is a reductive group, that is, all GL n - representations are completely reducible. Consider the unitary group U n = {A GL n A.A = n } where A is the Hermitian transpose of A. Clearly, U n is a compact Lie group. Any compact Lie group has a so called Haar measure which allows one to integrate continuous complex valued functions over the group in an invariant way. That is, there is a linear function assigning to every continuous function f : U n C its integral f f(g)dg C U n which is normalized such that U n dg = and is left and right invariant, which means that for all u U n we have the equalities f(gu)dg = U n f(g)dg = U n f(ug)dg. U n This integral replaces the classical idea in representation theory of averaging functions over a finite group. Proposition 4.29 Every U n -representation is completely reducible. Proof. Take a finite dimensional complex vectorspace V with a U n -action and assume that W is a subspace of V left invariant under this action. Extending a basis of W to V we get a linear map V φ W which is the identity on W. For any v V we have a continuous map U n W g g.φ(g.v) (use that W is left invariant) and hence we can integrate it over U n (integrate the coordinate functions). Hence we can define a map φ 0 : V W by φ 0 (v) = g.φ(g.v)dg. U n Clearly, φ 0 is linear and is the identity on W. Moreover, φ 0 (u.v) = g.φ(g u.v)dg = u. u g.φ(g u.v)dg U n U n =u. gφ(g.v)dg = u.φ 0 (v) U n where the starred equality uses the invariance of the Haar measure. Hence, V = W Ker φ 0 is a decomposition as U n -representations. Continuing whenever one of the components has a nontrivial subrepresentation we arrive at a decomposition of V into simple U n -representations.

149 4.6. SOME INVARIANT THEORY. 49 We claim that for any n, U n is Zariski dense in GL n. Let D n be the group of all diagonal matrices in GL n. The Cartan decomposition for GL n asserts that GL n = U n.d n.u n For, take g GL n then g.g is an Hermitian matrix and hence diagonalizable by unitary matrices. So, there is a u U n such that u.g.g.u = α... α n = s.g.s. s.g.s }{{}}{{} p p Then, each α i > 0 R as α i = n j= p ij 2. Let β i = α i and let d the diagonal matrix diag(β,..., β n ). Clearly, Indeed, we have g = u.d.(d.u.g) and we claim v = d.u.g U n. v.v =(d.u.g).(g.u.d ) = d.(u.g.g.u).d =d.d 2.d = n proving the Cartan decomposition. Now, D n = C... C and D n U n = U... U and because U = µ is Zariski dense (being infinite) in D = C, we have that D n is contained in the Zariski closure of U n. By the Cartan decomposition we then have that the Zariski closure of U n is GL n. Theorem 4.30 GL n is a reductive group. completely reducible. That is, all GL n -representations are Proof. Let V be a GL n -representation having a subrepresentation W. In particular, V and W are U n -representations, so by the foregoing proposition we have a decomposition V = W W as U n -representations. Consider the subgroup N = N GLn (W ) = {g GL n g.w W } then N is a Zariski closed subgroup of GL n containing U n. As the Zariski closure of U n is GL n we have N = GL n and hence that W is a representation of GL n. Continuing gives a decomposition of V in simple GL n -representations. Let S = S GLn be the set of isomorphism classes of simple GL n -representations. If W is a simple GL n -representation belonging to the isomorphism class s S, we say that W is of type s and denote this by W s. Let X be a complex vectorspace (not necessarily finite dimensional) with a linear action of GL n. We say that the action is locally finite on X if, for any finite dimensional subspace Y of X, there exists a finite dimensional subspace Y Y X which is a GL n -representation. The isotypical component of X of type s S is defined to be the subspace X (s) = {W W X, W s}. If V is a GL n -representation, we have seen that V is completely reducible. Then, V = V (s) and every isotypical component V (s) W es for W s and some number e s. Clearly, e s 0 for only finitely many classes s S. We call the decomposition V = s S V (s) the isotypical decomposition of V and we say that the simple representation W s occurs with multiplicity e s in V.

150 50 CHAPTER 4. RECONSTRUCTING ALGEBRAS. If V is another GL n -representation and if V φ V is a morphism of GL n - representations (that is, a linear map commuting with the action), then for any s S we have that φ(v (s) ) V (s). If the action of GL n on X is locally finite, we can reduce to finite dimensional GL n -subrepresentation and obtain a decomposition X = s S X (s), which is again called the isotypical decomposition of X. Let V be a GL n -representation of some dimension m. Then, we can view V as an affine space C m and we have an induced action of GL n on the polynomial functions f C[V ] by the rule g. V V f C... g.f that is (g.f)(v) = f(g.v) for all g GL n and all v V. If C[V ] = C[x,..., x m ] is graded by giving all the x i degree one, then each of the homogeneous components of C[V ] is a finite dimensional GL n -representation. Hence, the action of GL n on C[V ] is locally finite. Indeed, let {y,..., y l } be a basis of a finite dimensional subspace Y C[V ] and let d be the maximum of the deg(y i ). Then Y = d i=0 C[V ] i is a GL n -representation containing Y. Therefore, we have an isotypical decomposition C[V ] = s S C[V ] (s). In particular, if 0 S denotes the isomorphism class of the trivial GL n -representation (C triv = Cx with g.x = x for every g GL n ) then we have C[V ] (0) = {f C[V ] g.f = f, g GL n } = C[V ] GLn the ring of polynomial invariants, that is, of polynomial functions which are constant along orbits in V. Lemma 4.3 Let V be a GL n -representation.. Let I C[V ] be a GL n -stable ideal, that is, g.i I for all g GL n, then (C[V ]/I) GLn C[V ] GLn /(I C[V ] GLn ). 2. Let J C[V ] GLn be an ideal, then we have a lying-over property J = JC[V ] C[V ] GLn. is Noetherian, that is, every increasing chain of ideals stabi- Hence, C[V ] GLn lizes. 3. Let I j be a family of GL n -stable ideals of C[V ], then ( j I j ) C[V ] GLn = j (I j C[V ] GLn ). Proof. () : As I has the induced GL n -action which is locally finite we have the isotypical decomposition I = I (s) and clearly I (s) = C[V ] (s) I. But then also, taking quotients we have s (C[V ]/I) (s) = C[V ]/I = s C[V ] (s) /I (s).

151 4.6. SOME INVARIANT THEORY. 5 Therefore, (C[V ]/I) (s) = C[V ] (s) /I (s) and taking the special case s = 0 is the statement. (2) : For any f C[V ] GLn left-multiplication by f in C[V ] commutes with the GL n -action, whence f.c[v ] (s) C[V ] (s). That is, C[V ] (s) is a C[V ] GLn -module. But then, as J C[V ] GLn we have s (JC[V ]) (s) = JC[V ] = s JC[V ] (s). Therefore, (JC[V ]) (s) = JC[V ] (s) and again taking the special value s = 0 we obtain JC[V ] C[V ] GLn = (JC[V ]) (0) = J. The Noetherian statement follows from the fact that C[V ] is Noetherian (the Hilbert basis theorem). (3) : For any j we have the decomposition I j = s (I j ) (s). But then, we have s ( j I j ) (s) = j I j = j s (I j ) (s) = s (I j ) (s). Therefore, ( j I j) (s) = j (I j) (s) and taking s = 0 gives the required statement. j Theorem 4.32 Let V be a GL n -representation. Then, the ring of polynomial invariants C[V ] GLn is an affine C-algebra. Proof. Because the action of GL n on C[V ] preserves the gradation, the ring of invariants is also graded C[V ] GLn = R = C R R From lemma 4.3(2) we know that C[V ] GLn is Noetherian and hence the ideal R + = R R 2... is finitely generated R + = Rf Rf l by homogeneous elements f,..., f l. We claim that as a C-algebra C[V ] GLn is generated by the f i. Indeed, we have R + = l i= Cf i + R+ 2 and then also R 2 + = l Cf i f j + R+ 3 i,j= and iterating this procedure we obtain for all powers m that R+ m = Cf m... f m l l + R+ m+. P mi=m Now, consider the subalgebra C[f,..., f l ] of R = C[V ] GLn, then we obtain for any power d > 0 that C[V ] GLn = C[f,..., f l ] + R d +. For any i we then have for the homogeneous components of degree i R i = C[f,..., f l ] i + (R d +) i. Now, if d > i we have that (R+) d i = 0 and hence that R i = C[f,..., f l ] i. As this holds for all i we proved the claim. Choose generating invariants f,..., f l of C[V ] GLn, consider the morphism V φ C l defined by v (f (v),..., f l (v))

152 52 CHAPTER 4. RECONSTRUCTING ALGEBRAS. and define W to be the Zariski closure φ(v ) in C l. Then, we have a diagram V φ C l π W π and an isomorphism C[W ] C[V ] GL n. More general, let X be a closed GL n - stable subvariety of V, then X = V V (I) for some GL n -stable ideal I of C[V ]. From lemma 4.3() we obtain C[X] GLn = (C[V ]/I) GLn = C[V ] GLn /(I C[V ] GLn ) whence C[X] GLn is also an affine algebra (and generated by the images of the f i ). Define Y to be the Zariski closure of φ(x) in C l, then we have a diagram X φ C l π Y and an isomorphism C[Y ] π C[X] GL n. We call the morphism X π Y an algebraic quotient of X under GL n. We will now prove some important properties of this quotient. Proposition 4.33 (universal property) If X µ Z is a morphism which is constant along GL n -orbits in X, then there exists a unique factoring morphism µ X µ π Y Z... Proof. As µ is constant along GL n -orbits in X, we have an inclusion µ (C[Z]) C[X] GLn. We have the commutative diagram C[X] C[X] GLn µ µ π C[Z]... µ C[Y ] from which the existence and uniqueness of µ follows.

153 4.6. SOME INVARIANT THEORY. 53 As a consequence, an algebraic quotient is uniquely determined up to isomorphism (that is, we might have started from other generating invariants and still obtain the same quotient variety up to isomorphism). Proposition 4.34 (onto property) The algebraic quotient X π Y is surjective. Moreover, if Z X is a closed GL n -stable subset, then π(z) is closed in Y and the morphism π X Z : Z π(z) is an algebraic quotient, that is, C[π(Z)] C[Z] GLn. Proof. Let y Y with maximal ideal M y C[Y ]. By lemma 4.3(2) we have M y C[X] C[X] and hence there is a maximal ideal M x of C[X] containing M y C[X], but then π(x) = y. Let Z = V X (I) for a G-stable ideal I of C[X], then π(z) = V Y (I C[Y ]). That is, C[π(Z)] = C[Y ]/(I C[Y ]). However, we have from lemma 4.3() that C[Y ]/(C[Y ] I) (C[X]/I) GLn = C[Z] GLn and hence C[π(Z)] = C[Z] GLn. Finally, surjectivity of π Z is proved as above. An immediate consequence is that the Zariski topology on Y is the quotient topology of that on X. For, take U Y with π (U) Zariski open in X. Then, X π (U) is a GL n -stable closed subset of X. Then, π(x π (U)) = Y U is Zariski closed in Y. π Y separates dis- Proposition 4.35 (separation property) The quotient X joint closed GL n -stable subvarieties of X. Proof. Let Z j be closed GL n -stable subvarieties of X with defining ideals Z j = V X (I j ). Then, j Z j = V X ( j I j). Applying lemma 4.3(3) we obtain π( j Z j ) = V Y (( j I j ) C[Y ]) = V Y ( j (I j C[Y ])) = j V Y (I j C[Y ]) = j π(z j ) The onto property implies that π(z j ) = π(z j ) from which the statement follows. It follows from the universal property that the quotient variety Y determined by the ring of polynomial invariants C[Y ] GLn is the best algebraic approximation to the orbit space problem. From the separation property a stronger fact follows. Proposition 4.36 The algebraic quotient X π Y is the best continuous approximation to the orbit space. That is, points of Y parameterize the closed GL n -orbits in X. In fact, every fiber π (y) contains exactly one closed orbit C and we have π (y) = {x X C GL n.x} Proof. The fiber F = π (y) is a GL n -stable closed subvariety of X. Take any orbit GL n.x F then either it is closed or contains in its closure an orbit of strictly smaller dimension. Induction on the dimension then shows that G.x contains a closed orbit C. On the other hand, assume that F contains two closed orbits, then they have to be disjoint contradicting the separation property.

154 54 CHAPTER 4. RECONSTRUCTING ALGEBRAS. 4.7 Cayley-Hamilton algebras. A trace map on an (affine) C-algebra A is a C-linear map tr : A A satisfying the following three properties for all a, b A :. tr(a)b = btr(a), 2. tr(ab) = tr(ba) and 3. tr(tr(a)b) = tr(a)tr(b). Note that it follows from the first property that the image tr(a) of the trace map is contained in the center of A. Consider two algebras A and B equipped with a trace map which we will denote by tr A respectively tr B. A trace morphism φ : A B will be a C-algebra morphism which is compatible with the trace maps, that is, the following diagram commutes A φ B tr A tr B A φ B This definition turns algebras with a trace map into a category. We will say that an algebra A with trace map tr is trace generated by a subset of elements I A if the C-algebra generated by B and tr(b) is equal to A where B is the C-subalgebra generated by the elements of I. Note that A does not have to be generated as a C-algebra by the elements from I. Observe that for T the formal trace t : T N T is a trace map. Property () follows because N commutes with all elements of T, property (2) is the cyclic permutation property for t and property (3) is the fact that t is a N -linear map. The formal trace algebra T is trace generated by the variables {x, x 2,..., x i,...} but not as a C-algebra. Actually, T is the free algebra in the generators {x, x 2,..., x i,...} in the category of algebras with a trace map. That is, if A is an algebra with trace tr which is trace generated by {a, a 2,...}, then there is a trace preserving algebra epimorphism T π A. For example, define π(x i ) = a i and π(t(x i... x il )) = tr(π(x i )... π(x il )). Also, the formal trace algebra T m, that is the subalgebra of T trace generated by {x,..., x m }, is the free algebra in the category of algebras with trace that are trace generated by at most m elements. Given a trace map tr on A, we can define for any a A a formal Cayley- Hamilton polynomial of degree n. Indeed, express f(t) = n (t λ i ) i= as a polynomial in t with coefficients polynomial functions in the Newton functions n i= λk i. Replacing the Newton function λ k i by tr(a k ) we obtain the Cayley- Hamilton polynomial of degree n χ (n) a (t) A[t].

155 4.7. CAYLEY-HAMILTON ALGEBRAS. 55 Definition 4.37 An (affine) C-algebra A with trace map tr : A A is said to be a Cayley-Hamilton algebra of degree n if the following two properties are satisfied :. tr() = n, and 2. For all a A we have χ (n) a (a) = 0 in A. Observe that if R is a commutative C-algebra, then M n (R) is a Cayley-Hamilton algebra of degree n. The corresponding trace map is the composition of the usual trace with the inclusion of R M n (R) via scalar matrices. As a consequence, the infinite trace algebra T n has a trace map induced by the natural inclusion T n M n (C[M n ]) tr tr. C[Mn ] N n which has image tr(t n ) the infinite necklace algebra N n. Clearly, being a trace preserving inclusion, T n is a Cayley-Hamilton algebra of degree n. With this definition, we have the following categorical description of the trace algebra T n. Theorem 4.38 The trace algebra T n is the free algebra in the generic matrix generators {X, X 2,..., X i,...} in the category of Cayley-Hamilton algebras of degree n. For any m, the trace algebra T m n is the free algebra in the generic matrix generators {X,..., X m } in the category of Cayley-Hamilton algebras of degree n which are trace generated by at most m elements. Proof. Let F n be the free algebra in the generators {y, y 2,...} in the category of Cayley-Hamilton algebras of degree n, then by freeness of T there is a trace preserving algebra epimorphism T π Fn with π(x i ) = y i. By the universal property of F n, the ideal Ker π is the minimal ideal I of T such that T /I is Cayley-Hamilton of degree n. We claim that Ker π is substitution invariant. Consider a substitution endomorphism φ of T and consider the diagram T φ T... χ. T /Ker χ π Fn then Ker χ is an ideal closed under traces such that T /Ker χ is a Cayley- Hamilton algebra of degree n (being a subalgebra of F n ). But then Ker π Ker χ (by minimality of Ker π) and therefore χ factors over F n, that is, the substitution endomorphism φ descends to an endomorphism φ : F n Fn meaning that Ker π is left invariant under φ, proving the claim. Further, any formal Cayley-Hamilton polynomial χ (n) x (x) of degree n of x T maps to zero under π. By substitution invariance it follows that the ideal of trace relations Ker τ Ker π. We have

156 56 CHAPTER 4. RECONSTRUCTING ALGEBRAS. seen that T /Ker τ = T n is the infinite trace algebra which is a Cayley-Hamilton algebra of degree n. Thus, by minimality of Ker π we must have Ker τ = Ker π and hence F n T n. The second assertion follows immediately. Let A be a Cayley-Hamilton algebra of degree n which is trace generated by the elements {a,..., a m }. We have a trace preserving algebra epimorphism p A defined by p(x i ) = a i T m n p a A tr pa A tr A T m n and hence a presentation A T m n /T A where T A T m n is the ideal of trace relations holding among the generators a i. We recall that T m n is the ring of GL n -equivariant polynomial maps Mn m f M n, that is, M n (C[Mn m ]) GLn = T m n where the action of GL n is the diagonal action on M n (C[M m n ]) = M n C[M m n ]. Observe that if R is a commutative algebra, then any twosided ideal I M n (R) is of the form M n (J) for an ideal J R. Indeed, the subsets J ij of (i, j) entries of elements of I is an ideal of R as can be seen by multiplication with scalar matrices. Moreover, by multiplying on both sides with permutation matrices one verifies that J ij = J kl for all i, j, k, l proving the claim. Applying this to the induced ideal M n (C[M m n ]) T A M n (C[M m n ]) M n (C[M m n ]) we find an ideal N A C[M m n ] such that M n (C[M m n ]) T A M n (C[M m n ]) = M n (N A ) Observe that both the induced ideal and N A are stable under the respective GL n - actions. Assume that V and W are two (not necessarily finite dimensional) C- vectorspaces with a locally finite GL n -action and that V f W is a linear map commuting with the GL n -action. Decomposing V and W in their isotypical components and recalling that V (0) = V GLn respectively W (0) = W GLn we obtain a commutative diagram V f W R f0 W G V GLn where R is the Reynolds operator, that is, the canonical projection to the isotypical component of the trivial representation. Clearly, the Reynolds operator commutes with the GL n -action. Moreover, using complete decomposability we see that f 0 is surjective (resp. injective) if f is surjective (resp. injective). R

157 4.7. CAYLEY-HAMILTON ALGEBRAS. 57 Because N A situation is a GL n -stable ideal of C[M m n ] we can apply the above in the M n (C[M m n ]) π Mn (C[M m n ]/N A ) R T m n π 0 R Mn (C[Mn m ]/N A ) GLn and the bottom map factorizes through A = T m n /T A giving a surjection A M n (C[M m n ]/N A ) GLn. In order to verify that this map is injective (and hence an isomorphism) it suffices to check that M n (C[M m n ]) T A M n (C[M m n ]) T m n = T A. Using the functoriality of the Reynolds operator with respect to multiplication in M n (C[Mn ]) with an element x T m n or with respect to the trace map (both commuting with the GL n -action) we deduce the following relations : For all x T m n and all z M n (C[M n ]) we have R(xz) = xr(z) and R(zx) = R(z)x. For all z M n (C[M n ]) we have R(tr(z)) = tr(r(z)). Assume that z = i t ix i n i M n (C[M m n ]) T A M n (C[M m n ]) T m n with m i, n i M n (C[M m n ]) and t i T A. Now, consider X m+ T n. Using the cyclic property of traces we have tr(zx m+ ) = i tr(m i t i n i X m+ ) = i tr(n i X m+ m i t i ) and if we apply the Reynolds operator to it we obtain the equality tr(zx m+ ) = tr( i R(n i X m+ m i )t i ) For any i, the term R(n i X m+ m i ) is invariant so belongs to T m+ n X m+. Knowing the generating elements of T m+ n we can write and is linear in R(n i X m+ m i ) = j s ij X m+ t ij + k tr(u ik X m+ )v ik with all of the elements s ij, t ij, u ik and v ik in T m n. Substituting this information and again using the cyclic property of traces we obtain tr(zx m+ ) = tr(( i,j,k s ij t ij t i + tr(v ik t i ))X m+ ) and by the nondegeneracy of the trace map we again deduce from this the equality z = i,j,k s ij t ij t i + tr(v ik t i ) Because t i T A and T A is stable under taking traces we deduce from this that z T A as required.

158 58 CHAPTER 4. RECONSTRUCTING ALGEBRAS. Because A = M n (C[Mn m ]/N A ) GLn operator to the setting we can apply functoriality of the Reynolds M n (C[Mn m ]/N A ) tr C[M n ]/N A Concluding we also have the equality R R tr A A (C[M n ]/N A ) GLn tr A (A) = (C[M m n ]/J A ) GLn. Summarizing, we have proved the following invariant theoretic reconstruction result for Cayley-Hamilton algebras. Theorem 4.39 Let A be a Cayley-Hamilton algebra of degree n, with trace map tr A, which is trace generated by at most m elements. Then, there is a canonical ideal N A C[M m n ] from which we can reconstruct the algebras A and tr A (A) as invariant algebras A = M n (C[M m n ]/N A ) GLn and tr A (A) = (C[M m n ]/N A ) GLn A direct consequence of the above proof is the universal property of the embedding A i A M n (C[M m n ]/N A ). Let R be a commutative C-algebra, then M n (R) with the usual trace is a Cayley- Hamilton algebra of degree n. If f : A M n (R) is a trace preserving morphism, we claim that there exists a natural algebra morphism f : C[Mn m ]/N A R such that the diagram A f Mn (R) i A M n (C[Mn m ]/N A ) M n(f)... where M n (f) is the algebra morphism defined entrywise. To see this, consider the composed trace preserving morphism φ : T m n A f Mn (R). Its image is fully determined by the images of the trace generators X k of T m n which are say m k = (m ij (k)) i,j. But then we have an algebra morphism C[Mn m ] g R defined by sending the variable x ij (k) to m ij (k). Clearly, T A Ker φ and after inducing to M n (C[Mn m ]) it follows that N A Ker g proving that g factors through C[Mn m ]/J A R. This morphism has the required universal property. 4.8 Geometric reconstruction In this section we will give a geometric interpretation of the reconstruction result. Again, let A be a Cayley-Hamilton algebra of degree n, with trace map tr A, which is generated by at most m elements a,..., a m. We will give a functorial interpretation to the affine scheme determined by the canonical ideal N A C[M m n ]. First, let us

159 4.8. GEOMETRIC RECONSTRUCTION 59 identify the reduced affine variety V(N A ). A point m = (m,..., m m ) V(N A ) determines an algebra map f m : C[Mn m ]/N A C and hence an algebra map φm φ m A... Mn (C) M n (C[Mn m ]/N A ) M n(f m) which is trace preserving. Conversely, from the universal property it follows that any trace preserving algebra morphism A M n (C) is of this form by considering the images of the trace generators a,..., a m of A. Alternatively, the points of V(N A ) parameterize n-dimensional trace preserving representations of A. That is, n-dimensional representations for which the morphism A M n (C) describing the action is trace preserving. For this reason we will denote the variety V(N A ) by rep tr n A and call it the trace preserving reduced representation variety of A. Assume that A is generated as a C-algebra by a,..., a m (observe that this is no restriction as trace affine algebras are affine) then clearly I A (n) N A. That is, Lemma 4.40 For A a Cayley-Hamilton algebra of degree n generated by {a,..., a m }, the reduced trace preserving representation variety rep tr n A rep n A is a closed subvariety of the reduced representation variety. It is easy to determine the additional defining equations. For, write any trace monomial out in the generators tr A (a i... a ik ) = α j...j l a j... a jl then for a point m = (m,..., m m ) rep n A to belong to rep tr n all the relations of the form A, it must satisfy tr(m i... m ik ) = α j...j l m j... m jl with tr the usual trace on M n (C). These relations define the closed subvariety rep tr n (A). Usually, this is a proper subvariety. Example 4.4 Let A be a finite dimensional semi-simple algebra A = M d (C)... M dk (C), then A has precisely k distinct simple modules {M,..., M k } of dimensions {d,..., d k }. Here, M i can be viewed as column vectors of size d i on which the component M di (C) acts by left multiplication and the other factors act as zero. Because A is semi-simple every n-dimensional A-representation M is isomorphic to M = M e... M e k k for certain multiplicities e i satisfying the numerical condition n = e d e k d k That is, rep n A is the disjoint union of a finite number of (closed) orbits each determined by an integral vector (e,..., e k ) satisfying the condition called the dimension vector of M. G rep n A GL n/(gl e... GL ek ) (e,...,e k )

160 60 CHAPTER 4. RECONSTRUCTING ALGEBRAS. Let f i be natural numbers such that n = f d +... f k d k and consider the embedding of A into M n(c) defined by a a {z } f (a,..., a k ) A... M n(c) f k z } { 2 3 a k a k Via this embedding, A becomes a Cayley-Hamilton algebra of degree n when equipped with the induced trace tr from M n(c). Let M be the n-dimensional A-representation with dimension vector (e,..., e k ) and choose a basis compatible with this decomposition. Let E i be the idempotent of A corresponding to the identity matrix I di of the i-th factor. Then, the trace of the matrix defining the action of E i on M is clearly e i d i.i n. On the other hand, tr(e i ) = f i d i.i n, hence the only trace preserving n- dimensional A-representation is that of dimension vector (f,..., f k ). Therefore, rep tr n A consists of the single closed orbit determined by the integral vector (f,..., f k ). rep tr n rep tr n A GLn/(GL f... GL fk ) Consider the scheme structure of the trace preserving representation variety A. The corresponding functor assigns to a commutative affine C-algebra R An algebra morphism ψ : C[Mn m ]/N A n n matrices with coefficients in R by rep tr n (R) = Alg C(C[M m n ]/N A, R). r k = Composing with the canonical embedding R determines uniquely an m-tuple of ψ(x (k))... ψ(x n (k)).. ψ(x n (k))... ψ(x nn (k)) φ A... M n (R) M n (C[Mn m ]/N A ) M n(ψ) determines the trace preserving algebra morphism φ : A M n (R) where the trace map on M n (R) is the usual trace. By the universal property any trace preserving map A M n (R) is also of this form. Lemma 4.42 Let A be a Cayley-Hamilton algebra of degree n which is generated by {a,..., a m }. The trace preserving representation variety rep tr A represents the n functor rep tr n A(R) = {A φ M n (R) φ is trace preserving } Moreover, rep tr n A is a closed subscheme of rep n A.

161 4.8. GEOMETRIC RECONSTRUCTION 6 Recall that there is an action of GL n on C[M m n ] and from the definition of the ideals I A (n) and N A it is clear that they are stable under the GL n -action. That is, there is an action by automorphisms on the quotient algebras C[M m n ]/I A (n) and C[M m n ]/N A. But then, their algebras of invariants are equal to { C[repn A] GLn = (C[M m n ]/I A (n)) GLn = C[rep tr n A]GLn = (C[M m n ]/N A ) GLn = N m n (I A (n) N m n ) N m n (N A N m n ) That is, these rings of invariants define closed subschemes of the affine (reduced) variety associated to the necklace algebra N m n. We will call these schemes the quotient schemes for the action of GL n and denote them respectively by iss n A = rep n A/GL n and iss tr n A = rep tr n A/GL n. We have seen that the geometric points of the reduced variety iss n A of the affine quotient scheme iss n A parameterize the isomorphism classes of n-dimensional semisimple A-representations. Similarly, the geometric points of the reduced variety iss tr n A of the quotient scheme iss tr n A parameterize isomorphism classes of trace preserving n-dimensional semisimple A-representations. Proposition 4.43 Let A be a Cayley-Hamilton algebra of degree n with trace map tr A. Then, we have that tr A (A) = C[iss tr n A], the coordinate ring of the quotient scheme iss tr n A. In particular, maximal ideals of tr A (A) parameterize the isomorphism classes of trace preserving n-dimensional semi-simple A-representations. By definition, a GL n -equivariant map between the affine GL n -schemes rep tr n A f Mn = M n means that for any commutative affine C-algebra R the corresponding map rep tr n f(r) A(R) Mn (R) commutes with the action of GL n (R). Alternatively, the ring of all morphisms rep tr A M n n is the matrixalgebra M n (C[Mn m ]/N A ) and those that commute with the GL n action are precisely the invariants. That is, we have the following description of A. Proposition 4.44 Let A be a Cayley-Hamilton algebra of degree n with trace map tr A. Then, we can recover A as the ring of GL n -equivariant maps of affine GL n -schemes. A = {f : rep tr n A M n equivariant } Summarizing the results of this and the previous section we have Theorem 4.45 The functor which assigns to a Cayley-Hamilton algebra A of degree n the GL n -affine scheme rep tr A of trace preserving n-dimensional representations has a left inverse. n This left inverse functor assigns to a GL n -affine scheme X its witness algebra M n (C[X]) GLn which is a Cayley-Hamilton algebra of degree n.

162 62 CHAPTER 4. RECONSTRUCTING ALGEBRAS. Note however that this functor is not an equivalence of categories. For, there are many affine GL n -schemes having the same witness algebra. Example 4.46 Consider the action of GL n on M n by conjugation and take a nilpotent matrix A. All eigenvalues of A are zero, so the conjugacy class of A is fully determined by the sizes of its Jordan blocks. These sizes determine a partition λ(a) = (λ, λ 2,..., λ k ) of n with λ λ 2... λ k. Moreover, we have given an algorithm to determine whether an orbit O(B) of another nilpotent matrix B is contained in the orbit closure O(A), the criterium being that O(B) O(A) λ(b) λ(a). where λ denotes the dual partition. We see that the witness algebra of O(A) is equal to M n(c[o(a)]) GLn = C[X]/(X k ) where k is the number of columns of the Young diagram λ(a). Hence, the orbit closures of nilpotent matrices such that their associated Young diagrams have equal number of columns have the same witness algebras. For example, if n = 4 then the closures of the orbits corresponding to and have the same witness algebra, although the closure of the second is a proper closed subscheme of the closure of the first. Recall the orbitclosure diagram of conjugacy classes of nilpotent 8 8 matrices given by the Gerstenhaber-Hesselink theorem. In the picture below, the closures of orbits corresponding to connected nodes of the same colour have the same witness algebra.

163 Chapter 5 Etale Slices. Let A be an affine C-algebra. In the foregoing chapter we have found a geometric reconstruction of the approximation at level n of A { A@ n M n (C[rep n A]) GLn t(a@ n ) C[rep n A] GLn = C[iss n A] In this chapter we will use the GL n -geometry to determine the étale local structure of the Cayley-Hamilton algebra A@ n. By this we mean the following. Let m be a maximal ideal of the central subalgebra t(a@ n ), then we want to determine the m-adic completion (A@ n ) m of A@ n. We know that m determines a point ξ in the quotient variety iss n A and so there is an n-dimensional semi-simple representation M ξ of A with decomposition M ξ = S e... S e k k where the S i are distinct simple A-representations of dimension d i and occurring in M ξ with multiplicity e i, in particular n = k i= d ie i. To determine the local structure of A@ n in m we determine the GL n -local structure of rep n A in a neighborhood of the closed orbit O(M ξ ). This can be done by the theory of Luna s étale slices, or rather by the elegant extension of it to not necessarily reduced varieties, due to F. Knop [?], whose proof we will outline in section 4. When rep n A is smooth in M ξ (which is always the case when A is Quillen-smooth) then this local structure is determined by the normal space to the orbit, considered as a module over the stabilizer subgroup. In the case of representation varieties, this normal space can be identified with the vectorspace of self-extensions Ext A (M ξ, M ξ ) and the stabilizer subgroup with the centralizer of V ξ. The main result we will prove in this chapter is that this local data is encoded in a quiver-setting, or rather a marked quiver-setting where we allow some loops in the quiver to acquire a mark. We will prove that this marked quiver has k vertices (corresponding to the distinct simple components of M ξ ) and we have to consider α-dimensional representations of this quiver where α = (e,..., e k ), the multiplicities with which these simples occur in M ξ. In the next chapter we will show that the arrows and loops in the quiver are determined by (trace preserving) self-extensions of M ξ. The étale local structure of A@ n in m is then given by (A@ n ) m T α Q ξ 63

164 64 CHAPTER 5. ETALE SLICES. where Q ξ is the local quiver determined by M ξ, T α Q ξ is the ring of GL(α)- equivariant maps from rep α Q ξ to M n (C) and we take its completion at the maximal graded ideal of the corresponding ring of invariants. 5. C slices. Let A be an affine C-algebra and ξ iss n A a point in the quotient space corresponding to an n-dimensional semi-simple representation M ξ of A. In this chapter we will present a method to study the étale local structure of iss n A near ξ and the étale local GL n -structure of the representation variety rep n A near the closed orbit O(M ξ ) = GL n.m ξ. In this section we will outline the basic idea in the setting of differential geometry. Let M be a compact C -manifold on which a compact Lie group G acts differentially. By a usual averaging process we can define a G-invariant Riemannian metric on M. For a point m M we define The G-orbit O(m) = G.m of m in M, the stabilizer subgroup H = Stab G (m) = {g G g.m = m} and the normal space N m defined to be the orthogonal complement to the tangent space in m to the orbit in the tangent space to M. That is, we have a decomposition of H-vectorspaces T m M = T m O(m) N m The normal spaces N x when x varies over the points of the orbit O(m) define a vectorbundle N p O(m) over the orbit. We can identify the bundle with the associated fiber bundle N G H N m Any point n N in the normal bundle determines a geodesic { γ n : R γ n (0) = p(n) M defined by dγ n dt (0) = n Using this geodesic we can define a G-equivariant exponential map from the normal bundle N to the manifold M via N exp M where exp(n) = γn () γn n O(m) x M Now, take ε > 0 and define the C slice S ε to be Nx S ε = {n N m n < ε }

165 5.2. TANGENT SPACES. 65 then G H S ε is a neighborhood of the zero section in the normal bundle N = G H N m. But then we have a G-equivariant exponential G H S ε exp M which for small enough ε gives a diffeomorphism with a G-stable tubular neighborhood U of the orbit O(m) in M. U Nm m ε 0 G/H exp O(m) M Nx ε If we assume moreover that the action of G on M and the action of H on N m are such that the orbit-spaces are manifolds M/G and N m /H, then we have the situation G H exp S ε U M S ε /H U/G M/G giving a local diffeomorphism between a neighborhood of 0 in N m /H and a neighborhood of the point m in M/G corresponding to the orbit O(m). Returning to the setting of the orbit O(M ξ ) in rep n A we would equally like to define a GL n -equivariant morphism from an associated fiber bundle GL n GL(α) N ξ e repn A where GL(ξ) is the stabilizer subgroup of M ξ and N ξ is a normal space to the orbit O(M ξ ). Because we do not have an exponential-map in the setting of algebraic geometry, the map e will have to be an étale map. Before we come to the description of these étale slices we will first study the tangent spaces to rep n A and give a ringtheoretical interpretation of the normal space N ξ. 5.2 Tangent spaces. Let X be a not necessarily reduced affine variety with coordinate ring C[X] = C[x,..., x n ]/I. If the origin o = (0,..., 0) V(I), elements of I have no constant terms and we can write any p I as p = p (i) with p (i) homogeneous of degree i. i=

166 66 CHAPTER 5. ETALE SLICES. The order ord(p) is the least integer r such that p (r) 0. Define the following two ideals in C[x,..., x n ] I l = {p () p I} and I m = {p (r) p I and ord(p) = r}. The subscripts l (respectively m) stand for linear terms (respectively, terms of minimal degree). The tangent space to X in o, T o (X) is by definition the subscheme of C n determined by I l. Observe that I l = (a x a n x n,..., a l x a ln x n ) for some l n matrix A = (a ij ) i,j of rank l. That is, we can express all x k as linear combinations of some {x i,..., x in l }, but then clearly C[T o (X)] = C[x,..., x n ]/I l = C[x i,..., x in l ] In particular, T o (X) is reduced and is a linear subspace of dimension n l in C n through the point o. Next, consider an arbitrary geometric point x of X with coordinates (a,..., a n ). We can translate x to the origin o and the translate of X is then the scheme defined by the ideal (f (x + a,..., x n + a n ),..., f k (x + a,..., x n + a n )) Now, the linear term of the translated polynomial f i (x + a,..., x n + a n ) is equal to f i (a,..., a n )x f i (a,..., a n )x n x x n and hence the tangent space to X in x, T x (X) is the linear subspace of C n defined by the set of zeroes of the linear terms T x (X) = V( n j= f x j (x)x j,..., n j= f k x j (x)x j ) C n. In particular, the dimension of this linear subspace can be computed from the Jacobian matrix in x associated with the polynomials (f,..., f k ) dim T x (X) = n rk f x (x).... f k x (x)... f x n (x).. f k x n (x) We now give an alternative description of the tangent spaces using the associated functor of X. Let C[ε] be the algebra of dual numbers, that is, C[ε] C[y]/(y 2 ). Consider a C-algebra morphism C[x,..., x n ] φ C[ε] defined by xi a i + c i ε. Because ε 2 = 0 it is easy to verify that the image of a polynomial f(x,..., x n ) under φ is of the form φ(f(x,..., x n )) = f(a,..., a n ) + n j= f x j (a,..., a n )c j ε Therefore, φ factors through I, that is φ(f i ) = 0 for all i k, if and only if (c,..., c n ) T x (X). Hence, we can also identify the tangent space to X in x

167 5.2. TANGENT SPACES. 67 with the algebra morphisms C[X] φ C[ε] whose composition with the projection π : C[ε] C (sending ε to zero) is the evaluation in x = (a,..., a n ). That is, let ev x X(C) be the point corresponding to evaluation in x, then T x (X) = {φ X(C[ε]) X(π)(φ) = ev x }. Example 5. GL n (C[ε]) is the group of invertible n n matrices with coefficients in C[ε]. By the above we have for any g GL n that T g(gl n ) = {m M n(c) g + mε is invertible in M n(c[ε]) } = M n(c) because (g + mε) = g g.m.g ε for any m M n(c). This computation is consistent with the observation that GL n is an open subset of M n. For any affine algebraic group scheme G one defines the Lie algebra g of G to be the tangentspace T e(g) at G in the neutral element e. In particular, the Lie algebra gl n of GL n is the vectorspace M n(c). The following two examples compute the tangent spaces to the (trace preserving) representation varieties. Example 5.2 Let A be an affine C-algebra generated by {a,... a m} and ρ : A Mn(C) an algebra morphism, that is, ρ rep n A. We call a linear map A D Mn(C) a ρ-derivation if and only if for all a, a A we have that D(aa ) = D(a).ρ(a ) + ρ(a).d(a ). We denote the vectorspace of all ρ-derivations of A by Der ρ(a). Observe that any ρ-derivation is determined by its image on the generators a i, hence Der ρ(a) Mn m. We claim that T ρ(rep n A) = Der ρ(a). Indeed, we know that rep n A(C[ε]) is the set of algebra morphisms A φ Mn(C[ε]) By the functorial characterization of tangentspaces we have that T ρ(rep n A) is equal to {D : A Mn(C) linear ρ + Dε : A Mn(C[ε]) is an algebra map}. Because ρ is an algebra morphism, the algebra map condition is equivalent to D being a ρ-derivation. ρ(aa ) + D(aa )ε = (ρ(a) + D(a)ε).(ρ(a ) + D(a )ε) Example 5.3 Let A be a Cayley-Hamilton algebra of degree n with trace map tr A and trace generated by {a,..., a m}. Let ρ rep tr n A, that is, ρ : A Mn(C) is a trace preserving algebra morphism. Because rep tr A(C[ε]) is the set of all trace preserving algebra morphisms n A Mn(C[ε]) (with the usual trace map tr on M n(c[ε])) and the previous example one verifies that T ρ(rep tr A) = Dertr n ρ (A) Der ρ(a) the subset of trace preserving ρ-derivations D, that is, those satisfying A D Mn(C) D tr A = tr D tr A A tr D Mn(C) Again, using this property and the fact that A is trace generated by {a,..., a m} a trace preserving ρ-derivation is determined by its image on the a i so is a subspace of M m n. The tangent cone to X in o, T C o (X), is by definition the subscheme of C n determined by I m, that is, C[T C o (X)] = C[x,..., x n ]/I m.

168 68 CHAPTER 5. ETALE SLICES. It is called a cone because if c is a point of the underlying variety of T C o (X), then the line l = oc is contained in this variety because I m is a graded ideal. Further, observe that as I l I m, the tangent cone is a closed subscheme of the tangent space at X in o. Again, if x is an arbitrary geometric point of X we define the tangent cone to X in x, T C x (X) as the tangent cone T C o (X ) where X is the translated scheme of X under the translation taking x to o. Both the tangent space and tangent cone contain local information of the scheme X in a neighborhood of x. We will now present a ringtheoretical description of both using only the local algebra O x (X) of X in x. These descriptions have the additional advantage of providing a description of tangent space and tangent cone independent of the embedding of X. Let m x be the maximal ideal of C[X] corresponding to x (that is, the ideal of polynomial functions vanishing in x). Then, its complement S x = C[X] m x is a multiplicatively closed subset the local algebra O x (X) is the corresponding localization C[X] Sx. It has a unique maximal ideal m x with residue field O x (X)/m x = C. We equip the local algebra O x = O x (X) with the m x -adic filtration that is the Z- filtration F x :... m i m i... m O x = O x =... = O x =... with associated graded algebra gr(o x ) =... mi x m i+ x mi x m i x... m x m 2 C x Proposition 5.4 If x is a geometric point of the affine scheme X, then. C[T x (X)] is isomorphic to the polynomial algebra C[ mx m ]. 2 x 2. C[T C x (X)] is isomorphic to the associated graded algebra gr(o x (X)). Proof. After translating we may assume that x = o lies in V(I) C n. That is, C[X] = C[x,..., x n ]/I and m x = (x,..., x n )/I. () : Under these identifications we have m x m 2 x m x m 2 x (x,...,x n) (x,...,x n) 2 +I (x,...,x n) (x,...,x n) 2 +I l and as I l is generated by linear terms it follows that the polynomial algebra on m x m is isomorphic to the quotient algebra C[x 2,..., x n ]/I l which is by definition the x coordinate ring of the tangent space. (2) : Again using the above identifications we have gr(o x ) i=0 i=0 i=0 i=0 m i x m i+ x m i x m i+ x (x,...,x n) i (x,...,x n) i+ +(I (x,...,x n) i ) (x,...,x n) i (x,...,x n) i+ +I m(i)

169 5.3. NORMAL SPACES. 69 where I m (i) is the homogeneous part of I m of degree i. On the other hand, the i-th homogeneous part of C[x,..., x n ]/I m is equal to (x,..., x n ) i (x,..., x n ) i+ + I m (i) we obtain the required isomorphism. This gives a third interpretation of the tangent space as T x (X) = Hom C ( m x m 2, C) = Hom C ( m x x m 2, C). x Hence, we can also view the tangent space T x (X) as the space of point derivations Der x (O x ) on O x (X) (or of the point derivations Der x (C[X]) on C[X]). That is, C-linear maps D : O x C (or D : C[X] C) such that for all functions f, g we have D(fg) = D(f)g(x) + f(x)d(g). If we define the local dimension of an affine scheme X in a geometric point x, dim x X to be the maximal dimension of irreducible components of the reduced variety X passing through x, then dim x X = dim o T C x (X). We say that X is nonsingular at x (or equivalently, that x is a nonsingular point of X) if the tangent cone to X in x coincides with the tangent space to X in x. An immediate consequence is Proposition 5.5 If X is nonsingular at x, then O x (X) is a domain. That is, in a Zariski neighborhood of x, X is an irreducible variety. Proof. If X is nonsingular at x, then gr(o x ) C[T C x (X)] = C[T x (X)] the latter one being a polynomial algebra whence a domain. Now, let 0 a, b O x then there exist k, l such that a m k m k+ and b m l m l+, that is a is a nonzero homogeneous element of gr(o x ) of degree k and b one of degree l. But then, a.b m k+l m k+l hence certainly a.b 0 in O x. Now, consider the natural map φ : C[X] Ox. Let {P,..., P l } be the minimal prime ideals of C[X]. All but one of them, say P = φ (0), extend to the whole ring O x. Taking the product of f functions f i P i nonvanishing in x for 2 i l gives the Zariski open set X(f) containing x and whose coordinate ring is a domain, whence X(f) is an affine irreducible variety. When restricting to nonsingular points we reduce to irreducible affine varieties. From the Jacobian condition it follows that nonsingularity is a Zariski open condition on X and by the implicit function theorem X is a complex manifold in a neighborhood of a nonsingular point. 5.3 Normal spaces. Let X φ Y be a morphism of affine varieties corresponding to the algebra morphism C[Y ] φ C[X]. Let x be a geometric point of X and y = φ(x). As

170 70 CHAPTER 5. ETALE SLICES. φ (m y ) m x, φ induces a linear map my m 2 y the differential of φ in x which is a linear map m x m 2 x dφ x : T x (X) T φ(x) (Y ). and taking the dual map gives Assume X a closed subscheme of C n and Y a closed subscheme of C m and let φ be determined by the m polynomials {f,..., f m } in C[x,..., x n ]. Then, the Jacobian matrix in x f x (x)... J x (φ) =. f x n (x)... f m x (x). f m x n (x) defines a linear map from C n to C m and the differential dφ x is the induced linear map from T x (X) C n to T φ(x) (Y ) C m. Let D T x (X) = Der x (C[X]) and x D the corresponding element of X(C[ε]) defined by x D (f) = f(x) + D(f)ε, then x D φ Y (C[ε]) is defined by x D φ (g) = g(φ(x)) + (D φ )ε = g(φ(x)) + dφ x (D)ε giving us the ε-interpretation of the differential for all v T x (X). φ(x + vε) = φ(x) + dφ x (v)ε Proposition 5.6 Let X φ Y be a dominant morphism between irreducible affine varieties. There is a Zariski open dense subset U X such that dφ x is surjective for all x U. Proof. We may assume that φ factorizes into X ρ Y C d φ pr Y with φ a finite and surjective morphism. Because the tangent space of a product is the sum of the tangent spaces of the components we have that d(pr W ) z is surjective for all z Y C d, hence it suffices to verify the claim for a finite morphism φ. That is, we may assume that S = C[Y ] is a finite module over R = C[X] and let L/K be the corresponding extension of the function fields. By the principal element theorem we know that L = K[s] for an element s L which is integral over R with minimal polynomial Y F = t n + g n t n g t + g 0 with g i R Consider the ring S = R[t]/(F ) then there is an element r R such that the localizations S r and S r are isomorphic. By restricting we may assume that X = V(F ) Y C and that X = V(F ) Y C φ pr Y Y

171 5.3. NORMAL SPACES. 7 Let x = (y, c) X then we have (again using the identification of the tangent space of a product with the sum of the tangent spaces of the components) that T x (X) = {(v, a) T y (Y ) C c F t (x) + vg n c n vg c + vg 0 = 0}. But then, dφ x i surjective whenever F t (x) 0. This condition determines a nonempty open subset of X as otherwise F t would belong to the defining ideal of X in C[Y C] (which is the principal ideal generated by F ) which is impossible by a degree argument Example 5.7 Let X be a closed GL n-stable subscheme of a GL n-representation V and x a geometric point of X. Consider the orbitclosure O(x) of x in V. Because the orbit map µ : GL n GLn.x O(x) is dominant we have that C[O(x)] C[GL n] and therefore a domain, so O(x) is an irreducible affine variety. We define the stabilizer subgroup Stab(x) to be the fiber µ (x), then Stab(x) is a closed subgroup of GL n. We claim that the differential of the orbit map in the identity matrix e = n dµ e : gl n Tx(X) satisfies the following properties Ker dµ e = stab(x) and Im dµ e = T x(o(x)). By the proposition we know that there is a dense open subset U of GL n such that dµ g is surjective for all g U. By GL n-equivariance of µ it follows that dµ g is surjective for all g GL n, in particular dµ e : gl n Tx(O(x)) is surjective. Further, all fibers of µ over O(x) have the same dimension. But then it follows from the dimension formula of proposition that dim GL n = dim Stab(x) + dim O(x) (which, incidentally gives us an algorithm to compute the dimensions of orbitclosures). Combining this with the above surjectivity, a dimension count proves that Ker dµ e = stab(x), the Lie algebra of Stab(x). Let M and N two A-representations of dimensions say m and n. An A- representation P of dimension m + n is said to be an extension of N by M if there exists a short exact sequence of left A-modules e : 0 M P N 0 We define an equivalence relation on extensions (P, e) of N by M : (P, e) = (P, e ) if and only if there is an isomorphism P φ P of left A-modules such that the diagram below is commutative e : 0 M P N 0 id M φ id N e : 0 M P N 0 The set of equivalence classes of extensions of N by M will be denoted by Ext A (N, M). An alternative description of Ext A (N, M) is as follows. Let ρ : A Mm and σ : A Mn be the representations defining M and N. For an extension (P, e) we can identify the C-vectorspace with M N and the A-module structure on P gives a algebra map µ : A M m+n and we can represent the action of a on P by left multiplication of the block-matrix [ ] ρ(a) λ(a) µ(a) =, 0 σ(a)

172 72 CHAPTER 5. ETALE SLICES. where λ(a) is an m n matrix and hence defines a linear map λ : A Hom C (N, M). The condition that µ is an algebra morphism is equivalent to the condition λ(aa ) = ρ(a)λ(a ) + λ(a)σ(a ) and we denote the set of all liner maps λ : A HomC (N, M) by Z(N, M) and call it the space of cycles. (Observe already that if M = N and m = n then Z(M, M) is the vectorspace of ρ-derivations Der ρ (A) from A M n.) The extensions of N by M corresponding to two cycles λ and λ from Z(N, M) are equivalent if and only if we have an A-module isomorphism in block form [ ] idm β with β Hom 0 id C (N, M) N between them. A-linearity of this map translates into the matrix relation [ ] [ ] [ ] [ ] idm β ρ(a) λ(a) ρ(a) λ. = (a) idm β. for all a A 0 id N 0 σ(a) 0 σ(a) 0 id N or equivalently, that λ(a) λ (a) = ρ(a)β βσ(a) for all a A. We will now define the subspace of Z(N, M) of boundaries B(N, M) {δ Hom C (N, M) β Hom C (N, M) : a A : δ(a) = ρ(a)β βσ(a)}. We then have the description Ext Z(N,M) A (N, M) = B(N,M). Example 5.8 Let A be an affine C-algebra generated by {a,..., a m} and ρ : A Mn(C) an algebra morphism, that is, ρ rep n A determines an n-dimensional A-representation M. We claim to have the following description of the normal space to the orbitclosure C ρ = O(ρ) of ρ N ρ(rep n A) def = Tρ(rep n A) = Ext A (M, M). T ρ(c ρ) We have already seen that the space of cycles Z(M, M) is the space of ρ-derivations of A in M n(c), Der ρ(a), which we know to be the tangent space T ρ(rep n A). Moreover, we know that the differential dµ e of the orbit map GL n µ Cρ M m n dµ e : gl n = M n Tρ(C ρ) is surjective. Now, ρ = (ρ(a ),..., ρ(a m)) Mn m and the action of action of GLn is given by simultaneous conjugation. But then we have for any A gl n = M n that (I n + Aε).ρ(a i ).(I n Aε) = ρ(a i ) + (Aρ(a i ) ρ(a i )A)ε. Therefore, by definition of the differential we have that dµ e(a)(a) = Aρ(a) ρ(a)a for all a A. that is, dµ e(a) B(M, M) and as the differential map is surjective we have T ρ(c ρ) = B(M, M) from which the claim follows. Example 5.9 Let A be a Cayley-Hamilton algebra with trace map tr A and trace generated by {a,..., a m}. Let ρ rep tr n A, that is, ρ : A Mn(C) is a trace preserving algebra morphism. Any cycle λ : A Mn(C) in Z(M, M) = Der ρ(a) determines an algebra morphism ρ + λε : A Mn(C[ε]) We know that the tangent space T ρ(rep tr n derivations, that is, those satisfying λ(tr A (a)) = tr(λ(a)) A) is the subspace Dertr ρ (A) of trace preserving ρ- for all a A

173 5.4. LUNA S ÉTALE SLICES. 73 Observe that for all boundaries δ B(M, M), that is, such that there is an m M n(c) with δ(a) = ρ(a).m m.ρ(a) are trace preserving as δ(tr A (a)) = ρ(tr A (a)).m m.ρ(tr A (a)) = tr(ρ(a)).m m.tr(ρ(a)) = 0 = tr(m.ρ(a) ρ(a).m) = tr(δ(a)) Hence, we can define the space of trace preserving self-extensions Ext tr Dertr ρ (A) A (M, M) = B(M, M) and obtain as before that the normal space to the orbit closure C ρ = O(ρ) is equal to N ρ(rep tr n def A) 5.4 Luna s étale slices. = Tρ(reptr n A) T ρ(c ρ) = Ext tr A (M, M) The results of this section hold for any reductive algebraic group G. As we will use them only in the case G = GL n or GL(α) = GL a... GL ak we will restrict to the case of GL n. Also all affine GL n -varieties we will consider are representation varieties or associated fiber bundles. We fix the setting : X and Y are not necessarily reduced affine GL n -varieties, ψ is a GL n -equivariant map x = ψ(y) X ψ Y y π X X/GL n and we assume the following restrictions : ψ is étale in y, π Y Y /GL n the GL n -orbits O(y) in Y and O(x) in X are closed. That is, in representation varieties we restrict to semi-simple representations, the stabilizer subgroups are equal Stab(x) = Stab(y). In the case of representation varieties, for a semi-simple n-dimensional representation with decomposition M = S e... S e k k into distinct simple components, this stabilizer subgroup is GL(α) = GL e (C f )... GLek (C f k ) GL n where f i = dim S i. In particular, the stabilizer subgroup is again reductive. In algebraic terms : consider the coordinate rings R = C[X] and S = C[Y ] and the dual morphism R ψ S. Let I R be the ideal describing O(x) and J S the ideal describing O(y). With R R we will denote the I-adic completion lim I of R n and with Ŝ the J-adic completion of S. Lemma 5.0 The morphism ψ induces for all n an isomorphism In particular, R Ŝ. R I n ψ S J n

174 74 CHAPTER 5. ETALE SLICES. Proof. Let Z be the closed GL n -stable subvariety of Y where ψ is not étale. By the separation property, there is an invariant function f S GLn vanishing on Z such that f(y) = because the two closed GL n -subschemes Z and O(y) are disjoint. Replacing S by S f we may assume that ψ is an étale morphism. Because O(x) is smooth, ψ O(x) is the disjoint union of its irreducible components and restricting Y if necessary we may assume that ψ O(x) = O(y). But then J = ψ (I)S and as O(y) O(x) we have R I S J so the result holds for n =. Because étale maps are flat, we have ψ (I n )S = I n R S = J n and an exact sequence But then we have 0 I n+ R S I n R S In I n+ R S 0 I n I n+ = In I n+ S R/I J = In I n+ R S J n J n+ and the result follows from induction on n and the commuting diagram 0 In I n+ R I n+ R I n 0 0 J n J n+. S J n+ S J n 0 As in the previous chapter we will denote for any irreducible GL n -representation s and any locally finite GL n -module X its s-isotypical component by X (s). Lemma 5. Let s be an irreducible GL n -representation. There are natural numbers m (independent of s) and n 0 such that for all k N we have I mk+n R (s) (I GL n ) k R (s) I k R (s) Proof. Consider A = i=0 In t n R[t], then A GLn is affine so certainly finitely generated as R GLn -algebra say by {r t m,..., r z t mz } with r i R and m i. Further, A (s) is a finitely generated A GLn -module, say generated by {s t n,..., s y t ny } with s i R (s) and n i 0. Take m = max m i and n = max n i and r I mk+n R (s), then rt mk+n A (s) and rt mk+n = j p j (r t m,..., r z t mz )s j t nj with p j a homogeneous polynomial of t-degree mk + n n j mk. But then each monomial in p j occurs at least with ordinary degree mk m = k and therefore is contained in (I GLn ) k R (s) t mk+n. Let R GLn be the I GLn -adic completion of the invariant ring R GLn and let ŜGLn be the J GLn -adic completion of S GLn.

175 5.4. LUNA S ÉTALE SLICES. 75 Lemma 5.2 The morphism ψ induces an isomorphism R R GLn R GLn S S GLnŜGLn Proof. Let s be an irreducible GL n -module, then the I GLn -adic completion of R (s) is equal to R (s) = R (s) R R GLn GLn. Moreover, R (s) = lim ( R I k ) (s) = lim R (s) (I k R (s) ) which is the I-adic completion of R (s). coincide on R (s) and therefore By the foregoing lemma both topologies R (s) = R (s) and similarly Ŝ (s) = Ŝ(s) Because R Ŝ it follows that R (s) foregoing holds for all s. Ŝ(s) from which the result follows as the Theorem 5.3 (Luna s fundamental lemma) Consider a GL n -equivariant map Y ψ X, y Y, x = ψ(y) and ψ étale in y. Assume that the orbits O(x) and O(y) are closed and that ψ is injective on O(y). Then, there is an affine open subset U Y containing y such that. U = π Y (π Y (U)) and π Y (U) = U/GL n. 2. ψ is étale on U with affine image. 3. The induced morphism U/GL n ψ X/GLn is étale. 4. The diagram below is commutative U ψ X π U U/GL n π X ψ X/GLn Proof. By the foregoing lemma we have R GLn which means that ψ is étale ŜGLn in π Y (y). As étaleness is an open condition, there is an open affine neighborhood V of π Y (y) on which ψ is étale. If R = R R GLn S GLn then the above lemma implies that R S GLn S ŜGLn S GLn ŜGLn Let S GLn loc be the local ring of S GLn in J GLn, then as the morphism S GLn loc is faithfully flat we deduce that Ŝ GLn R S GLn S GLn loc S S GLn S GLn loc but then there is an f S GLn J GLn such that R f S f. Now, intersect V with the open affine subset where f 0 and let U be the inverse image under π Y of this set. Remains to prove that the image of ψ is affine. As U ψ X is étale, its image is open and GL n -stable. By the separation property we can find an invariant h R GLn such that h is zero on the complement of the image and h(x) =. But then we take U to be the subset of U of points u such that h(u) 0.

176 76 CHAPTER 5. ETALE SLICES. Theorem 5.4 (Luna s slice theorem) Let X be an affine GL n -variety with quotient map X π X/GLn. Let x X be such that its orbit O(x) is closed and its stabilizer subgroup Stab(x) = H is reductive. Then, there is a locally closed affine subscheme S X containing x with the following properties. S is an affine H-variety, 2. the action map GL n S X induces an étale GLn -equivariant morphism GL n H S ψ X with affine image, 3. the induced quotient map ψ/gl n is étale 4. the diagram below is commutative (GL n H S)/GL n S/H ψ/gln X/GLn GL n H S ψ X S/H ψ/gl n π X/GLn If we assume moreover that X is smooth in x, then we can choose the slice S such that also the following properties are satisfied. S is smooth, 2. there is an H-equivariant morphism S φ Tx S = N x with φ(x) = 0 having an affine image, 3. the induced morphism is étale S/H φ/h Nx /H 4. the diagram below is commutative GL n H S GL n Hφ GLn H N x S/H φ/h Nx /H Proof. Choose a finite dimensional GL n -subrepresentation V of C[X] that generates the coordinate ring as algebra. This gives a GL n -equivariant embedding X i W = V

177 5.5. GROTHENDIECK SMOOTHNESS. 77 Choose in the vectorspace W an H-stable complement S 0 of gl n.i(x) = T i(x) O(x) and denote S = i(x)+s 0 and S 2 = i (S ).Then, the diagram below is commutative GL n H S 2 GLn H S ψ ψ 0 X i W By construction we have that ψ 0 induces an isomorphism between the tangent spaces in (, i(x)) GL n H S 0 and i(x) W which means that ψ 0 is étale in i(x), whence ψ is étale in (, x) GL n H S 2. By the fundamental lemma we ge an affine neighborhood U which must be of the form U = GL n H S giving a slice S with the required properties. Assume that X is smooth in x, then S is transversal to X in i(x) as T i(x) i(x) + S 0 = W Therefore, S is smooth in x. Again using the separation property we can find an invariant f C[S] H such that f is zero on the singularities of S (which is a H-stable closed subscheme) and f(x) =. Then replace S with its affine reduced subvariety of points s such that f(s) 0. Let m be the maximal ideal of C[S] in x, then we have an exact sequence of H-modules 0 m 2 m α N x 0 Choose a H-equivariant section φ : Nx m C[S] of α then this gives an H-equivariant morphism S φ N x which is étale in x. Applying again the fundamental lemma to this setting finishes the proof. 5.5 Grothendieck smoothness. In this section we prove that an affine variety X is smooth if and only if its coordinate ring C[X] satisfies a certain lifting property in the category of all commutative C-algebras. This allows us to define formally smooth algebras in other categories such as the category of Cayley-Hamilton algebras of degree n or the category of all C-algebras. Let X be a possibly non-reduced affine variety and x a geometric point of X. As we are interested in local properties of X near x, we may assume (after translation) that x = o in C n and that we have a presentation C[X] = C[x,..., x n ]/I with I = (f,..., f m ) and m x = (x,..., x n )/I. Denote the polynomial algebra P = C[x,..., x n ] and consider the map d : I (P dx... P dx n ) P C[X] = C[X]dx... C[X]dx n where the dx i are a formal basis of the free module of rank n and the map is defined by d(f) = ( f,..., f ) mod I. x x n This gives a C[X]-linear mapping I I 2 d C[X]dx... C[X]dx n.

178 78 CHAPTER 5. ETALE SLICES. Extending to the local algebra O x at x and then quotient out the maximal ideal m x we get a C = O x /m x - linear map I I 2 d(x) Cdx... Cdx n Clearly, x is a nonsingular point of X if and only if the C-linear map d(x) is injective. This is equivalent to the existence of a C-section and by the Nakayama lemma also to the existence of a O x -linear splitting s x of the induced O x -linear map d x I I 2 d x s x O x dx... O x dx n satisfying s x d x = id I I 2 A C-algebra epimorphism (between commutative algebras) R π S with square zero kernel is called an infinitesimal extension of S. It is called a trivial infinitesimal extension if π has an algebra section σ : S R satisfying π σ = id S. An infinitesimal extension R π S of S is said to be versal if for any other infinitesimal extension R π S of S there is a C-algebra morphism R... g π S π R making the diagram commute. From this universal property it is clear that versal infinitesimal extensions are uniquely determined up to isomorphism. Moreover, if a versal infinitesimal extension is trivial, then so is any infinitesimal extension. Definition 5.5 A commutative C-algebra S is said to be Grothendieck smooth if and only if it has the following universal property. Let T be a commutative C- algebra and I a nilpotent ideal of T. Then, any C-algebra morphism κ : S T/I T λ... T/I T making the diagram commu- can be lifted to a C-algebra morphism λ : S tative. S κ Clearly, by iterating, S is Grothendieck smooth if and only if it has the lifting property with respect to nilpotent ideals I with square zero. Therefore, assume we have a test object (T, I) with I 2 = 0, then we have a commuting diagram S T/I T pr S pr 2 κ T p T/I

179 5.5. GROTHENDIECK SMOOTHNESS. 79 where we define the pull-back algebra S T/I T = {(s, t) S T κ(s) = p(t)}. Observe that pr : S T/I T S is a C-algebra epimorphism with kernel 0 T/I I having square zero, that is, it is an infinitesimal extension of S. Moreover, the existence of a lifting λ of κ is equivalent to the existence of a C-algebra section σ : S S T/I T defined by s (s, λ(s)). Hence, S is Grothendieck smooth if and only if a versal infinitesimal extension of S is trivial. Returning to the situation of interest to us, we claim that the algebra epimorphism c x Ox O x (C n )/Ix 2 is a versal infinitesimal extension of O x. Indeed, consider any other infinitesimal extension R π Ox then we define a C-algebra morphism O x (C n )/Ix 2 R as follows : let r i R such that π(r i ) = c x (x i ) and define an algebra morphism C[x,..., x n ] R by sending the variable x i to r i. As the image of any polynomial non-vanishing in x is a unit in R, this algebra map extends to one from the local algebra O x (C n ) and it factors over O x (C n )/Ix 2 as the image of I x lies in the kernel of π which has square zero, proving the claim. Hence, O x is Grothendieck smooth if and only if there is a C-algebra section satisfying c x r x = id Ox. O x (C n )/Ix 2 c x r x O x Proposition 5.6 The affine scheme X is non-singular at the geometric point x if and only if the local algebra O x (X) is Grothendieck smooth. Proof. The result will follow once we prove that there is a natural one-to-one correspondence between O x -module splittings s x of d x and C-algebra sections r x of c x. This correspondence is given by assigning to an algebra section r x the map s x defined by s x (dx i ) = (x i r x c x (x i )) mod Ix 2 If X is an affine scheme which is smooth in all of its geometric points, then we have seen before that X = X must be reduced, that is, an affine variety. Restricting to its disjoint irreducible components we may assume that C[X] = x X O x. Clearly, if C[X] is Grothendieck smooth, so is any of the local algebras O x. Conversely, if all O x are Grothendieck smooth and C[X] = C[x,..., x n ]/I one knows that the algebra epimorphism C[x,..., x n ]/I 2 c C[X] has local sections in every x, but then there is an algebra section. Because c is clearly a versal infinitesimal deformation of C[X], it follows that C[X] is Grothendieck smooth. Proposition 5.7 Let X be an affine scheme. Then, C[X] is Grothendieck smooth if and only if X is non-singular in all of its geometric points. In this case, X is a reduced affine variety.

180 80 CHAPTER 5. ETALE SLICES. 5.6 Cayley smoothness. Observe that the commutative C-algebras are precisely the Cayley-Hamilton algebras of degree one, so we recover the notion of Grothendieck smoothness for commutative algebras from the following one. Definition 5.8 A Cayley-Hamilton algebra A of degree n with trace map tr A is said to be Cayley smooth if it satisfies the following lifting property. Let T be a Cayley-Hamilton algebra of degree n with trace map tr T and I a twosided nilpotent ideal of T such that tr T (I) I. Assume there is a trace preserving C- algebra morphism κ : A T/I, then there is a trace preserving C-algebra lift λ : A T A making the diagram commutative. T T/I λ... Let B be a Cayley-Hamilton algebra of degree n with trace map tr B and trace generated by m elements say {b,..., b m }. Then, we can write κ B = T m n /T B with T B closed under traces. Now, consider the extended ideal E B = M n (C[M m n ]).T B.M n (C[M m n ]) = M n (N B ) and we have seen that C[rep tr n result. B] = C[M m n ]/N B. We need the following technical Lemma 5.9 With notations as above, we have for all k that E kn2 B T m n T k B. Proof. Let T m n be the trace algebra on the generic n n matrices {X,..., X m } and T l+m n the trace algebra on the generic matrices {Y,..., Y l, X,..., X m }. Let {U,..., U l } be elements of T m n and consider the trace preserving map T l+m u n T m n induced by the map defined by sending Y i to U i. Then, by the universal property we have a commutative diagram of Reynold operators M n (C[M l+m n ]) ũ Mn (C[M m n ]) R T l+m n u T m n Now, let A,..., A l+ be elements from M n (C[Mn m ]), then we can calculate R(A U A 2 U 2 A 3... A l U l A l+ ) by first computing R r = R(A Y A 2 Y 2 A 3... A l Y l A l+ ) and then substituting the Y i with U i. The Reynolds operator preserves the degree in each of the generic matrices, therefore r will be linear in each of the Y i and is a.

181 5.6. CAYLEY SMOOTHNESS. 8 sum of trace algebra elements. By our knowledge of the generators of necklaces and the trace algebra we can write each term of the sum as an expression tr(m )tr(m 2 )... tr(m z )M z+ where each of the M i is a monomial of degree n 2 in the generic matrices {Y,..., Y l, X,..., X m }. Now, look at how the generic matrices Y i are distributed among the monomials M j. Each M j contains at most n 2 of the Y i s, hence the monomial M z+ contains at least l vn 2 of the Y i where v z is the number of M i with i z containing at least one Y j. Now, assume all the U i are taken from the ideal T B T m n which is closed under taking traces, then it follows that R(A U A 2 U 2 A 3... A l U l A l+ ) T v+(l vn2 ) B T k B if we take l = kn 2 as v + (k v)n 2 k. But this finishes the proof of the required inclusion. Let B be a Cayley-Hamilton algebra of degree n with trace map tr B and I a twosided ideal of B which is closed under taking traces. We will denote by E(I) the extended ideal with respect to the universal embedding, that is, E(I) = M n (C[rep tr n B])IM n(c[rep tr n B]). Then, for all powers k we have the inclusion E(I) kn2 B I k. Theorem 5.20 Let A be a Cayley-Hamilton algebra of degree n with trace map tr A. Then, A is Cayley smooth if and only if the trace preserving representation variety rep tr A is non-singular in all points (in particular, reptr A is reduced). n n Proof. Let A be Cayley smooth, then we have to show that C[rep tr A] is n Grothendieck smooth. Take a commutative test-object (T, I) with I nilpotent and an algebra map κ : C[rep tr A] T/I. Composing with the universal embedding n i A we obtain a trace preserving morphism µ 0 µ A... M n (T ) i A M n (C[rep tr n A]) µ 0 M n(κ) M n (T/I) Because M n (T ) with the usual trace is a Cayley-Hamilton algebra of degree n and M n (I) a trace stable ideal and A is Cayley smooth there is a trace preserving algebra map µ. But then, by the universal property of the embedding i A there exists a C- algebra morphism λ : C[rep tr n A] T such that M n (λ) completes the diagram. The morphism λ is the required lift. Conversely, assume that C[rep tr A] is Grothendieck smooth. Assume we have a Cayley-Hamilton algebra of degree n with trace map tr T and a trace-stable nilpotent ideal I of T and a trace preserving C-algebra map κ : A T/I. If we combine

182 82 CHAPTER 5. ETALE SLICES. this test-data with the universal embeddings we obtain a diagram i A A M n (C[rep tr n κ T T/I? λ... A]) i T /I... M n(α) i T M n (C[rep tr n T ]) M n (C[rep tr T/I]) =M n (C[rep tr T ]/J) Here, J = M n (C[rep tr T ])IM n n(c[rep tr T ]) and we know already that J T = I. n By the universal property of the embedding i A we obtain a C-algebra map n n C[rep tr A] α n C[rep tr T ]/J n which we would like to lift to C[rep tr T ]. This does not follow from Grothendieck n smoothness of C[rep tr A] as J is usually not nilpotent. However, as I is a nilpotent n ideal of T there is some h such that I h = 0. As I is closed under taking traces we know by the remark preceding the theorem that E(I) hn2 T I h = 0. Now, by definition E(I) = M n (C[rep tr T ])IM n n(c[rep tr T ]) which is equal to n M n (J). That is, the inclusion can be rephrased as M n (J) hn2 T = 0, whence there is a trace preserving embedding T M n (C[rep tr T ]/J hn2 ). Now, we have n the following situation i A A M n (C[rep tr n κ T T/I λ... A]) M n(α) M n (C[rep tr n T ]/J kn2 ) i T /I Mn (C[rep tr T ]/J)... This time we can lift α to a C-algebra morphism C[rep tr n A] C[rep tr n T ]/J hn2. This in turn gives us a trace preserving morphism A λ Mn (C[rep tr n T ]/J hn2 ) the image of which is contained in the algebra of GL n -invariants. Because T M n (C[rep tr T ]/J hn2 ) and by surjectivity of invariants under surjective n maps, the GL n -equivariants are equal to T, giving the required lift λ. n

183 5.7. QUILLEN SMOOTHNESS Quillen smoothness. In this section we introduce Quillen smooth algebras which are the basic building blocks to construct noncommutative manifolds. Definition 5.2 A C-algebra A is said to be Quillen smooth if it satisfies the following lifting property. Let T be a C-algebra and I T a nilpotent ideal. If there is a C-algebra morphism A κ T/I then there exists a C-algebra lift A λ T making the diagram commutative. T A T/I λ... This definition is rather restrictive. In particular, a commutative (Grothendieck) smooth algebra does not have to satisfy the lifting property in the category of all C- algebras. Example 5.22 consider the polynomial algebra C[x,..., x d ] and the 4-dimensional noncommutative local algebra T = C x, y (x 2, y 2 = C Cx Cy Cxy, xy + yx) Consider the one-dimensional nilpotent ideal I = C(xy yx) of T, then the 3-dimensional quotient T I is commutative and we have a morphism C[x φ,..., x d ] T I by x x, x 2 y and x i 0 for i 2. This morphism admits no lift to T as for any potential lift the commutator κ [ φ(x), φ(y)] 0 in T. Therefore, C[x,..., x d ] can only be Quillen smooth if d =. In fact, we will see in chapter 9 that the only commutative affine Quillen smooth algebras are the coordinate rings of a disjoint union of points and affine smooth curves. Still, the world of Quillen smooth algebras is rather exotic containing many algebras determined by universal constructions. There is a fairly innocent class of Quillen smooth algebras determined by combinatorial data : path algebras of quivers. Consider the commutative C-algebra C k = C[e,..., e k ]/(e 2 i e i, e i e j, k e i ). C k is the universal C-algebra in which is decomposed into k orthogonal idempotents, that is, if R is any C-algebra such that = r r k with r i R idempotents satisfying r i r j 0, then there is an embedding C k R sending e i to r i. Observe that as a C-algebra C k C... C }{{} k or equivalently, the coordinate ring of k distinct points. Proposition 5.23 C k is Quillen smooth. That is, if I be a nilpotent ideal of a C- algebra T and if = e +...+e k is a decomposition of into orthogonal idempotents e i T/I. Then, we can lift this decomposition to = e e k for orthogonal idempotents e i T such that π(e i ) = e i where T π T/I is the canonical projection. i=

184 84 CHAPTER 5. ETALE SLICES. Proof. as Assume that I l = 0, clearly any element i with i I is invertible in T ( i)( + i + i i l ) = i l =. If e is an idempotent of T/I and x T such that π(x) = e. Then, x x 2 I whence ( ) 0 = (x x 2 ) l = x l lx l+ l + x l ( ) l x 2l 2 ( l and therefore x l = ax l+ where a = l x ( ) 2) l x l and so ax = xa. If we take e = (ax) l, then e is an idempotent in T as e 2 = (ax) 2l = a l (a l x 2l ) = a l x l = e the next to last equality follows from x l = ax l+ = a 2 x l+2 =... = a l x 2l. Moreover, π(e) = π(a) l π(x) l = π(a) l π(x) 2l = π(a l x 2l ) = π(x) l = e. If f is another idempotent in T/I such that ef = 0 = fe then as above we can lift f to an idempotent f of T. As f e I we can form the element f = ( e)( f e) f ( f e). Because f ( f e) = f ( e) one verifies that f is idempotent, π(f) = f and e.f = 0 = f.e. Assume by induction that we have already lifted the pairwise orthogonal idempotents e,..., e k to pairwise orthogonal idempotents e,..., e k of R, then e = e e k is an idempotent of T such that ee k = 0 = e k e. Hence, we can lift e k to an idempotent e k T such that ee k = 0 = e k e. But then also e i e k = (e i e)e k = 0 = e k (ee i ) = e k e i. Finally, as e e k = i I we have that e e k = (e e k ) l = i l = 0 finishing the proof. Let Q be a quiver, that is, a directed graph determined by a finite set Q v = {v,..., v k } of vertices, and a finite set Q a = {a,..., a l } of arrows where we allow multiple arrows between vertices and loops in vertices. For now, we will depict vertex v i by i and an arrow a from vertex v i to v j by a j. i Note however than once we come to dimension vectors, we will encircle the vector components and will indicate the ordering of the vertices by subscripts when necessary. The path algebra CQ has as underlying C-vectorspace basis the set of all oriented paths in Q, including those of length zero corresponding to the vertices v i. Multiplication in CQ is induced by (left) concatenation of paths. More precisely, = v v k is a decomposition of into mutually orthogonal idempotents and further we define v j.a is always zero unless a.v i is always zero unless a j in which case it is the path a, a in i which case it is the path a,

185 5.7. QUILLEN SMOOTHNESS. 85 a i.a j is always zero unless path a i a j. a i a j in which case it is the Proposition 5.24 For any quiver Q, the path algebra CQ is Quillen smooth. Proof. Take an algebra T with a nilpotent twosided ideal I T and consider T T I...? φ φ The decomposition = φ(v ) φ(v k ) into mutually orthogonal idempotents in T I can be lifted up the nilpotent ideal I to a decomposition = φ(v ) φ(v k ) into mutually orthogonal idempotents in T. But then, taking for every arrow a CQ a j an i arbitrary element φ(a) φ(vj )(φ(a) + I) φ(v i ) gives a required lifted algebra morphism CQ φ T. A Quillen smooth algebra A determines for every integer n a Cayley smooth algebra A@ n. Let alg tr be the category of all C-algebras equipped with a trace map and with trace preserving morphisms. The forgetful functor alg tr alg has a left adjoint alg τ alg tr that is, given an algebra A we construct an algebra A τ by formally adjoining traces (as in the case of T given before). If tr : A τ A τ is the trace map on A τ we define for given n a Cayley-Hamilton algebra A@ n to be the quotient A@ n = A τ (tr() n, χ (n) a (a) a A) In general it may happen that A@ n = 0 for example if A has no n-dimensional representations. The characteristic feature of A@ n is that any C-algebra map A B with B a Cayley-Hamilton algebra of degree n factors through A@ n A can φ A@ n B φ n... with φ n a trace preserving algebra morphism. From this universal property the next result is immediate Proposition 5.25 If A is Quillen smooth, then for every integer n, the Cayley- Hamilton algebra of degree n, A@ n, is Cayley smooth. Moreover, is a smooth affine variety. rep n A rep tr n A@ n

186 86 CHAPTER 5. ETALE SLICES. This result allows us to study Quillen smooth algebras. We know that the algebra n is given by the GL n -equivariant maps from rep tr A@ n n to M n (C). Because this representation variety is smooth we will apply the Luna étale slices to determine the local structure of the GL n -variety rep tr A@ n n and hence of A@ n. The major result we will prove in a moment is that this local structure is fully determined by a quiver situation. That is, the local study of arbitrary Quillen smooth algebras can be reduced to that of the better understood subclass of path algebras of quivers. 5.8 Local structure. Let A be an affine C-algebra generated by m elements {a,..., a m }. The Cayley- Hamilton algebra of A@ n of degree n is then trace generated by m elements, that is, there is a trace preserving epimorphism T m ψ n A@n. That is, we have a GL n -equivariant closed embedding of affine schemes rep n A = rep tr n A@ n ψ rep tr n Tm n = M m n Take a point ξ of the quotient scheme iss n A = rep tr A@ n n/gl n. We know that ξ determines the isomorphism class of a semi-simple n-dimensional representation of A, say M ξ = S e... S e k k where the S i are distinct simple A-representations, say of dimension d i and occurring in M ξ with multiplicity e i. These numbers determine the representation type τ(ξ) of ξ (or of the semi-simple representation M ξ ), that is τ(ξ) = (e, d ; e 2, d 2 ;... ; e k, d k ) Choosing a basis of M ξ adapted to this decomposition we find a point x = (X,..., X m ) in the orbit O(M ξ ) Mn m such that each n n matrix X i is of the form m (i) e m (i) X i = 2 e m (i) k ek where each m (i) j M dj (C). Using this description we can compute the stabilizer subgroup Stab(x) of GL n consisting of those invertible matrices g GL n commuting with every X i. That is, Stab(x) is the multiplicative group of units of the centralizer of the algebra generated by the X i, that is M d (C) e M d2 (C) e M dk (C) ek It is easy to verify that this group is isomorphic to Stab(x) GL e GL e2... GL ek with the embedding Stab(x) GL n given by GL e (C d ) GL e2 (C d2 ) GL ek (C dk )

187 5.8. LOCAL STRUCTURE. 87 Clearly, a different choice of point in the orbit O(M ξ ) gives a subgroup of GL n conjugated to Stab(x). Consider the vector α = (e, e 2,..., e k ), then we see that Stab(x) GL(α) = GL e GL e2... GL ek We will compute the normal space Nx big to the orbit O(M xi ) in Mn m = rep tr n Tm n. This is an elaborate book-keeping operation involving GL(α)-representations. As x = (X,..., X m ) the tangent space T x O(M xi ) in Mn m to the orbit is equal to the image of the linear map gl n = M n Mn... M n = T x Mn m A ([A, X ],..., [A, X m ]) Observe that the kernel of this map is the centralizer of the subalgebra generated by the X i, so we have an exact sequence of Stab(x) = GL(α)-modules 0 gl(α) = Lie GL(α) gln = M n Tx O(x) 0 As GL(α) is a reductive group every GL(α)-module is completely reducible and so the sequence splits. But then, the normal space in M m n = T x M m n to the orbit is isomorphic as GL(α)-module to N big x = M n... M }{{ n gl(α) } m with the action of GL(α) (embedded as above in GL n ) is given by simultaneous conjugation. If we consider the GL(α)-action on M n d d 2 } {{ }}{{} d d 2 we see that it decomposes into a direct sum of subrepresentations for each i k we have d 2 i copies of the GL(α)-module M e i on which GL ei acts by conjugation and the other factors of GL(α) act trivially, for all i, j k we have d i d j copies of the GL(α)-module M ei e j on which GL ei GL ej acts via g.m = g i mg j and the other factors of GL(α) act trivially. These GL(α) components are precisely the modules appearing in representation spaces of quivers, which are defined in chapter or more precisely in chapter 6. Theorem 5.26 Let ξ be of representation type τ = (e, d ;... ; e k, d k ) and let α = (e,..., e k ). Then, the GL(α)-module structure of the normal space Nx big in rep tr n Tm n = Mn m to the orbit of the semi-simple n-dimensional representation O(M ξ ) is isomorphic to rep α Q ξ

188 88 CHAPTER 5. ETALE SLICES. where the quiver Q ξ has k vertices and the subquiver on any two vertices v i, v j for i, j k has the following shape (m )d2 i + (m )d i d j e i e j (m )d 2 j + (m )d i d j That is, in each vertex v i there are (m )d 2 i + -loops and there are (m )d id j arrows from vertex v i to vertex v j for all i, j k. Example 5.27 If m = 2 and n = 3 and the representation type is τ = (, ;, ;, ) (that is, M ξ is the direct sum of three distinct one-dimensional simple representations) then the quiver Q ξ is We say that A is smooth at ξ iss n A if the representation variety rep tr A@ n n is smooth in M ξ. Before we can apply the Luna slice theorem we have to control the normal space Nx sm to the orbit O(M ξ ) in rep tr A@ n n. We have GL n -equivariant embeddings O(M ξ ) rep tr A@ n n rep tr n Tm n = Mn m and corresponding embeddings of the tangent spaces in x T x O(M ξ ) T x rep tr n A@ n T x M m n Because GL(α) is reductive we then obtain for the normal spaces to the orbit N x big O(M ξ ) x rep A n N sm x = T x rep tr A@ n n T x O(M ξ ) N sm x N big x = T x M m n T x O(M ξ )

189 5.8. LOCAL STRUCTURE. 89 a direct summand as GL(α)-modules. As we know the isotypical decomposition of Nx big as the GL(α)-module rep α Q ξ this allows us to control Nx sm. We only have to observe that arrows in Q ξ correspond to simple GL(α)-modules, whereas a loop at vertex v i decomposes as GL(α)-module into the simples M ei = M 0 e i C triv where C triv is the one-dimensional simple with trivial GL(α)-action and Me 0 i is the space of trace zero matrices in M ei. Again, we can represent the GL(α)-module structure of Nx sm graphically, this time by a marked quiver using the dictionary a loop at vertex v i corresponds to the GL(α)-module M ei on which GL ei acts by conjugation and the other factors act trivially, a marked loop at vertex v i corresponds to the simple GL(α)-module Me 0 i which GL ei acts by conjugation and the other factors act trivially, on an arrow from vertex v i to vertex v j corresponds to the simple GL(α)-module M ei e j on which GL ei GL ej acts via g.m = g i mg j and the other factors act trivially, Theorem 5.28 Let ξ be such that M ξ is a smooth point on rep n A of representation type τ = (e, d ;... ; e k, d k ) and let α = (e,..., e k ). Then, the GL(α)-module structure of the normal space N x to the orbit is isomorphic to the GL(α)-module of representations of the marked quiver rep α Q ξ on k vertices {v,..., v k } such that the marked subquiver on any two vertices v i, v j with i, j k has the form a ii m ii e i a ij e j a jj a ji m jj where these numbers satisfy a ij (m )d i d j and a ii + m ii (m )d 2 i +. Under the assumptions of the theorem, the étale slice result enables us with a slice S φ x N sm x and a commutative diagram GL n GL(α) N sm x GL n GL(α) φ GL n GL(α) S x S x /GL(α) ψ rep n A N sm x /GL(α) φ/gl(α) ψ/gl n iss n A

190 90 CHAPTER 5. ETALE SLICES. where the vertical maps are the quotient maps, all diagonal maps are étale and the upper ones are GL n -equivariant. Hence, the GL n -local structure of the representation variety rep n A = rep tr A@ n n in a neighborhood of the orbit of x is the same as that of the associated fiber bundle GL n GL(α) Nx sm in a neighborhood of the orbit of ( n, 0). Further, the local structure of the quotient scheme iss n A in a neighborhood of ξ is the same as that of the quotient variety of the marked quiver representations Nx sm /GL(α) in a neighborhood of the trivial representation 0. Let m C[iss n A] be the maximal ideal corresponding to ξ. As the ring of polynomial invariants C[iss n A] is via the diagonal embedding a central subalgebra of the ring of GL n -equivariant maps from rep n A to M n (C) we can localize this ring of equivariant maps A@ n at m and also take its m-adic completion which we denote by (A@ n ) m. Let m 0 be the maximal ideal of the ring of GL(α)-polynomial invariants of the marked quiver representation space C[rep α Q ξ ]GL(α) = C[Nx sm /GL(α)]. Let T α Q ξ denote the ring of GL(α)-equivariant maps from rep α Q ξ to M n(c) and denote the m 0 -adic filtration of it with T α Q ξ m0. The above diagram then implies Theorem 5.29 Let ξ correspond to a semi-simple n-dimensional representation of A such that the representation variety is smooth along this closed orbit. Then, with notations as before we have an isomorphism of complete local algebras (A@ n ) m T α Q ξ m0 In the following sections we will determine the algebra structure of T α Q ξ. 5.9 Finite dimensional algebras. Let A be a Cayley-Hamilton algebra of degree n wit trace map tr, then we can define a norm-map on A by defining N(a) = σ n (a) for all a A. Recall that the elementary symmetric function σ n is a polynomial function f(t, t 2,..., t n ) in the Newton functions t i = n j= xi j. Then, σ(a) = f(tr(a), tr(a 2 ),..., tr(a n )). Because, we have a trace preserving embedding A M n (C[rep tr A]) and the norm map N coincides with the determinant in n this matrix-algebra, we have that N() = and a, b A N(ab) = N(a)N(b). Furthermore, the norm-map extends to a polynomial map on A[t] and we have that χ (n) a (t) = N(t a), in particular we can obtain the trace by polarization of the norm map. Consider a finite dimensional semi-simple C-algebra A = M d (C)... M dk (C), then all the Cayley-Hamilton structures of degree n on A with trace values in C are given by the following result. Lemma 5.30 Let A be a semi-simple algebra as above and tr a trace map on A making it into a Cayley-Hamilton algebra of degree n. Then, there exist a dimension vector α = (m,..., m k ) N k + such that n = k i= m id i and for any a = (A,..., A k ) A with A i M di (C) we have that tr(a) = m T r(a ) m k T r(a k ) where T r are the usual trace maps on matrices.

191 5.9. FINITE DIMENSIONAL ALGEBRAS. 9 Proof. The norm-map N on A defined by the trace map tr induces a group morphism on the invertible elements of A N : A = GL d (C)... GL dk (C) C that is, a character. Now, any character is of the following form, let A i GL di (C), then for a = (A,..., A k ) we must have N(a) = det(a ) m det(a 2 ) m2... det(a k ) m k for certain integers m i Z. Since N extends to a polynomial map on the whole of A we must have that all m i 0. By polarization it then follows that tr(a) = m T r(a ) +... m k T r(a k ) and it remains to show that no m i = 0. Indeed, if m i = 0 then tr would be the zero map on M di (C), but then we would have for any a = (0,..., 0, A, 0,..., 0) with A M di (C) that χ (n) a (t) = t n whence χ (n) a (a) 0 whenever A is not nilpotent. This contradiction finishes the proof. We can extend this to all finite dimensional C-algebras. Let A be a finite dimensional algebra with radical J and assume there is a trace map tr on A making A into a Cayley-Hamilton algebra of degree n and such that tr(a) = C. We claim that the norm map N : A C is zero on J. Indeed, any j J satisfies j l = 0 for some l whence N(j) l = 0. But then, polarization gives that tr(j) = 0 and we have that the semisimple algebra A ss = A/J = M d (C)... M dk (C) is a semisimple Cayley-hamilton algebra of degree n on which we can apply the foregoing lemma. Finally, note that A A ss J as C-vectorspaces. This concludes the proof of Proposition 5.3 Let A be a finite dimensional C-algebra with radical J and semisimple part A ss = A/J = M d (C)... M dk (C). If tr : A C A is a trace map such that A is a Cayley-Hamilton algebra of degree n, there exists a dimension vector α = (m,..., m k ) N k + such that for all a = (A,..., A k, j) with A i M di (C) and j J we have tr(a) = m T r(a ) +... m k T r(a k ) with T r the usual traces on M di (C) and i m id i = n. However, there can be other trace maps on A making A into a Cayley-Hamilton algebra of degree n. For example let C be a finite dimensional commutative C-algebra with radical N, then A = M n (C) is finite dimensional with radical J = M n (N) and the usual trace map tr : M n (C) C makes A into a Cayley-Hamilton algebra of degree n such that tr(j) = N 0. Still, if A is semi-simple, the center Z(A) = C... C (as many terms as there are matrix components in A) and any subring of Z(A) is of the form C... C. In particular, tr(a) has this form and composing the trace map with projection on the j-th component we have a trace map tr j on which we can apply the lemma.

192 92 CHAPTER 5. ETALE SLICES. With C n we will denote the universal algebra in the category of Cayley- Hamilton algebras of degree n such that has a decomposition into k orthogonal idempotents {e,..., e k }. That is, take generic n n matrices X l = x (l)... x n (l).. x n (l)... x nn (l) for the idempotents e l for l k. As e i e j = 0 if i j and e 2 i = e i, the only nonvanishing traces of monomials in the e i (up to cyclic permutation) form the polynomial algebra P = C[t,..., t k ]/(t t k n) where t i = tr(x i ) = n j= x jj(i). Now, consider the quotient R of the polynomial algebra C[x ij (l) i, j n, l k] by the ideal of all entries coming from the matrix identities Xi 2 = X i X i X j = 0 for i j X X k = I n χ (n) X i (X i ) = 0 and observe that all the coefficients of the Cayley-Hamilton polynomial of X i are polynomials in t i. Then, C n is the subalgebra of M n (R) generated by the images of the X l and t l. Theorem 5.32 With notations as above, we have :. C n is a smooth Cayley-Hamilton algebra of degree n. 2. rep tr n C k@ n is the disjoint union of the homogeneous varieties GL n /(GL m... GL mk ) where α = (m,..., m k ) N k + is a dimension vector such that m +...+m k = n. 3. The Cayley-Hamilton algebra corresponding to the component determined by α = (m,..., m k ) is the semi-simple algebra C k (α) which is the subalgebra of M n (C) generated by the images e i P i k= mi j= P i k= m k+ e jj M n (C) Proof. () : Let T be a Cayley-Hamilton algebra of degree n with trace map tr and I a twosided nilpotent ideal of T such that tr(i) I. Assume there is a trace preserving algebra map C φ n T/I which is determined by the φ(x l ) = f l which are idempotents in T/I such that = f f k. By the lemma above we can lift this decomposition to = f f k where f i are orthogonal idempotents of T. Clearly, there is a C-algebra morphism ψ : C n T lifting φ by sending X l to f l and t l to tr(f l ). Observe that this is possible as the only relation holding among the t l is t t k = n and because T is of degree n we have that tr(f ) tr(f k ) = n.

193 5.0. INVARIANT AND EQUIVARIANT MAPS. 93 (2) : By the previous part we know that the trace preserving representation variety rep tr C n k@ n is smooth and hence reduced. Therefore, it suffices to describe the points. Take a trace preserving n-dimensional representation M determined by the trace preserving algebra morphism C n χ Mn (C) The image χ(c n ) is a finite dimensional semi-simple commutative Cayleyhamilton algebra with trace image C and hence one of the algebras C k (α) for some dimension vector α = (m,..., m k ) with m i = n. Therefore, M is a semi-simple representation and the multiplicities of the simple components S l corresponding to the l-th factor of C k (α) is equal to the dimension of e l.m which is the number m l. Hence, there is a unique trace preserving representation factoring via C k (α). Therefore, rep tr C n k@ n is the disjoint union of finitely many closed orbits (since they are semi-simple) one for each dimension vector α = (m,..., m k ) with m i = n. The stabilizer group of M is the group of module automorphisms and hence equal to GL m... GL mk proving the assertion. (3) : We have already seen that there is a unique isomorphism class of trace preserving representation for C k (α) so the reduced variety of rep tr C n k(α) is the orbit GL n /(GL m... GL mk ). Moreover, C k (α) C n /(t m,..., t k m k ) = C n (α) We claim that C[rep tr C n k@ n (α)] is formally smooth. Let C be a commutative algebra and I C a nilpotent ideal. Any algebra morphism C[rep tr C n k@ n (α)] φ C/I determines a decomposition = c c k into orthogonal idempotents c l M n (C/I) with tr(c l ) = m l. This decomposition can be lifted to = c c k where c l are orthogonal idempotents in M n (C). The entries of the v l determine an algebra morphism C[rep tr C n k@ n (α)] C lifting φ proving formal smoothness. Therefore rep tr C n k(α) is smooth and reduced hence is the orbit. 5.0 Invariant and equivariant maps. Let A be a Cayley-Hamilton algebra of degree n with trace map tr. Assume we have a decomposition = a a k of as a sum of orthogonal idempotents a l in A. Then, we have a trace preserving embedding C n φ A defined by X i a i, t i tr(a i ). On the level of representation schemes this embedding gives rise to a morphism between the representation varieties rep tr A π n rep tr C n k@ n defined by composition. By the foregoing result we have a decomposition of rep tr C n k@ n as a disjoint union of finitely many orbits O(α) determined by a dimension vector α = (m,..., m k ) N k + such that m i = n. Therefore, we can similarly decompose A = απ O(α) rep tr n into a disjoint union of finitely many closed and open subschemes. We will denote the component π O(α) by rep α A. Observe that of course some of these

194 94 CHAPTER 5. ETALE SLICES. components may be empty. On the level of coordinate algebras this decomposition translates itself into C[rep tr A] = n αc[rep α A] and M n (C[rep tr A]) = n αm n (C[rep α A]) Since each of the components rep α A is stable under the GL n -action we have that M n (C[rep tr n A])GLn = α M n (C[rep α A]) GLn as the left term equals A this finishes the proof of the following result. Proposition 5.33 Let A be a Cayley-Hamilton algebra of degree n having a decomposition = a a k into orthogonal idempotents, then. A = α A α the sum ranging over all dimension vectors α = (m,..., m k ) N k + satisfying m i = n. 2. The projection of a i in the component A α has trace m i where α = (m,..., m k ). Again, observe that usually most components in the above direct sum decomposition are zero. We will now concentrate on one of the components A α, that is we assume that A is a Cayley-Hamilton algebra of degree n with decomposition = a a n into orthogonal idempotents such that tr(a i ) = m i N + and mi = n. Then, we have a trace preserving embedding C k (α) i A making A into a C k (α) = k i=c-algebra. We have constructed a trace preserving embedding C k (α) i M n (C) by sending the idempotent e i to the diagonal idempotent d i M n (C) with ones from position i j= m j to i j= m i. This calls for the introduction of a restricted representation scheme of all trace preserving algebra morphisms χ such that the diagram below is commutative A i χ Mn (C) i C k (α) that is, such that χ(a i ) = d i. This again determines an affine scheme rep res A α which is in fact a closed subscheme of rep tr A. The functorial description of the n restricted module scheme is as follows. Let C be any commutative C-algebra, then M n (C) is a C k (α)-algebra and the idempotents d i allow for a block decomposition M n (C) = i,j d i M n (C)d j = d M n (C)d... d M n (C)d k... d k M n (C)d... d k M n (C)d k The scheme rep res A assigns to the algebra C the set of all trace preserving algebra α maps A φ M n (B) such that φ(a i ) = d i. Equivalently, the idempotents a i decompose A into block form A = i,j a i Aa j and then rep res A(C) are the trace preserving algebra morphisms A M α n (B) compatible with the block decompositions.

195 5.0. INVARIANT AND EQUIVARIANT MAPS. 95 The embedding C k (α) i Mn (C) sending e i to d i gives a special point p of the homogeneous variety rep tr n C k(α) = GL n /(GL m... GL mk ). Still another description of the restricted representation scheme is therefore that rep res A is the scheme theoretic fiber π (p) of the point p under the GL α n - equivariant morphism rep tr A π n rep tr C n k(α). Hence, the stabilizer subgroup of p acts on rep res A. This stabilizer is the subgroup α GL(α) = GL m... GL mk embedded in GL n along the diagonal GL(α) = GL m... GL mk GL n. Clearly, GL(α) acts via this embedding by conjugation on M n (C). The main implication of the existence of a decomposition of into orthogonal idempotents is the following reduction result both of the affine scheme and of the acting group. Theorem 5.34 Let A be a Cayley-Hamilton algebra of degree n such that = a +...+a k is a decomposition into orthogonal idempotents with tr(a i ) = m i N +. Then, A is isomorphic to the ring of GL(α)-equivariant maps rep res α A M n. Proof. We know that A is the ring of GL n -equivariant maps rep tr A Mn. n Further, we have a GL n -equivariant map rep tr n A π repn tr C k (α) = GL n.p GL n /GL(α) Thus, the GL n -equivariant maps from rep tr A to M n n coincide with the Stab(p) = GL(α)-equivariant maps from the fiber π (p) = rep res A to M α n. That is, we have a block matrix decomposition for A. Indeed, we have A (C[rep res α A] M n(c)) GL(α) and this isomorphism is clearly compatible with the block decomposition and thus we have for all i, j that a i Aa j (C[rep res α A] M m i m j (C)) GL(α) where M mi m j (C) is the space of rectangular m i m j matrices M with coefficients in C on which GL(α) acts via g.m = g i Mg j where g = (g,..., g k ) GL(α). Another consequence of a idempotent decomposition is. Theorem 5.35 Let A be a Cayley-Hamilton algebra of degree n such that = a a k is a decomposition into orthogonal idempotents with tr(a i ) = m i N +. If the restricted representation scheme is a smooth GL(α)-variety, then A is a smooth Cayley-Hamilton algebra.

196 96 CHAPTER 5. ETALE SLICES. Proof. Consider again the projection rep tr A π rep tr C n n k(α). As the base is a homogeneous variety it is smooth. Moreover all the fibers are isomorphic to rep res A α whence smooth by assumption. Then, the total space rep tr A is also smooth whence n is A a smooth algebra. If Q be a quiver on k vertices, then the vertex idempotents e i give a decomposition = e e k of one into orthogonal idempotents and make the path algebra CQ into a C k -algebra. Fix a dimension vector α = (d,..., d k ) N k and let n = d i. Observe that we may assume that α N k + (and hence that the map C φ k CQ is an embedding. If not, we can restrict to the full subquiver of Q on the vertices v i such that d i 0. The algebra embedding C φ k CQ determines a morphism rep n CQ π repn C k = β O(β) where the disjoint union is taken over all the dimension vectors β = (b,..., b k ) such that n = b i. Again, consider the point p O(α) determined by sending the idempotents e i to the canonical diagonal idempotents of M n (C). As rep α Q can be identified with the variety of n-dimensional representations of CQ in block form determined by these idempotents we see that rep α Q = π (p). We construct the algebra T α Q as follows : first adjoin formally all traces to the path algebra CQ, that is, consider the path algebra of Q over the polynomial algebra R in the variables t p where p is a word in the arrows a j Q a and is determined only up to cyclic permutation. As a consequence we only retain the variables t p where p is an oriented cycle in Q (as all the others have a cyclic permutation which is the zero element in CQ). This way we put a formal trace map on R CQ by defining tr(p) = t p is p is an oriented cycle in Q and tr(p) = 0 otherwise. The algebra T α Q is obtained from this formal trace algebra R CQ by dividing out the substitution invariant twosided ideal generated by all the evaluations of the formal Cayley-Hamilton algebras of degree n, χ (n) a (a) for a R CQ together with the additional relations that tr(e i ) = d i. T α Q is a Cayley-Hamilton algebra of degree n with a decomposition = e e k into orthogonal idempotents such that tr(e i ) = d i. Consequently, the restricted representation scheme rep res α T α Q rep α Q as GL(α)-varieties. Summarizing the results proved before in this special we obtain Theorem 5.36 With notations as before,. The algebra T α Q is a smooth Cayley-Hamilton algebra. 2. T α Q is the algebra of GL(α)-equivariant maps from rep α Q to M n, that is, T α Q = M n (C[rep α Q]) GL(α) 3. The ring of GL(α)-polynomial invariants of rep α Q, N α Q = C[rep α Q] GL(α) is generated by traces along oriented cycles in the quiver Q of length bounded by n 2 +.

197 5.0. INVARIANT AND EQUIVARIANT MAPS. 97 A realization of these algebras is as follows. To an arrow j corresponds i a d j d i matrix of variables from C[rep α Q] x (a) x di (a) M a =.. x dj(a) x djd i (a) where x ij (a) are the coordinate functions of the entries of V a of a representation V rep α Q. Let p = a a 2... a r be an oriented cycle in Q, then we can compute the following matrix M p = M ar... M a2 M a over C[rep α Q]. As we have that s(a r ) = t(a ) = v i, this is a square d i d i matrix with coefficients in C[rep α Q] and we can take its ordinary trace T r(m p ) C[rep α Q]. Then, N α Q is the C-subalgebra of C[rep α Q] generated by these elements. Consider the block structure of M n (C[rep α Q]) with respect to the idempotents e i M d (S) M d d k (S).... M dj d i (S). M dk d i (S) M dk (S) where S = C[rep α Q]. Then, we can also view the matrix M a for an arrow j as i a block matrix in M n (C[rep α Q]) a M a Then, T α Q is the C k (α)-subalgebra of M n (C[rep α Q]) generated by N α Q and these block matrices for all arrows a Q a. T α Q itself has a block decomposition P P k T α Q =... P ij. P k P kk where P ij is the N α Q-module spanned by all matrices M p where p is a path from v i to v j of length bounded by n 2. Example 5.37 This result proves the claims we made in chapter on the algebras related to the study of Calogero particles. For consider the path algebra M of the quiver a x e y u v f

198 98 CHAPTER 5. ETALE SLICES. and take as dimension vector α = (n, ). The total dimension is in this case n = n + and we fix the embedding C 2 = C C M given by the decomposition = e + f. Then, the above realization of T α M consists in taking the following n n matrices x... x n 0 e n = fn = xn = x n... x nn y... y n u y n = y n... y nn 0 5 un = u 5 vn = n v... v n 0 In order to determine the ring of GL(α)-polynomial invariants of rep α M we have to consider the traces along oriented cycles in the quiver. Any nontrivial such cycle must pass through the vertex e and then we can decompose the cycle into factors x, y and uv (observe that if we wanted to describe circuits based at the vertex f they are of the form c = vc u with c a circuit based at e and we can use the cyclic property of traces to bring it into the claimed form). That is, all relevant oriented cycles in the quiver can be represented by a necklace word w w where each bead is one of the elements = x = y and = uv In calculating the trace, we first have to replace each occurrence of x, y, u or v by the relevant n n-matrix above. This results in replacing each of the beads in the necklace by one of the following n n matrices x... x n y... y n u v... u v n 6 = = = x n... x nn y n... y nn u nv... u nv n and taking the trace of the n n matrix obtained after multiplying these bead-matrices cyclicly in the indicated orientation. This concludes the description of the invariant ring N α Q. The algebra T α M of GL(α)-equivariant maps from rep α M to M n, that is, T α M = M(n) defined in chapter, is then the subalgebra of M n (C[rep α M]) generated as C 2 (α)-algebra (using the idempotent n n matrices corresponding to e and f) by N α M and the n n-matrices corresponding to x, y, u and v. We will have to extend these results to a marked quiver Q. Let {l,..., l m } be the marked loops in Q, then we define { Nα Q N = α Q T α Q = (tr(l ),...,tr(l m)) T α Q (tr(l ),...,tr(l m)) Clearly, T α Q is a Cayley-Hamilton algebra of degree n having a decomposition = e e k into orthogonal idempotents and such that rep res α T(Q, α) = rep α Q Theorem The algebra T α Q is a smooth Cayley-Hamilton algebra.

199 5.0. INVARIANT AND EQUIVARIANT MAPS T α Q is the algebra of GL(α)-equivariant maps from rep α Q to M n (C), that is, T α Q = M n (C[rep α Q ]) GL(α) 3. The algebra N α Q is the ring of GL(α)-polynomial invariants of rep α Q, N α Q = C[rep α Q ] GL(α) and is generated by traces along oriented cycles in the quiver Q of length bounded by n 2 +.

200 200 CHAPTER 5. ETALE SLICES. Combinatorics.

201 5.0. INVARIANT AND EQUIVARIANT MAPS. 20

202 202 CHAPTER 5. ETALE SLICES.

203 Chapter 6 Local Classification. Every commutative smooth variety of dimension d is locally in the étale topology isomorphic to affine d-space A d. In this chapter we study the corresponding problem for Cayley-smooth orders A of degree n. In the foregoing chapter we have described the étale local structure of A near a point ξ iss n A by a marked quiver setting (Q ξ, α ξ). In this chapter we will classify those quiver settings which can occur for given n and given dimension d = dim iss n A of the central variety. We will show that for fixed d and n only a finite number of such settings do arise, that is, Cayley-smooth algebras have a finite number of possible étale local behaviour. We prove this by simplifying a quiver-setting by shrinking (identifying arrow-connected vertices with the same vertex-dimension) to one of a finite list of settings in reduced form. For dimension d 4, the complete list is where the boxed value is the dimension d. To arrive at this result we need to classify the dimension vectors of simple representations of (marked) quivers and be able to compute the dimension of the corresponding quotient varieties. Further, we show that the local quiver setting (Q ξ, α ξ) contains enough information to determine the 203

204 204 CHAPTER 6. LOCAL CLASSIFICATION. quiver-settings in nearby points and even to give the dimension of the strata of points of equal type. We then apply these results to the local characterization of Cayley-smooth orders over curves and surfaces. Smooth curve orders are proved to coincide with hereditary orders and smooth surface orders must have a smooth surface as their center, a ramification divisor having as worst singularities normal crossings and must be étale splittable. If we combine these results with the Artin- Mumford sequence, describing the Brauer group of the functionfield of a smooth projective surface X, we are able to prove that any central simple algebra of degree n over C(X) contains a Cayley-Hamilton order A having at worst a finite number of noncommutative singularities, all of which are étale isomorphic to those appearing in the origin of a quantum plane. Finally, we classify all central simple algebras over C(X) admitting a Cayley-smooth model. Whereas we restrict attention mainly to orders, it is clear that the strategy can be extended to Cayley-Hamilton algebras of degree n which are finite modules over a central subring C, provided we have some control on the commutative extension C Z(A). For an application of this to the theory of quantum groups, the reader may consult [9]. 6. Marked quivers. In this section we recall the basics on representations of (marked) quivers. Recall that a quiver Q is a directed graph determined by a finite set Q v = {v,..., v k } of vertices, and a finite set Q a = {a,..., a l } of arrows where we allow multiple arrows between vertices and loops in vertices. Every arrow a j has i a starting vertex s(a) = i and a terminating vertex t(a) = j. Multiplication in the path algebra CQ is induced by (left) concatenation of paths. More precisely, = v v k is a decomposition of into mutually orthogonal idempotents and further we define v j.a is always zero unless a j in which case it is the path a, a.v i is always zero unless a in i which case it is the path a, a i.a j is always zero unless path a i a j. a i a j in which case it is the The description of the quiver Q can be encoded in an integral k k matrix χ... χ k χ Q =.. with χ ij = δ ij # { j i } χ k... χ kk Example 6. Consider the quiver Q 2 3 Then, with the indicated ordering of the vertices we have that the integral matrix is 2 χ Q =

205 6.. MARKED QUIVERS. 205 and the path algebra of Q is isomorphic to the block-matrix algebra 2 CQ 4 C C C C C[x] C[x] where x is the loop in vertex v 3. The subspace CQv i has as basis the paths starting in vertex v i and because CQ = i CQv i, CQv i is a projective left ideal of CQ. Similarly, v i CQ has as basis the paths ending at v i and is a projective right ideal of CQ. The subspace v i CQv j has as basis the paths starting at v j and ending at v i and CQv i CQ is the twosided ideal of CQ having as basis all paths passing through v i. If 0 f CQv i and 0 g v i CQ, then f.g 0 for let p be a longest path occurring in f and q a longest path in g, then the coefficient of p.q in f.g cannot be zero. As a consequence we have Lemma 6.2 The projective left ideals CQv i are indecomposable and paired nonisomorphic. Proof. If CQv i is not indecomposable, then there exists a projection idempotent f Hom CQ (CQv i, CQv i ) v i CQv i. But then, f 2 = f = f.v i whence f.(f v i ) = 0, contradicting the remark above. Further, for any left CQ-module M we have that Hom CQ (CQv i, M) v i M. So, if CQv i CQv j then the isomorphism gives elements f v i CQv j and g v j CQv i such that f.g = v i and g.f = v j. But then, v i CQv j CQ, a contradiction unless i = j as this space has basis all paths passing through v j. Example 6.3 Let Q be a quiver, then the following properties hold :. CQ is finite dimensional if and only if Q has no oriented cycles. 2. CQ is prime (that is, I.J 0 for all twosided ideals I, J 0) if and only if Q is strongly connected, that is, for all vertices v i and v j there is a path from v i to v j. 3. CQ is Noetherian (that is, satisfies the ascending chain condition on left (or right) ideals) if and only if for every vertex v i belonging to an oriented cycle there is only one arrow starting at v i and only one arrow terminating at v i. 4. The radical of CQ has as basis all paths from v i to v j for which there is no path from v j to v i. 5. The center of CQ is of the form C... C C[x]... C[x] with one factor for each connected component C of Q (that is, connected component for the underlying graph forgetting the orientation) and this factor is isomorphic to C[x] if and only if C is one oriented cycle. Recall that a representation V of the quiver Q is given by a finite dimensional C-vector space V i for each vertex v i Q v, and a linear map V j Va V i for every arrow a in j i Q a. If dim V i = d i we call the integral vector α = (d,..., d k ) N k the dimension vector of V and denote it with dim V. A morphism V φ W between two representations V and W of Q is determined by a set of linear maps V i φ i Wi for all vertices v i Q v satisfying the following compatibility conditions for every arrow a j in i Q a V i V a Vj φ i φ j Wa Wj W i

206 206 CHAPTER 6. LOCAL CLASSIFICATION. Clearly, composition of morphisms V φ W ψ X is given by the rule that (ψ φ) i = ψ i ψ i and one readily verifies that this is again a morphism of representations of Q. In this way we form a category rep Q of all finite dimensional representations of the quiver Q. Proposition 6.4 The category rep Q is equivalent to the category of finite dimensional CQ-representations (left modules). Proof. Let M be an n-dimensional CQ-representation. Then, we construct a representation V of Q by taking V i = v i M, and for any arrow a j in i Q a define V a : V i Vj by V a (x) = v j ax. Observe that the dimension vector dim(v ) = (d,..., d k ) satisfies d i = n. If φ : M N is CQ-linear, then we have a linear map V i = v i M φi W i = v i N which clearly satisfies the compatibility condition. Conversely, let V be a representation of Q with dimension vector dim(v ) = (d,..., d k ). Then, consider the n = d i -dimensional space M = i V i which we turn into a CQ-representation as follows. Consider the canonical injection and projection maps V ij j M π j Vj. Then, define the action of CQ by fixing the action of the algebra generators v j and a l to be { v j m = i j (π j (m)) a l m = i j (V a (π i (m))) a l for all arrows j. i A computation verifies that these two operations are inverse to each other and induce an equivalence of categories. The Euler form of the quiver Q is the bilinear form on Z k χ Q (.,.) : Z k Z k Z defined by χ Q (α, β) = α.χ Q.β τ for all row vectors α, β Z k. Theorem 6.5 Let V and W be two representations of Q, then dim C Hom CQ (V, W ) dim C Ext CQ(V, W ) = χ Q (dim(v ), dim(w )) Proof. We claim that there exists an exact sequence of C-vectorspaces 0 HomCQ (V, W ) γ vi Q v Hom C (V i, W i ) δ δ a Qa Hom C (V s(a), W t(a) ) ɛ Ext CQ (V, W ) 0 Here, γ(φ) = (φ,..., φ k ) and δ maps a family of linear maps (f,..., f k ) to the linear maps µ a = f t(a) V a W a f s(a) for any arrow a in Q, that is, to the obstruction of the following diagram to be commutative V s(a) V a Vt(a) µa f s(a) W s(a)... f t(a) Wa Wt(a)

207 6.. MARKED QUIVERS. 207 By the definition of morphisms between representations of Q it is clear that the kernel of δ coincides with Hom CQ (V, W ). Further, the map ɛ is defined by sending a family of maps (g,..., g s ) = (g a ) a Qa to the equivalence class of the exact sequence 0 W i E p V 0 where for all v i Q v we have E i = W i V i and the inclusion i and projection map p are the obvious ones and for each generator a Q a the action of a on E is defined by the matrix [ ] Wa g E a = a : E 0 V s(a) = W s(a) V s(a) Wt(a) V t(a) = E t(a) a Clearly, this makes E into a CQ-module and one verifies that the above short exact sequence is one of CQ-modules. Remains to prove that the cokernel of δ can be identified with Ext CQ (V, W ). For this, we need to look back at the description of Ext in terms of cycles and boundaries. A set of algebra generators of CQ is given by {v,..., v k, a,..., a l }. A cycle is given by a linear map λ : CQ Hom C (V, W ) such that for all f, f CQ we have the condition λ(ff ) = ρ(f)λ(f ) + λ(f)σ(f ) where ρ determines the action on W and σ that on V. First, consider v i then the condition says λ(vi 2) = λ(v i) = p W i λ(v i ) + λ(v i )p V i whence λ(v i ) : V i Wi but then applying again the condition we see that λ(v i ) = 2λ(v i ) so λ(v i ) = 0. Similarly, using the condition on a = v t(a) a = av s(a) we deduce that λ(a) : V s(a) W t(a). That is, we can identify a Qa Hom C (V s(a), W t(a) ) with Z(V, W ) under the map ɛ. Moreover, the image of δ gives under δ rise to a family of morphisms λ(a) = f t(a) V a W a f s(a) for a linear map f = (f i ) : V W so this image coincides precisely to the subspace of boundaries B(V, W ) proving that indeed the cokernel of δ is Ext CQ (V, W ) finishing the proof of exactness of the long sequence of vectorspaces. But then, if dim(v ) = (r,..., r k ) and dim(w ) = (s,..., s k ), we have that dim Hom(V, W ) dim Ext (V, W ) is equal to dim Hom C (V i, W i ) dim Hom C (V s(a), W t(a) ) v i Q v a Q a = r i s i r s(a) s t(a) v i Q v a Q a = (r,..., r k )M Q (s,..., s k ) τ = χ Q (dim(v ), dim(w )) finishing the proof. Fix a dimension vector α = (d,..., d k ) N k and consider the set rep α Q of all representations V of Q such that dim(v ) = α. Because V is completely determined by the linear maps V a : V s(a) = C d s(a) C d t(a) = V t(a) we see that rep α Q is the affine space rep α Q = M dj d i (C) C r a j i

208 208 CHAPTER 6. LOCAL CLASSIFICATION. where r = a Q a d s(a) d t(a). On this affine space we have an action of the algebraic group GL(α) = GL d... GL dk by conjugation. That is, if g = (g,..., g k ) GL(α) and if V = (V a ) a Qa then g.v is determined by the matrices (g.v ) a = g t(a) V a g s(a). If V and W in rep α Q are isomorphic as representations of Q, such an isomorphism is determined by invertible matrices g i : V i Wi GL di such that for every arrow a j we i have a commutative diagram V i V a V j g i g j Wa W j W i or equivalently, g j V a = W a g i. That is, two representations are isomorphic if and only if they belong to the same orbit under GL(α). In particular, we see that Stab GL(α) V Aut CQ V and the latter is an open subvariety of the affine space End CQ (V ) = Hom CQ (V, V ) whence they have the same dimension. The dimension of the orbit O(V ) of V in rep α Q is equal to dim O(V ) = dim GL(α) dim Stab GL(α) V. But then we have a geometric reformulation of the above theorem. Lemma 6.6 Let V rep α Q, then dim rep α Q dim O(V ) = dim End CQ (V ) χ Q (α, α) = dim Ext CQ(V, V ) Proof. We have seen that dim rep α Q dim O(V ) is equal to d s(a) d t(a) ( d 2 i dim End CQ (V )) = dim End CQ (V ) χ Q (α, α) a i and the foregoing theorem asserts that the latter term is equal to dim Ext CQ (V, V ). In particular it follows that the orbit O(V ) is open in rep α Q if and only if V has no self-extensions. Moreover, as rep α Q is irreducible there can be at most one isomorphism class of a representation without self-extensions. For our purposes we have to generalize the setting slightly. A marked quiver Q has an underlying quiver Q such that some of its loops can acquire a marking. Such a marked loop will be depicted by A representation V of a marked quiver Q is a representation of the underlying quiver Q such that the matrices corresponding to marked loops have trace equal to zero. If we fix a dimension vector α, rep α Q = {V rep α Q tr(v a ) = 0 if a is a marked loop in Q }

209 6.2. SIMPLE DIMENSION VECTORS. 209 rep α Q is an affine subspace of rep α Q of codimension equal to the number of marked loops in Q. This subspace is stable under the action of GL(α) on rep α Q and GL(α)-orbits in rep α Q correspond to isomorphism classes of representations of Q. The Euler form of the marked quiver Q is the Euler form of the underlying quiver Q. We denote χ ii = u i m i where u i is the number of unmarked loops at v i and m i is the number of marked loops at v i. 6.2 Simple dimension vectors. In this section we characterize the dimension vectors α such that the marked quiver Q has a simple representation V (that is, contains no proper subrepresentations) with dim(v ) = α. Consider the underlying quiver Q with vertex set Q v = {v,..., v k }. To a subset S Q v we associate the full subquiver Q S of Q, that is, Q S has as set of vertices the subset S and as set of arrows all arrows a j in i Q a such that v i and v j belong to S. A full subquiver Q S is said to be strongly connected if and only if for all v i, v j V there is an oriented cycle in Q S passing through v i and v j. We can partition Q v = S... S s such that the Q Si are maximal strongly connected components of Q. Clearly, the direction of arrows in Q between vertices in S i and S j is the same by the maximality assumption and can be used to define an orientation between S i and S j. The strongly connected component quiver SC(Q) is then the quiver on s vertices {w,..., w s } with w i corresponding to S i and there is one arrow from w i to w j if and only if there is an arrow in Q from a vertex in S i to a vertex in S j. Observe that when the underlying graph of Q is connected, then so is the underlying graph of SC(Q) and SC(Q) is a quiver without oriented cycles. Example 6.7 Consider the connected quiver Q then the partitioning of Q v into maximal strongly connected components is Q v = {v } {v 2, v 3, v 4 } {v 5 } and the strictly connected component quiver SC(Q) of Q has the following form We will give names to vertices with very specific in- and out-going arrows source sink

210 20 CHAPTER 6. LOCAL CLASSIFICATION. prism focus For example, for the quiver Q of the above example, v is a source, there are no sinks, v 2 and v 5 are focuses and v 4 is a prism (observe that a loop gives one incoming and one outgoing arrow). If α = (d,..., d k ) is a dimension vector, then supp(α) = {v i Q v d i 0}. Lemma 6.8 If α is the dimension vector of a simple representation of Q, then Q supp(α) is a strongly connected subquiver. Proof. If not, we consider the strongly connected component quiver SC(Q supp(α) ) and by assumption there must be a sink in it corresponding to a proper subset S Qv. If V rep α Q we can then construct a representation W by W i = V i for v i S and W i = 0 if v i / S, W a = V a for an arrow a in Q S and W a = 0 otherwise. One verifies that W is a proper subrepresentation of V, so V cannot be simple, a contradiction. The second necessary condition involves the Euler form of Q. With ɛ i be denote the dimension vector of the simple representation having a one-dimensional space at vertex v i and zero elsewhere and all arrows zero matrices. Lemma 6.9 If α is the dimension vector of a simple representation of Q, then { χ Q (α, ɛ i ) 0 χ Q (ɛ i, α) 0 for all v i supp(α). Proof. Let V be a simple representation of Q with dimension vector α = (d,..., d k ). One verifies that χ Q (ɛ i, α) = d i j a i d j Assume that χ Q (ɛ i, α) > 0, then the natural linear map j a i V a : V i j a i V j has a nontrivial kernel, say K. But then we consider the representation W of Q determined by W i = K and W j = 0 for all j i, W a = 0 for all a Q a.

211 6.2. SIMPLE DIMENSION VECTORS. 2 It is clear that W is a proper subrepresentation of V, a contradiction. Similarly, assume that χ Q (α, ɛ i ) = d i d j > 0, then the linear map a j i a j V a : i a j V j i V i has an image I which is a proper subspace of V i. determined by The representation W of Q W i = I and W j = V j for j i, W a = V a for all a Q a. is a proper subrepresentation of V, a contradiction finishing the proof. Example 6.0 The necessary conditions of the foregoing two lemmas are not sufficient. Consider the extended Dynkin quiver of type Ãk with cyclic orientation. and dimension vector α = (a,..., a). a a a a a a For a simple representation all arrow matrices must be invertible but then, under the action of GL(α), they can be diagonalized. Hence, the only simple representations (which are not the trivial simples concentrated in a vertex) have dimension vector (,..., ). Nevertheless, we will show that these are the only exceptions. A vertex v i is said to be large with respect to a dimension vector α = (d,..., d k ) whenever d i is maximal among the d j. The vertex v i is said to be good if v i is large and has no direct successor which is a large prism nor a direct predecessor which is a large focus. Lemma 6. Let Q be a strongly connected quiver, not of type Ãk, then one of the following hold. Q has a good vertex, or, 2. Q has a large prism having no direct large prism successors, or 3. Q has a large focus having no direct large focus predecessors. Proof. If neither of the cases hold, we would have an oriented cycle in Q consisting of prisms (or consisting of focusses). Assume (v i,..., v il ) is a cycle of prisms, then the unique incoming arrow of v ij belongs to the cycle. As Q Ãk there is at least one extra vertex v a not belonging to the cycle. But then, there can be no oriented path from v a to any of the v ij, contradicting the assumption that Q is strongly connected. If we are in one of the two last cases, let a be the maximum among the components of the dimension vector α and assume that α satisfies χ Q (α, ɛ i ) 0 and

212 22 CHAPTER 6. LOCAL CLASSIFICATION. χ Q (ɛ i, α) 0 for all i k, then we have the following subquiver in Q a large focus a large prism We can reduce to a quiver situation with strictly less vertices. Lemma 6.2 Assume Q is strongly connected and we have a vertex v i which is a prism with unique predecessor the vertex v j which is a focus. Consider the dimension vector α = (d,..., d k ) with d i = d j = a 0. Then, α is the dimension of a simple representation of Q if and only if α = (d,..., d i, d i+,..., d k ) N k is the dimension vector of a simple representation of the quiver Q on k vertices, obtained from Q by identifying the vertices v i and v j, that is, the above subquiver in Q is simplified to the one below in Q a Proof. If b is the unique arrow from v j to v i and if V rep α Q is a simple representation then V b is an isomorphism, so we can identify V i with V j and obtain a simple representation of Q. Conversely, if V rep α Q is a simple representation, define V rep α Q by V i = V j and V z = V z for z i, V b = V b for all arrows b b and V b = a. Clearly, V is a simple representation of Q. Theorem 6.3 α = (d,..., d k ) is the dimension vector of a simple representation of Q if and only if one of the following two cases holds. supp(α) = Ãk, the extended Dynkin quiver on k vertices with cyclic orientation and d i = for all i k 2. supp(α) Ãk. Then, supp(α) is strongly connected and for all i k we have { χ Q (α, ɛ i ) 0 χ Q (ɛ i, α) 0

213 6.2. SIMPLE DIMENSION VECTORS. 23 Proof. We will use induction, both on the number of vertices k in supp(α) and on the total dimension n = i d i of the representation. If supp(α) does not possess a good vertex, then the above lemma finishes the proof by induction on k. Observe that the Euler-form conditions are preserved in passing from Q to Q as d i = d j. Hence, assume v i is a good vertex in supp(α). If d i = then all d j = for v j supp(α) and we can construct a simple representation by taking V b = for all arrows b in supp(α). Simplicity follows from the fact that supp(α) is strongly connected. If d i >, consider the dimension vector α = (d,..., d i, d i, d i+,..., d k ). Clearly, supp(α ) = supp(α) is strongly connected and we claim that the Euler-form conditions still hold for α. the only vertices v l where things might go wrong are direct predecessors or direct successors of v i. Assume for one of them χ Q (ɛ l, α) > 0 holds, then d l = d l > d m d i = d i a m l But then, d l = d i whence v l is a large vertex of α and has to be also a focus with end vertex v i (if not, d l > d i ), contradicting goodness of v i. Hence, by induction on n we may assume that there is a simple representation W rep α Q. Consider the space rep W of representations V rep α Q such that V α = W. That is, for every arrow a j V i a = W a v... v dj a i V j a = v W a. v dj Hence, rep W is an affine space consisting of all representations degenerating to W S i where S i is the simple one-dimensional representation concentrated in v i. As χ Q (α, ɛ i ) < 0 and χ Q (ɛ i, α ) < 0 we have that Ext (W, S i ) 0 Ext (S i, W ) so there is an open subset of representations which are not isomorphic to W S i. As there are simple representations of Q having a one-dimensional component at each vertex in supp(α) and as the subset of simple representations in rep α Q is open, we can choose W such that rep W contains representations V such that a trace of an oriented cycle differs from that of W S i. Hence, by the description of the invariant ring C[rep α Q] GL(α) as being generated by traces along oriented cycles and by the identification of points in the quotient variety as isomorphism classes of semi-simple representations, it follows that the Jordan-Hölder factors of V are different from W and S i. In view of the definition of rep W, this can only happen if V is a simple representation, finishing the proof of the theorem. From this result we can easily deduce the characterization of dimension vectors of simple representations of a marked quiver Q. For, if l is a marked loop in a vertex v i with d i >, then we may replace the matrix V l of a simple representation V rep α Q by V l = V l d di i and retain the property that V is a simple representation. Things are different, however, for a marked loop in a vertex v i with d i = as this -matrix factor is removed from the representation space. That is, we have the following characterization result.

214 24 CHAPTER 6. LOCAL CLASSIFICATION. Theorem 6.4 α = (d,..., d k ) is the dimension vector of a simple representation of a marked quiver Q if and only if α = (d,..., d k ) is the dimension vector of a simple representation of the quiver Q obtained from the underlying quiver Q of Q after removing the loops in Q which are marked in Q in all vertices v i such that d i =. 6.3 The local quiver. Consider the underlying quiver Q and a fixed dimension vector α. Closed GL(α)- orbits is rep α Q correspond to isomorphism classes of semi-simple representations of Q of dimension vector α. We have a quotient map rep α Q π repα Q/GL(α) = iss α Q and we have seen that the coordinate ring C[iss α Q] is generated by traces along oriented cycles in the quiver Q. Consider a point ξ iss α Q and assume that the corresponding semi-simple representation V ξ has a decomposition V ξ = V e... Vz ez into distinct simple representations V i of dimension vector say α i and occurring in V ξ with multiplicity e i. We then say that ξ is a point of representation-type τ = t(ξ) = (e, α ;..., e z, α z ) with α = z e i α i We want to apply the Luna slice theorem to obtain the étale GL(α)-local structure of the representation space rep α Q in a neighborhood of V ξ and the étale local structure of the quotient variety iss α Q in a neighborhood of ξ. That is, we have to calculate the normal space N ξ to the orbit O(V ξ ) as a representation over the stabilizer subgroup GL(α) ξ = Stab GL(α) (V ξ ). Denote a i = k j= a ij where α i = (a i,..., a ik ), that is, a i = dim V i. We will choose a basis of the underlying vectorspace i= vi Q v C di of V ξ = V e... Vz ez as follows : the first e a vectors give a basis of the vertex spaces of all simple components of type V, the next e 2 a 2 vectors give a basis of the vertex spaces of all simple components of type V 2, and so on. If n = k i= e id i is the total dimension of V ξ, then with respect to this basis, the subalgebra of M n (C) generated by the representation V ξ has the following block-decomposition M a (C) e M a2 (C) e M az (C) ez But then, the stabilizer subgroup Stab GL(α) (V ξ ) GL e... GL ez embedded in GL(α) with respect to this particular basis as GL e (C a ) GL e2 (C a2 ) GL ez (C az )

215 6.3. THE LOCAL QUIVER. 25 The tangentspace to the GL(α)-orbit in V ξ is equal to the image of the natural linear map Lie GL(α) repα Q sending a matrix m Lie GL(α) M d... M dk to the representation determined by the commutator [m, V ξ ] = mv ξ V ξ m. By this we mean that the matrix [m, V ξ ] a corresponding to an arrow a is obtained as the commutator in M n (C) using the canonical embedding with respect to the above choice of basis. The kernel of this linear map is the centralizer subalgebra. That is, we have an exact sequence of GL(α) ξ -modules 0 CMn(C)(V ξ ) Lie GL(α) TVξ O(V ξ ) 0 where M e (C a ) M e2 (C a2 ) 0 C Mn(C)(V ξ ) = M ez (C az ) where the action of GL(α) Vξ is given by conjugation on M n (C) via the above embedding. We will now engage in a book-keeping operation counting the factors of the relevant GL(α) ξ -spaces. We identify the factors by the action of the GL ei - components of GL(α) ξ. The centralizer C Mn(C)(V ξ ) decomposes as a GL(α) ξ -module into one factor M ei on which GL e acts via conjugation and the other factors act trivially, one factor M ez on which GL ez acts via conjugation and the other factors act trivially. 2. Recall the notation α i = (a i,..., a ik ),then the Lie algebra Lie GL(α) decomposes as a GL(α) ξ -module into k j= a2 j factors M e on which GL e acts via conjugation and the other factors act trivially, k j= a2 zj factors M e z on which GL ez acts via conjugation and the other factors act trivially, k j= a ja 2j factors M e e 2 on which GL e GL e2 acts via γ.m.γ2 and the other factors act trivially,.. k j= a zja z j factors M ez e z on which GL ez GL ez acts via γ z.m.γz and the other factors act trivially. 3. The representation space rep α Q decomposes as a GL(α) ξ -modulo into the following factors, for every arrow a j in i Q (or every loop in v i by setting i = j in the expressions below) we have.

216 26 CHAPTER 6. LOCAL CLASSIFICATION. a i a j factors M e on which GL e acts via conjugation and the other factors act trivially, a i a 2j factors M e e 2 on which GL e GL e2 acts via γ.m.γ 2 and the other factors act trivially, a zi a z j factors M ez e z on which GL ez GL ez act via γ z.m.γ z and the other factors act trivially, a zi a zj factors M ez on which GL ez acts via conjugation and the other factors act trivially. Removing the factors of. from those of 2. we obtain a description of the tangentspace to the orbit T Vξ O(V ξ ). But then, removing these factors from those of 3. we obtain the description of the normal space N Vξ as a GL(α) ξ -module as there is an exact sequence of GL(α) ξ -modules 0 TVξ O(V ξ ) repα Q N Vξ 0 The upshot of this book-keeping is that we have proved that the normal space to the orbit in V ξ depends only on the representation type τ = t(ξ) of the point ξ and can be identified with the representation space of a specific local quiver Q τ. Theorem 6.5 Let ξ iss α Q be a point of representation type τ = t(ξ) = (e, α ;..., e z, α z ) Then, the normal space N Vξ to the orbit, as a module over the stabilizer subgroup, is identical to the representation space of a local quiver situation N Vξ rep ατ Q τ where Q τ is the quiver on z vertices (the number of distinct simple components of V ξ ) say {w,..., w z } such that in Q τ # a = j i χ Q (α i, α j ) for i j, and. # i = χ Q (α i, α i ) and such that the dimension vector α τ = (e,..., e z ) (the multiplicities of the simple components in V ξ ). We can repeat this argument in the case of a marked quiver Q. the only difference lies in the description of the factors of rep α Q where we need to replace the factors M ej in the description of a loop in v i by Me 0 i (trace zero matrices) in case the loop gets a mark in Q. Recall the notation of u i as the number of unmarked loops in v i and m i the number of marked loops in v i. We define u χ 2... χ k m χ Q = χ 2 u 2... χ 2k..... and χ 2 Q. = m 2... χ k χ k2... u k mk such that χ Q = χ Q + χ2 Q where Q is the underlying quiver of Q.

217 6.3. THE LOCAL QUIVER. 27 Theorem 6.6 Let ξ iss α Q be a point of representation type τ = t(ξ) = (e, α ;..., e z, α z ) Then, the normal space N Vξ to the orbit, as a module over the stabilizer subgroup, is identical to the representation space of a local marked quiver situation N Vξ rep ατ Q τ where Q τ is the quiver on z vertices (the number of distinct simple components of V ξ ) say {w,..., w z } such that in Q τ # a = j i χ Q (α i, α j ) for i j, and # i = χ Q (α i, α i ) # i = χ 2 Q (α i, α i ) and such that the dimension vector α τ = (e,..., e z ) (the multiplicities of the simple components in V ξ ). Proposition 6.7 If α = (d,..., d k ) is the dimension vector of a simple representation of Q, then the dimension of the quotient variety iss α Q is equal to χ Q (α, α) Proof. There is a Zariski open subset of iss α Q consisting of points ξ such that the corresponding semi-simple module V ξ is simple, that is, ξ has representation type τ = (, α). But then the local quiver setting (Q τ, α τ ) is a b where a = χ Q (α, α) and b = χ2 Q (α, α). The corresponding representation space has coordinate ring C[rep ατ Q τ ] = C[x,..., x a ] on which GL(α τ ) = C acts trivially. That is, the quotient variety is rep ατ Q τ /GL(α τ ) = rep ατ Q τ C a As iss α Q has the same local structure near ξ as this quotient space near the origin, by the Luna slice result, the result follows.

218 28 CHAPTER 6. LOCAL CLASSIFICATION. 6.4 The stratification. In this section we will draw some consequences from the description of the local quiver. usually, the quotient varieties iss α Q have lots of singularities. Still, we can decompose these quotient varieties in smooth pieces according to the representation types of its points. Proposition 6.8 Let iss α Q (τ) be the set of points ξ iss α Q of representation type τ = (e, α ;... ; e z, α z ) Then, iss α Q (τ) is a locally closed smooth subvariety of iss α Q and iss α Q = τ iss α Q (τ) is a finite smooth stratification of the quotient variety. Proof. Let Q τ be the local marked quiver in ξ. Consider a nearby point ξ. If some trace of an oriented cycles of length > in Q τ is non-zero in ξ, then ξ cannot be of representation type τ as it contains a simple factor composed of vertices of that cycle. That is, locally in ξ the subvariety iss α Q (τ) is determined by the traces of unmarked loops in vertices of the local quiver Q τ and hence is locally in the étale topology an affine space whence smooth. All other statements are direct. Given a stratification of a topological space, one always wants to determine which strata make up the boundary of a given stratum. In the stratification of iss α Q given by the above result, we have a combinatorial solution to this problem. Two representation types τ = (e, α ;... ; e z, α z ) and τ = (e, α ;... ; e z, α z ) are said to be direct successors τ < τ if and only if one of the following two cases occurs (splitting of one simple) : z = z + and for all but one i z we have that (e i, α i ) = (e j, α j ) for a uniquely determined j and for the remaining i 0 we have that the remaining couples of τ are (e i, α u; e i, α v) with α i = α u + α v (combining two simple types) : z = z and for all but one i z we have that (e i, α i ) = (e j, α j ) for a uniquely determined j and for the remaining i we have that the remaining couples of τ are (e u, α i; e v, α i) with e u + e v = e i This direct successor relation < induces an ordering which we will denote with <<. Observe that τ << τ if and only if the stabilizer subgroup GL(α) τ is conjugated to a subgroup of GL(α) τ. The following result either follows from general theory, see for example [29], or from the description of the local marked quivers. Proposition 6.9 The stratum iss α iss α Q if and only if τ << τ. Q (τ ) lies in the closure of the stratum Using the dimension of the quotient variety iss α Q given in the precious section when α is the dimension vector of a simple representation can be used to determine the dimensions of the different strata iss α Q (τ) in general.

219 6.5. THE CAYLEY-SMOOTH LOCUS. 29 Proposition 6.20 Let τ = (e, α ;... ; e z, α z ) a representation type of α. Then, dim iss α Q = z ( χ Q (α j, α j ) j= Because rep α Q and hence iss α Q is irreducible, there is a unique representation type τ gen such that iss α Q (τ gen ) is Zariski open. We call τ gen the generic representation type for rep α Q. The generic representation type can be determined as follows. Let Q be the full marked subquiver of Q on the support of α and consider its strongly connected component quiver SC(Q ). Let V rep α Q be in general position, then a simple subrepresentation S V must have its support in a strongly connected component G of Q which is a sink in SC(Q ). Restrict attention to this subquiver G say on l vertices. As (,..., ) N l is the dimension vector of a simple representation of G, there exists a dimension vector β with support equal to G satisfying the following properties. β is the dimension vector of a simple representation of G. 2. If α is the restriction of the dimension vector α to G, then a representation of rep α G in general position has a subrepresentation of dimension vector β. 3. β is minimal among all dimension vectors (,..., ) β α satisfying. and 2. A representation in rep α Q will then contain a simple subrepresentation of dimension vector β. Continue the process with starting dimension vector α β until this difference is the zero vector. We will ten have found the generic decomposition α = β β z into dimension vectors of simple representations. Calculate χ Q (β i, β i ). If it is zero, then β i occurs with multiplicity e i in the generic representation type τ gen if e i is the number of components β j in the generic decomposition which are equal to β i. This determines the generic representation type. The difficult part in the procedure is determining when a representation in general position has a subrepresentation of given dimension vector. In the next chapter we will prove a combinatorial procedure to verify this, due to A. Schofield [27]. 6.5 The Cayley-smooth locus. Let A be a Cayley-Hamilton algebra of degree n equipped with a trace map A and consider the quotient map tr A rep t n A π iss t n A Let ξ be a geometric point of he quotient scheme iss t n A with corresponding n- dimensional trace preserving semi-simple representation V ξ with decomposition V ξ = S e... S e k k where the S i are distinct simple representations of A of dimension d i such that n = k i= d ie i.

220 220 CHAPTER 6. LOCAL CLASSIFICATION. Definition 6.2 The Cayley-smooth locus of A is the subset of iss t n A Sm n A = {ξ iss t n A isst n A is smooth along π (ξ) } As the singular locus of iss t n A is a GL n -stable closed subscheme of iss t n A this is equivalent to Sm n A = {ξ iss t n A iss t n A is smooth in V ξ } We will give some conditions on ξ to be in the smooth locus Sm n A. To begin, rep t A is sooth in V n ξ if and only if the dimension of the tangent space in V ξ is equal to the local dimension of rep t A in V n ξ. In the previous chapter we have calculated the tangent space to be the set of trace preserving derivations A D M n (C) satisfying D(aa ) = D(a)ρ(a ) + ρ(a)d(a ) where A ρ Mn (C) is the C-algebra morphism determined by the action of A on V ξ and such that D is compatible with the traces, that is, the diagram A D M n (C) tr A tr A D Mn (C) is commutative. The C-vectorspace of such derivations is denoted by Derρ t A. Therefore, ξ Sm n A dim C Derρ t A = dim Vξ rep t A n Further, if ξ Sm n A, then we know from the Luna slice theorem that the local GL n -structure of rep t A near V n ξ is determined by a marked quiver setting (Q ξ, α ξ) where Q ξ is a marked quiver on k vertices (the number of distinct simple components of V ξ ) and α ξ = (e,..., e k ) (the multiplicities with which these simples occur in V ξ ). Recall that GL(α ξ ) is the stabilizer subgroup in GL n of V ξ and can be embedded in GL n after a suitable choice of basis via GL e (C d ) GL e2 (C d2 )... 0 GL(α ξ ) = GL ek (C GL n dk ) and we have an isomorphism of GL(α ξ )-modules rep αξ Q ξ N Vξ O(V ξ ) Ext t A(V ξ, V ξ ), the last isomorphism was proved in the previous chapter. This fact also allows us to determine the quiver Q ξ. Indeed, Ext A(V ξ, V ξ ) = k i,j=ext A(S i, S j ) eiej That is, as a GL(α ξ )-module Ext A (V ξ, V ξ ) is isomorphic to a quiver setting rep αξ Q big ξ where the arrows and loops in the quiver Q big ξ are given by # a = j i dim C Ext A(S i, S j ) if i j, and # i = dim C Ext A(S i, S i )

221 6.5. THE CAYLEY-SMOOTH LOCUS. 22 Proposition 6.22 With notations as before, Q ξ is a marked subquiver of the extension-quiver Q big ξ determined by the GL(α ξ )-submodule Ext t A (V ξ, V ξ ) Ext A (V ξ, V ξ ). It follows from the definition of trace preserving extensions Ext t A (V ξ, V ξ ) that Q ξ and Qbig ξ have the same number of arrows a j when i i j, but some of the loops in Q big ξ may vanish or get a marking in Q ξ. Observe that we can define this quiver Q ξ to any point ξ rept A whether ξ Sm n n A or not. However, if ξ Sm n A then by the Luna slice theorem, we have local étale isomorphisms between the varieties GL n GL(α ξ) rep αξ Q ξ et rep t A and rep n α ξ Q et ξ/gl(α ξ ) iss t n A Which gives us the following numerical restrictions on ξ Sm n A : Proposition 6.23 ξ Sm n A if and only if the following two equalities hold { dim Vξ rep t n A = n2 (e e 2 k ) + dim C Ext t A (V ξ, V ξ ) dim ξ iss t n A = dim 0 rep αξ Q ξ /GL(α ξ) Moreover, if ξ Sm n A, then rep t n A is a normal variety in a neighborhood of ξ Proof. The last statement follows from the fact that C[rep αξ closed and this property is preserved under the étale map. Q ξ ]GL(αξ) is integrally In general, the difference between these numbers gives a measure for the noncommutative singularity of A in ξ. In the next section we will refine these conditions under the extra assumption that A is an order in a central simple algebra. Example 6.24 Consider the affine C-algebra A = C x,y (xy+yx) then u = x2 and v = y 2 are central elements of A and A is a free module of rank 4 over C[u, v]. In fact, A is a C[u, v]-order in the quaternion division algebra «u v = C(u, v) and the reduced trace map on makes A into a Cayley-Hamilton algebra of degree 2. precisely, tr is the linear map on A such that ( tr(x i y j ) = 0 if either i or j are odd, and tr(x i y j ) = 2x i y j if i and j are even. More In particular, a trace preserving 2-dimensional representation is determined by a couple of 2 2 matrices»»»» x x ρ = ( 2 x4 x, 5 x x ) with tr( 2 x4 x. 5 ) = 0 x 3 x x 6 x 4 x 3 x x 6 x 4 That is,rep t 2 A is the hypersurface in C6 determined by the equation rep t 2 A = V(2x x 4 + x 2 x 6 + x 3 x 5 ) C 6 and is therefore irreducible of dimension 5 with an isolated singularity at p = (0,..., 0). image of the trace map is equal to the center of A which is C[u, v] and the quotient map The rep t 2 A π iss t 2 A = C 2 π(x,..., x 6 ) = (x 2 + x 2x 3, x x 5x 6 ) There are three different representation types to consider. Let ξ = (a, b) C 2 = iss t 2 A with ab 0, then π (ξ) is a closed GL 2 -orbit and a corresponding simple A-module is given by the matrixcouple»» i a 0 ( 0 i 0 b, a ) b 0

222 222 CHAPTER 6. LOCAL CLASSIFICATION. That is, ξ is of type (, 2) and the stabilizer subgroup are the scalar matrixes C 2 GL2. So, the action on both the tangentspace to rep t A and the tangent space to the orbit are trivial. 2 As they have respectively dimension 5 and 3, the normalspace corresponds to the quiver setting N ξ = which is compatible with the numerical restrictions. Next, consider a point ξ = (0, b) (or similarly, (a, 0)), then ξ is of type (, ;, ) and the corresponding semi-simple representation is given by the matrices (» 0 0, 0 0» i b 0 0 i ) b The stabilizer subgroup is in this case the maximal torus of diagonal matrices C C GL 2. The tangent space in this point to rep t 2 A are the 6-tuples (a,..., a 6 ) such that» 0 0 tr ( + ɛ 0 0»»» a a 2 i b 0 ).( a 3 a 0 i b4 b + ɛ 5 ) = 0 where ɛ b b 6 b 2 = 0 4 This leads to the condition a = 0, so the tangentspace are the matrix couples»»» 0 a2 a4 a (, 5 λ 0 ) on which the stabilizer a 3 0 a 6 a 4 0 µ acts via conjugation. That is, the tangentspace corresponds to the quiver setting Moreover, the tangentspace to the orbit is the image of the linear map (»»» m m 2 + ɛ b 0 ).(, m 3 m ), (» m m 2 2 ) b m 3 m 4 which is equal to» 0 0 ( 0 0»» b 0, 0 + ɛ b 0 2m 2 b 2m 3 b 0 ) on which the stabilizer acts again via conjugation giving the quiver setting Therefore, the normal space to the orbit corresponds to the quiver setting which is again compatible with the numerical restrictions. Finally, consider ξ = (0, 0) which is of type (2, ) and whose semi-simple representation corresponds to the zero matrix-couple. The action fixes this point, so the stabilizer is GL 2 and the tangent space to the orbit is the trivial space. Hence, the tangent space to rep t A coincides with the normalspace to the orbit and both 2 spaces are acted on by GL 2 via simultaneous conjugation leading to the quiver setting N ξ = 2 This time, the data is not compatible with the numerical restriction as 5 = dim rep t 2 A n2 e 2 + dim rep α Q = consistent with the fact that the zero matrix-couple is a (in fact, the only) singularity on rep t 2 A. 6.6 Cayley-smooth orders. Let X be a normal affine variety with coordinate ring C[X] and functionfield C(X). Let be a central simple C(X)-algebra of dimension n 2 which is a Cayley-Hamilton algebra of degree n using the reduced trace map tr. Let A be a C[X]-order in,

223 6.6. CAYLEY-SMOOTH ORDERS. 223 that is, the center of A is C[X] and A C[X] C(X). Because C[X] is integrally closed, the restriction of the reduced trace tr to A has its image in C[X], that is, A is a Cayley-Hamilton algebra of degree n and tr(a) = C[X] Consider the quotient morphism for the representation variety rep t n A π iss t n A then the above argument shows that X iss t n scheme is reduced. A and in particular the quotient Proposition 6.25 Let A be a Cayley-Hamilton order of degree n over C[X]. Then, its smooth locus Sm n A is a nonempty Zariski open subset of X. In particular, the set az A of Azumaya points, that is, of points x X = iss n A of representation type (, n) is a non-empty Zariski open subset of X and its intersection with the Zariski open subset X reg of smooth points of X satisfies X az X reg Sm n A Proof. Because AC(X) =, there is an f C[X] such that A f = A C[X] C[X] f is a free C[X] f -module of rank n 2 say with basis {a,..., a n 2}. Consider the n 2 n 2 matrix with entries in C[X] f R = tr(a a )... tr(a a n 2).. tr(a n 2a )... tr(a n 2a n 2) The determinant d = det R is nonzero in C[X] f. For, let K be the algebraic closure of C(X) then A f C[X]f K M n (K) and for any K-basis of M n (K) the corresponding matrix is invertible (for example, verify this on the matrixes e ij ). As {a,..., a n 2} is such a basis, d 0. Next, consider the Zariski open subset U = X(f) X(d) X. For any x X with maximal ideal m x C[X] we claim that A Am x A M n(c) Indeed, the images of the a i give a C-basis in the quotient such that the n 2 n 2 - matrix of their product-traces is invertible. This property is equivalent to the quotient being M n (C). Such points x are called Azumaya points of A. The corresponding semi-simple representation of A is simple, proving that az A is a non-empty Zariski open subset of X. But then, over U the restriction of the quotient map rep t n A π (U) U is a principal P GL n -fibration. In fact, this restricted quotient map determines an element in Het(U, P GL n ) determining the class of the central simple C(X)-algebra in Het(C(X), P GL n ). Restrict this quotient map further to U X reg, then the P GL n -fibration rep t A n π (U X reg ) U Xreg has a smooth base and therefore also the total space is smooth. But then, U X reg is a non-empty Zariski open subset of Sm n A.

224 224 CHAPTER 6. LOCAL CLASSIFICATION. We will now determine the étale local structure of A in points ξ Sm n A. Observe that the normality assumption on X is no restriction as the quotient scheme is locally normal in a point of Sm n A. Our next result drastically limits the local dimension vectors α ξ. Proposition 6.26 Let A be a Cayley-Hamilton order and ξ Sm n A such that the normal space to the orbit of the corresponding semi-simple n-dimensional representation is N ξ = rep αξ Q ξ Then, α ξ is the dimension vector of a simple representation of Q ξ. Proof. Let V ξ be the semi-simple representation of A determined by ξ. Let S ξ be the slice variety in V ξ then we have by the Luna slice theorem the following diagram of étale GL n -equivariant maps GL n GL(α ξ) S ξ et GL n GL(α ξ) rep αξ Q ξ et rep t n A linking a neighborhood of V ξ with one of ( n, 0). Because A is an order, every Zariski neighborhood of V ξ in rep t A contains simple n-dimensional representations, that n is, closed GL n -orbits with stabilizer subgroup isomorphic to C. Transporting this property via the GL n -equivariant étale maps, every Zariski neighborhood of ( n, 0) contains closed GL n -orbits with stabilizer C. By the correspondence of orbits is associated fiber bundles, every Zariski neighborhood of the trivial representation 0 rep αξ Q ξ contains closed GL(α ξ)-orbits with stabilizer subgroup C. We have seen that closed GL(α ξ )-orbits correspond to semi-simple representations of Q ξ. However, if the stabilizer subgroup of a semi-simple representation is C this representation must be simple. These two results allow us to refine the numerical characterization of smooth points given in the previous section. Theorem 6.27 Let A be a Cayley-Hamilton order of degree n with center C[X] where X is a normal variety of dimension d. For ξ X = iss t n A with corresponding semi-simple representation V ξ = S e... S e k k and normal space to the orbit O(V ξ ) isomorphic to rep αξ Q ξ as GL(α ξ)-modules where α ξ = (e,..., e k ). Then, ξ Sm n A if and only if the following two conditions are met { α ξ is the dimension vector of a simple representation of Q, and d = χ Q (α ξ, α ξ ) k i= m i where Q is the underlying quiver of Q ξ and m i is the number of marked loops in Q ξ in vertex v i.

225 6.6. CAYLEY-SMOOTH ORDERS. 225 Proof. By the Luna slice theorem we have étale maps rep αξ Q ξ/gl(α ξ ) et S ξ /GL(α ξ ) et iso t n A = X connecting a neighborhood of ξ X with one of the trivial semi-simple representation 0. By definition of the Euler-form of Q we have that χ Q (α ξ, α ξ ) = i j e i e j χ ij + i e 2 i ( u i m i ) On the other hand we have the following dimensions dim rep α Q α ξ = e i e j χ ij + i j i e 2 i (u i + m i ) i m i dim GL(α ξ ) = i e 2 i As any Zariski open neighborhood of ξ contains an open set where the quotient map is a P GL(α ξ ) = GL(α ξ) C -fibration we see that the quotient variety rep αξ Q ξ has dimension equal to dim rep αξ Q ξ dim GL(α ξ ) + and plugging in the above information we see that this is equal to χ Q (α ξ, α ξ ) i m i. Example 6.28 The quantum plane. We will generalize the discussion of example 6.24 to the algebra A = C x, y (yx qxy) where q is a primitive n-th root of unity. Let u = x n and v = y n then it is easy to see that A is a free module of rank n 2 over its center C[u, v] and is a Cayley-Hamilton algebra of degree n with the trace determined on the basis ( tr(x i y j 0 when either i or j is not a multiple of n, ) = nx i y j when i and j are multiples of n, Let ξ iss n A = C 2 be a point (a n, b) with a.b 0, then ξ is of representation type (, n) as the corresponding (semi)simple representation V ξ is determined by (if m is odd, for even n we replace a by ia and b by b) a qa ρ(x) = and ρ(y) = q n a b One computes that Ext A (V ξ, V ξ ) = C 2 where the algebra map A to (α, β) is given by ( φ(x) = ρ(x) + ε α n φ(y) = ρ(y) + ε β n φ Mn(C[ε]) corresponding and all these algebra maps are trace preserving. That is, Ext A (V ξ, V ξ ) = Ext t A (V ξ, V ξ ) and as the stabilizer subgroup is C the marked quiver-setting (Q ξ, α ξ) is and d = χ Q (α, α) P i m i as 2 = ( ) + 0, compatible with the fact that over these points the quotient map is a principal P GL n-fibration. Next, let ξ = (a n, 0) with a 0 (or, by a similar argument (0, b n ) with b 0). Then, the representation type of ξ is (, ;... ;, ) because V ξ = S... S n

226 226 CHAPTER 6. LOCAL CLASSIFICATION. where the simple one-dimensional representation S i is given by ( ρ(x) = q i a ρ(y) = 0 One verifies that Ext A (S i, S i ) = C and Ext A (S i, S j ) = δ i+,j C and as the stabilizer subgroup is C... C, the Ext-quiver setting is φ The algebra map A Mn(C[ε]) corresponding to the extension (α, β,..., α n, β n) Ext A (V ξ, V ξ ) is given by a + ε α qa + ε α 2 φ(x) = >< >: φ(y) q n a + ε α 2 3 n 0 β β 2 0 = ε β n 5 β n The conditions tr(x j ) = 0 for i < n impose n linear conditions among the α j, whence the space of trace preserving extensions Ext t A (V ξ, V ξ ) corresponds to the quiver setting The Euler-form of this quiver Q is given by the n n matrix giving the numerical restriction as α ξ = (,..., ) χ Q (α, α) X i m i = ( ) 0 = 2 = dim iss t n A so ξ Sm n A. Finally, the only remaining point is ξ = (0, 0). This has representation type (n, ) as the corresponding semi-simple representation V ξ is the trivial one. The stabilizer subgroup is GL n and the (trace preserving) extensions are given by Ext A (V ξ, V ξ ) = M n M n and Ext t A (V ξ, V ξ ) = M 0 n M 0 n

227 6.7. SMOOTH LOCAL TYPES. 227 determined by the algebra maps A φ Mn(C[ε]) given by ( φ(x) = ε m φ(y) = ε m 2 That is, the relevant quiver setting (Q ξ, α ξ) is in this point n This time, ξ / Sm n A as the numerical condition fails χ Q (α, α) X i m i = ( n 2 ) 0 2 = dim iss t n A unless n =. That is, Sm n A = C 2 {(0, 0)}. 6.7 Smooth local types. If we want to study the local structure of Cayley-Hamilton orders A of degree n over a central normal variety X of dimension d, we have to compile a list of admissible marked quiver settings, that is couples (Q, α) satisfying the two properties { α is the dimension vector of a simple representation of Q, and d = χ Q (α, α) i m i In this section, we will give the first steps in such a classification project The basic idea that we use is to shrink a marked quiver-setting to its simplest form and classify these simplest forms for given d. By shrinking we mean the following process. Assume α = (e,..., e k ) is the dimension vector of a simple representation of Q and let v i and v j be two vertices connected with an arrow such that e i = e j = e. That is, locally we have the following situation u i χpi e χ iq m i χ ij χ ji u j χ rj e χjs m j We will use one of the arrows connecting v i with v j to identify the two vertices. That is, we form the shrinked marked quiver-setting (Q s, α s ) where Q s is the marked quiver on k vertices {v,..., ˆv i,..., v k } and α s is the dimension vector with e i removed. That is, Q s has the following form in a neighborhood of the contracted vertex u i + u j + χ ij + χ ji χpi + χ pj χ rj + χ ri e χ iq + χ jq χjs + χ is m i + m j That is, in Q s we have for all k, l i that χ s kl = χ kl. Moreover, the number of arrows and (marked) loops connected to v j are determined as follows χ s jk = χ ik + χ jk χ s kj = χ ki + χ kj u s j = u i + u j + χ ij + χ ji

228 228 CHAPTER 6. LOCAL CLASSIFICATION. m s j = m i + m j Lemma 6.29 α is the dimension vector of a simple representation of Q if and only if α s is the dimension vector of a simple representation of Q s. Moreover, dim rep α Q /GL(α) = dim rep αs Q s/gl(α s ) Proof. Fix an arrow a. j i As e i = e j = e there is a Zariski open subset U rep α Q of points V such that V a is invertible. By basechange in either v i or v j we can find a point W in its orbit such that W a = e. If we think of W a as identifying C ei with C ej we can view the remaining maps of W as a representation in rep αs Q s and denote it by W s. The map U repαs Q s is well-defined and maps GL(α)-orbits to GL(α s )-orbits. Conversely, given a representation W rep αs Q s we can uniquely determine a representation W U mapping to W. Both claims follow immediately from this observation. It is clear that any marked quiver-setting can uniquely be reduced to its simplest form, which has the characteristic property that no arrow-connected vertices can have the same dimension. The shrinking process has a converse operation which we will call splitting of a vertex. However, this splitting operation is usually not uniquely determined. Before we can compile lists of marked-quiver settings in simplified form for a specific base-dimension d, we need to bound the components of the occurring dimension vectors α. We will do this in the case of quivers and leave the extension to marked quivers an an exercise. Proposition 6.30 Let α = (e,..., e k ) be the dimension vector of a simple representation of Q and let χ Q (α, α) = d = dim rep α Q /GL(α). Then, if e = max e i, we have that d e +. Proof. By the above lemma we may assume that (Q, α) is brought in its simplest form, that is, no two arrow-connected vertices have the same dimension. Let χ ii denote the number of loops in a vertex v i, then { i e i ( j χ Q (α, α) = χ ije j e i ) i e i ( j χ jie j e i ) and observe that the bracketed terms are positive by the requirement that α is the dimension vector of a simple representation. We call them the incoming in i, respectively outgoing out i, contribution of the vertex v i to d. Let v m be a vertex with maximal vertex-dimension e. in m = e( j m χ jm e j + (χ ii )e) and out m = e( j m χ ij e j + (χ ii )e) If there are loops in v m, then in m 2 or out m 2 unless the local structure of Q is e in which case in m = e = out m. Let v i be the unique incoming vertex of v m, then we have out i e. But then, d = χ Q (α, α) = + j out j 2e

229 6.7. SMOOTH LOCAL TYPES. 229 If v m has no loops, consider the incoming vertices {v i,..., v is }, then s in m = e( χ ijme ij e) j= which is e unless χ ijme ij = e, but in that case we have s out ij j= e 2 s e 2 i j j= e the last inequality because all e ij < e. In either case we have that d = χ Q (α, α) = + i out i = + i in i e +. This result allows us to compile a list of all possible marked quiver-settings in simplest form for small values of d. In such a list we are only interested in rep α Q as GL(α)-module and we call two setting equivalent if they determine the same GL(α)-module. For example, the marked quiver-settings 2 and 2 determine the same C GL 2 -module, hence are equivalent. Theorem 6.3 Let A be a Cayley-Hamilton order of degree n over a central normal variety X of degree d. Then, the local quiver of A in a point ξ X = iss t n A belonging to the smooth locus Sm n A can be shrinked to one of a finite list of equivalence classes of marked quiver-settings. For d 4, the complete lists are given below where the boxed values are the dimension d of X

230 230 CHAPTER 6. LOCAL CLASSIFICATION. An immediate consequence is the following analog of the fact that commutative smooth varieties have only one type of analytic (or étale) local behavior. Theorem 6.32 There are only finitely many types of étale local behaviour of smooth Cayley-Hamilton orders of degree n over a central variety of dimension d. Proof. The foregoing reduction shows that for fixed d there are only a finite number of marked quiver-settings shrinked to their simplest form. As e i n, we can only apply the splitting operations on vertices a finite number of times. 6.8 Curve orders. W. Schelter has proved in [26] that in dimension one, smooth orders are hereditary. In this section we will give an alternative proof of this result using the étale local classification. The result below can also be proved by the splitting operation and the above classification. We give this direct proof as an illustration of the stratification result of 4. Theorem 6.33 Let A be a Cayley-Hamilton order of degree n over an affine curve X = iss t n A. If ξ Sm n A, then the étale local structure of A in ξ is determined by a marked quiver-setting which is an oriented cycle on k vertices with k n. and an unordered partition p = (d,..., d k ) having precisely k parts such that i d i = n determining the dimensions of the simple components of V ξ. Proof. Let (Q, α) be the corresponding local marked quiver-setting. Because Q is strongly connected, there exist oriented cycles in Q. Fix one such cycle of length s k and renumber the vertices of Q such that the first s vertices make up the cycle. If α = (e,..., e k ), then there exist semi-simple representations in rep α Q with composition α = (,...,, 0,..., 0) ɛ e }{{}}{{}... ɛs es s k s ɛ es+ s+... ɛ e k k where ɛ i stands for the simple one-dimensional representation concentrated in vertex v i. There is a one-dimensional family of simple representations of dimension vector α, hence the stratum of semi-simple representations in iss α Q of representation type τ = (, α ; e, ɛ ;... ; e s, ɛ s ; e s+, ɛ s+ ; e k, ɛ k ) is at least one-dimensional. However, as dim iss α Q = this can only happen if this semi-simple representation is actually simple. That is, when α = α and k = s.

231 6.8. CURVE ORDERS. 23 Hence, if V ξ is the semi-simple n-dimensional representation of A corresponding to ξ, then V ξ = S... S k with dim S i = d i That is, the stabilizer subgroup is GL(α) = C... C embedded in GL n via the diagonal embedding (λ,..., λ k ) diag(λ,..., λ }{{},..., λ k,..., λ k ) }{{} d d k Further, using basechange in rep α Q we can bring every simple α-dimensional representation of Q α in standard form x where x C is the arrow from v k to v. That is, C[rep α Q ] GL(α) C[x] proving that the quotient (or central) variety X must be smooth in ξ by the Luna slice result. Moreover, as Âξ T α Q we have, using the numbering conventions of the vertices) the following block decomposition M d (C[[x]]) M d d 2 (C[[x]])... M d d k (C[[x]]) M d2 d (xc[[x]]) M d2 (C[[x]])... M d2 d k (C[[x]]) Â ξ M dk d (xc[[x]]) M dk d 2 (xc[[x]])... M dk (C[[x]]) and from the local description of hereditary orders given in [25, Thm. 39.4] we deduce that A ξ is an hereditary order. That is, we have the following characterization of the smooth locus Proposition 6.34 Let A be a Cayley-Hamilton order of degree n over a central affine curve X. Then, Sm n A is the locus of points ξ X such that A ξ is an hereditary order (in particular, ξ must be a smooth point of X). Globalizing this result, we obtain the following characterization of noncommutative smooth models in dimension one. Theorem 6.35 Let A be a Cayley-Hamilton central O X -order of degree n where X is a projective curve. Equivalent are. A is a sheaf of Cayley-smooth orders 2. X is smooth and A is a sheaf of hereditary O X -orders

232 232 CHAPTER 6. LOCAL CLASSIFICATION. 6.9 Surface orders. The result below can equally be proved using the splitting operation and the classification result. Theorem 6.36 Let A be a Cayley-Hamilton order of degree n over an affine surface X = iss t n A. If ξ Sm n A, then the étale local structure of A in ξ is determined by a marked local quiver-setting A klm on k + l + m n vertices 2 k+l+m k+ k+l+ k+l. k- k and an unordered partition p = (d,..., d k+l+m ) of n with k + l + m non-zero parts determined by the dimensions of the simple components of V ξ. Proof. Let (Q, α) be the marked quiver-setting on r vertices with α = (e,..., e r ) corresponding to ξ. As Q is strongly connected and the quotient variety is twodimensional, Q must contain more than one oriented cycle, hence it contains a subquiver of type A klm, possibly degenerated with k or l equal to zero. Order the first k+ l + m vertices of Q as indicated. One verifies that A klm has simple representations of dimension vector (,..., ). Assume that A klm is a proper subquiver and denote s = k+l+m+ then Q has semi-simple representations in rep α Q with dimensionvector decomposition α = (,...,, 0,..., 0) ɛ e }{{}... ɛ e k+l+m k+l+m k+l+m ɛ es s... ɛ er r Applying the formula for the dimension of the quotient variety shows that iss (,...,) A klm has dimension 2 so there is a two-dimensional family of such semisimple representation in the two-dimensional quotient variety iss α Q. This is only possible if this semi-simple representation is actually simple, whence r = k + l + m, Q = A klm and α = (,..., ). If V ξ is the semi-simple n-dimensional representation of A corresponding to ξ, then V ξ = S... S r with dim S i = d i

233 6.9. SURFACE ORDERS. 233 and the stabilizer subgroup GL(α) = C... C embedded diagonally in GL n (λ,..., λ r ) diag(λ,..., λ }{{},..., λ r,..., λ r ) }{{} d d r By basechange in rep α A klm we can bring every simple α-dimensional representation in the following standard form x y with x, y C and as C[iss α A klm ] = C[rep α A klm ] GL(α) is the ring generated by traces along oriented cycles in A klm, it is isomorphic to C[x, y]. From the Luna slice results one deduces that ξ must be a smooth point of X and because Âξ T α A klm we deduce it must have the following block-decomposition  ξ () (y) (x) () (x) (y) () () Mn (C[[x, y]]) (x) } {{ } k () (y) (x, y) }{{}}{{} l m where at spot (i, j) with i, j k + l + m there is a block of dimension d i d j with entries the indicated ideal of C[[x, y]]. Definition 6.37 Let A be a Cayley-Hamilton central C[X]-order of degree n in a central simple C(X)- algebra of dimension n 2.. A is said to be étale locally split in ξ if and only if Âξ is a central Ô X,x -order in M n (ÔX,x OX,x C(X)). 2. The ramification locus ram A of A is the locus of points ξ X such that A m ξ Am ξ M n (C) The complement X ram A is called the Azumaya locus az A of A.

234 234 CHAPTER 6. LOCAL CLASSIFICATION. Theorem 6.38 Let A be a Cayley-smooth central O X -order of degree n over a projective surface X. Then,. X is smooth. 2. A is étale locally split in all points of X. 3. The ramification divisor ram A X is either empty or consists of a finite number of isolated (possibly embedded) points and a reduced divisor having as its worst singularities normal crossings. Proof. () and (2) follow from the above local description of A. As for (3) we have to compute the local quiver-settings in proper semi-simple representations of rep α A klm. As simples have a strongly connected support, the decomposition types of these proper semi-simple can be depicted by one of the following two situations x y with x, y C. By the description of local quivers given in section 3 we see that they are respectively of the following form A 0l A k0 and the associated unordered partitions are defined in the obvious way, that is, to the looped vertex one assigns the sum of the d i belonging to the loop-contracted circuit and the other components of the partition are preserved. Using the étale local isomorphism between X in a neighborhood of ξ and of iss α A klm in a neighborhood of the trivial representation, we see that the local picture of quiver-settings of A in

235 6.9. SURFACE ORDERS. 235 a neighborhood of ξ can be represented by A klm A k0 A 0l A 00 The Azumaya points are the points in which the quiver-setting is A 00 (the twoloop quiver). From this local description the result follows if we take care of possibly degenerated cases. For example, an isolated point in ξ can occur if the quiver-setting in ξ is of type A 00m with m 2, that is, In the next section we will characterize those central simple C(X)-algebras allowing a Cayley-smooth model. We first need to perform a local calculation. Consider the ring of algebraic functions in two variables C{x, y} and let X loc = Spec C{x, y}. There is only one codimension two subvariety : m = (x, y). Let us compute the coniveau spectral sequence for X loc. If K is the field of fractions of C{x, y} and if we denote with k p the field of fractions of C{x, y}/p where p is a

236 236 CHAPTER 6. LOCAL CLASSIFICATION. height one prime, we have as its first term H 2 (K, µ n ) p H (k p, Z n ) µ n 0... H (K, µ n ) p Z n µ n Because C{x, y} is a unique factorization domain, we see that the map H et(k, µ n ) = K /(K ) n γ p Z n is surjective. Moreover, all fields k p are isomorphic to the field of fractions of C{z} whose only cyclic extensions are given by adjoining a root of z and hence they are all ramified in m. Therefore, the component maps Z n = H et(k p, Z n ) β L µ are isomorphisms. But then, the second (and limiting) term of the spectral sequence has the form Ker α Ker β/im α Ker γ µ n Finally, we use the fact that C{x, y} is strict Henselian whence has no proper étale extensions. But then, H i et(x loc, µ n ) = 0 for i and substituting this information in the spectral sequence we obtain that the top sequence of the coniveau spectral sequence 0 Brn K α p Z n Zn 0 is exact. From this sequence we immediately obtain the following Lemma 6.39 With notations as before, we have

237 6.0. NONCOMMUTATIVE SMOOTH SURFACES Let U = X loc V (x), then Br n U = 0 2. Let U = X loc V (xy), then Br n U = Z n with generator the quantum-plane algebra C u, v C ζ [u, v] = (vu ζuv) where ζ is a primitive n-th root of one 6.0 Noncommutative smooth surfaces. Let be a central simple algebra of dimension n 2 over a field of transcendence degree 2 say L. We want to determine when admits a Cayley-smooth order A, that is, a sheaf of Cayley-smooth O X -algebras where X is a projective surface with functionfield C(X) = L. In the previous section we have seen that if such a model exists, then X has to be a smooth projective surface. So we may assume that X is a commutative smooth model for L. But then we know from the Artin-Mumford exact sequence, proved in chapter 2, that the class of in Br n C(X) is determined by the following geo-combinatorial data A finite collection C = {C,..., C k } of irreducible curves in X. A finite collection P = {P,..., P l } of points of X where each P i is either an intersection point of two or more C i or a singular point of some C i. For each P P the branch-data b P = (b,..., b ip ) with b i Z n = Z/nZ and {,..., i P } the different branches of C in P. These numbers must satisfy the admissibility condition b i = 0 Z n for every P P for each C C we fix a cyclic Z n -cover of smooth curves i D C of the desingularization C of C which is compatible with the branch-data. We have seen in chapter 2 that if A is a maximal O X -order in, then the ramification locus ram A coincides with the collection of curves C. We fix such a maximal O X -order A and investigate its smooth locus. Proposition 6.40 Let A be a maximal O X -order in with X a projective smooth surface and with geo-combinatorial data (C, P, b, D) determining the class of in Br n C(X). If ξ X lies in X C or if ξ is a non-singular point of C, then A is smooth in ξ. Proof. If ξ / C, then A ξ is an Azumaya algebra over O X,x. As X is smooth in ξ, A is Cayley-smooth in ξ. Alternatively, we know that Azumaya algebras are split by étale extensions, whence Âξ M n (C[[x, y]]) which shows that the behaviour of A near ξ is controlled by the local data }. {{.. } n

238 238 CHAPTER 6. LOCAL CLASSIFICATION. and hence ξ Sm n A. Next, assume that ξ is a nonsingular point of the ramification divisor C. Consider the pointed spectrum X ξ = Spec O X,ξ {m ξ }. The only prime ideals are of height one, corresponding to the curves on X passing through ξ and hence this pointed spectrum is a Dedekind scheme. Further, A determines a maximal order over X ξ. But then, tensoring A with the strict henselization O sh X,ξ C{x, y} determines a sheaf of hereditary orders on the pointed spectrum ˆX ξ = Spec C{x, y} {(x, y)} and we may choose the local variable x such that x is a local parameter of the ramification divisor C near ξ. Using the characterization result for hereditary orders over discrete valuation rings, given in [25, Thm. 39.4] we know the structure of this extended sheaf if hereditary orders over every height one prime of ˆXξ. Because A ξ is a reflexive (even a projective) O X,ξ -module this height one information determines A sh ξ or Âξ. This proves that A sh ξ must be isomorphic to the following blockdecomposition M d (C{x, y}) M d d 2 (C{x, y})... M d d k (C{x, y}) M d2 d (xc{x, y}) M d2 (C{x, y})... M d2 d k (C{x, y}) M dk d (xc{x, y}) M dk d 2 (xc{x, y})... M dk (C{x, y}) for a certain partition p = (d,..., d k ) of n having k parts. In fact, as we started out with a maximal order A one can even show that all these integers d i must be equal. Anyway, this local form corresponds to the following quiver-setting p = (d,..., d k ) A k0 whence ξ Sm n A as this is one of the allowed surface settings. Concluding, a maximal O X -order in can have at worst noncommutative singularities in the singular points of the ramification divisor C. We have seen that a Cayley-smooth order over a surface has as ramification-singularities at worst normal crossings. We are always able to reduce to normal crossings by the following classical result on commutative surfaces, see for example [9, V.3.8]. Theorem 6.4 (Embedded resolution of curves in surfaces) Let C be any curve on the surface X. Then, there exists a finite sequence of blow-ups X = X s Xs... X0 = X

239 6.0. NONCOMMUTATIVE SMOOTH SURFACES. 239 and, if f : X X is their composition, then the total inverse image f (C) is a divisor with normal crossings. Fix now a series of blow-ups X f X such that the inverse image f (C) is a divisor on X having as worst singularities normal crossings. We will now replace the Cayley-Hamilton O X -order A by a Cayley-Hamilton O X -order A where A is a sheaf of O X -maximal orders in. In order to determine the ramification divisor of A we need to be able to keep track how the ramification divisor C of changes if we blow up a singular point p P. Lemma 6.42 Let X X be the blow-up of X at a singular point p of C, the ramification divisor of on X. Let C be the strict transform of C and E the exceptional line on X. Let C be the ramification divisor of on the smooth model X. Then,. Assume the local branch data at p distribute in an admissible way on C, that is, b i,p = 0 for all q E C i at q where the sum is taken only over the branches at q. Then, C = C. 2. Assume the local branch data at p do not distribute in an admissible way, then C = C E. Proof. Clearly, C C C E. By the Artin-Mumford sequence applied to X we know that the branch data of C must add up to zero at all points q of C E. We investigate the two cases. : Assume E C. Then, the E-branch number at q must be zero for all q C E. But there are no non-trivial étale covers of P = E so ram( ) gives the trivial element in Het(C(E), µ n ), a contradiction. Hence C = C. E a a p a a a a 2. : If at some q C E the branch numbers do not add up to zero, the only remedy is to include E in the ramification divisor and let the E-branch number be such that the total sum is zero in Z n. Theorem 6.43 Let be a central simple algebra of dimension n 2 over a field L of transcendence degree two. Then, there exists a smooth projective surface S with functionfield C(S) = L such that any maximal O S -order A S in has at worst a finite number of isolated noncommutative singularities. Each of these singularities is locally étale of quantum-plane type. Proof. We take any projective smooth surface X with functionfield C(X) = L. By the Artin-Mumford exact sequence, the class of determines a geo-combinatorial set of data (C, P, b, D)

240 240 CHAPTER 6. LOCAL CLASSIFICATION. as before. In particular, C is the ramification divisor ram( ) and P is the set of singular points of C. We can separate P in two subsets P unr = {P P where all the branch-data b P = (b,..., b ip ) are trivial, that is, all b i = 0 in Z n } = (b,..., b ip ) are non- P ram = {P P where some of the branch-data b P trivial, that is, some b i 0 in Z n } After a finite number of blow-ups we get a birational morphism S π X such that π (C) has as its worst singularities normal crossings and all branches in points of P are separated in S. Let C be the ramification divisor of in S. By the foregoing argument we have If P P unr, then we have that C π (P ) consists of smooth points of C, If P P ram, then π (P ) contains at least one singular points Q of C with branch data b Q = (a, a) for some a 0 in Z n. In fact, after blowing-up singular points Q in π (P ) with trivial branch-data we obtain a smooth surface S S X such that the only singular points of the ramification divisor C of have non-trivial branch-data (a, a) for some a Z n. Then, take a maximal O S -order A in. By the local calculation of Br n C{x, y} performed in the last section we know that locally étale A is of quantum-plane type in these remaining singularities. As the quantum-plane is not étale locally split, A is not Cayley-smooth in these finite number of singularities. In fact, the above proof gives also a complete classification of those central simple algebras admitting a Cayley-smooth model. Theorem 6.44 Let be a central simple C(X)-algebra of dimension n 2 determined by the geo-combinatorial data (C, P, b, D) given by the Artin-Mumford sequence. Then, admits a Cayley-smooth model if and only if all branch-data are trivial. Proof. If all branch-data are trivial, the foregoing proof constructs a Cayley-smooth model of. Conversely, if A is a Cayley-smooth O S -order in with S a smooth projective model of C(X), then A is locally étale split in every point s S. But then, so is any maximal O S -order A max containing A. By the foregoing arguments this can only happen if all branch-data are trivial. 6. Higher dimensional orders. The strategy we used to characterize the central simple algebras over a surface admitting a Cayley-smooth model can also be applied (at least in principle) to higher dimensional varieties. First, one uses the classification result of marked quiversettings to compile a list of allowed étale local behaviour of Cayley-smooth orders and of their ramification. Next, if a subclass of central simple algebras is determined by ramification data, the obtained local behaviour puts restrictions on those admitting a smooth model. We have seen that in the case of curves and surfaces, the central variety X of a Cayley-smooth model A had to be smooth and that A is étale locally split in every point ξ X. Both of these properties are no longer valid in higher dimensions. Lemma 6.45 For dimension d 3, the center Z of a Cayley-smooth order of degree n can have singularities.

241 6.. HIGHER DIMENSIONAL ORDERS. 24 Proof. Consider the following marked quiver-setting b a c d which is allowed for dimension d = 3 and degree n = 2. The quiver-invariants are generated by the traces along oriented cycles, that is, are generated by ac, ad, bc and bd. That is, C[x, y, z, v] C[iss α Q] (xv yz) which has a singularity in the origin. This example can be extended to dimensions d 3 by adding loops in one of the vertices. b a d 3 c d Lemma 6.46 For dimension d 3, a Cayley-smooth algebra does not have to be locally étale split in every point of its central variety. Proof. Consider the following allowable quiver-setting for d = 3 and n = 2 2 The corresponding Cayley-smooth algebra A is generated by two generic 2 2 trace zero matrices, say A and B. Using our knowledge of T 2 2 we see that its center is generated by A 2 = x, B 2 = z and AB + BA = z. Alternatively, we can identify A with the Clifford-algebra over C[x, y, z] of the non-degenerate quadratic form [ ] x y y z This is a noncommutative domain and remains to be so over the formal power series C[[x, y, z]]. That is, A cannot be split by an étale extension in the origin. More generally, whenever the local marked quiver contains vertices with dimension 2, the corresponding Cayley-smooth algebra cannot be split by an étale extension as the local quiver-setting does not change and for a split algebra all vertex-dimensions have to be equal to. In particular, the Cayley-smooth algebra of degree 2 corresponding to the quiver-setting k 2 cannot be split by an étale extension in the origin. Its corresponding dimension is d = 3k + 4l 3 l whenever k + l 2 and so all dimensions d 3 are reached.

242 242 CHAPTER 6. LOCAL CLASSIFICATION.

243 Chapter 7 Moduli Spaces. In the study of the Hilbert scheme Hilb n of n points in C 2 we ran into the quiver setting (Q, α) n In fact, we proved that Hilb n is the orbit space of the GL(α)-orbits on triples (A, B, u) rep α Q = M n M n C n such that [A, B] = 0 and u is a cyclic vector for (A, B). None of these triples (A, B, u) determines a closed GL(α)-orbit in rep α Q because lim (, t n ).(A, B, v) = (A, B, 0) t 0 Still, a cyclic triple does determine a closed GL(α)-orbit in some Zariski open subset rep (σ) determined by a Hilbert stair σ, as the dimension of all GL(α)-orbits in rep (σ) is equal to n 2. Such situations, where a shortage of closed orbits is compensated when restricted to suitable open subsets, often occur such as in the study of linear dynamical systems as we will see in the first sections. For a general quiver setting (Q, α) and a character χ θ : GL(α) C we will study a moduli space Mα ss (Q, θ) classifying closed orbits in the Zariski open subset of so called θ-semistable representations of rep α Q. These moduli spaces were introduced and studied by A. King in [2]. The intuition we have formed on algebraic quotient varieties is helpful in studying these moduli spaces provided we use the following dictionary iss α Q closed orbits in rep α Q simple representation semi-simple representation polynomial invariants M ss α (Q, θ) closed orbits in rep ss α (Q, θ) θ-stable representation direct sum of θ-stable representations semi-invariants of weight θ A first important problem is to determine which of these moduli spaces are nonempty, that is for which triples (Q, α, θ) do there exist θ-(semi)stable representations in rep α Q. A beautiful inductive combinatorial answer to this problem was discovered by A. Schofield [27]. His characterization of the dimension vectors α allowing θ-stables is as fundamental to the study of the moduli spaces Mα ss (Q, θ) as the description of the dimension vectors of simple representations is to the study of the quotient varieties iss α Q. These moduli spaces are defined to be the projective varieties of certain graded algebras of semi-invariant functions. Hence, we need to find generators of semi-invariants precisely as we needed to control the polynomial 243

244 244 CHAPTER 7. MODULI SPACES. invariants to study the quotient varieties. Of the many independent description of these semi-invariants we follow here the approach due to A. schofield and M. Van den Bergh in [28]. In the next chapter we will see that the investigation of these moduli spaces is crucial in the study of fibers of the representation spaces rep n A iss n A and of the Brauer-Severi fibration BS n A iss n A for Cayley-smooth algebras A of degree n. 7. Dynamical systems. A linear time invariant dynamical system Σ is determined by the following system of differential equations dx = Bx + Au dt (7.) y = Cx. Here, u(t) C m is the input or control of the system at tome t, x(t) C n the state of the system and y(t) C p the output of the system Σ. Time invariance of Σ means that the matrices A M n m (C), B M n (C) and C M p n (C) are constant. The system Σ can be represented as a black box u(t) y(t) x(t) which is in a certain state x(t) that we can try to change by using the input controls u(t). By reading the output signals y(t) we can try to determine the state of the system. Recall that the matrix exponential e B of any n n matrix B is defined by the infinite series e B = n + B + B2 2! Bm m! +... The importance of this construction is clear from the fact that e Bt is the fundamental matrix for the homogeneous differential equation dx dt = Bx. That is, the columns of e Bt are a basis for the n-dimensional space of solutions of the equation dx dt = Bx. Motivated by this, we look for a solution to equation (7.) as the form x(t) = e Bt g(t) for some function g(t). Substitution gives the condition dg τ dt = e Bt Au whence g(τ) = g(τ 0 ) + e Bt Au(t)dt. τ 0 Observe that x(τ 0 ) = e Bτ0 g(τ 0 ) and we obtain the solution of the linear dynamical system Σ = (A, B, C) : { x(τ) = e (τ τ0)b x(τ 0 ) + τ τ 0 e (τ t)b Au(t)dt y(τ) = Ce B(τ τ0) x(τ 0 ) + τ τ 0 Ce (τ t)b Au(t)dt. Differentiating we see that this is indeed a solution and it is the unique one having a prescribed starting state x(τ 0 ). Indeed, given another solution x (τ) we have that x (τ) x(τ) is a solution to the homogeneous system dx dt = Bt, but then x (τ) = x(τ) + e τb e τ0b (x (τ 0 ) x(τ 0 )).

245 7.. DYNAMICAL SYSTEMS. 245 We call the system Σ completely controllable if we can steer any starting state x(τ 0 ) to the zero state by some control function u(t) in a finite time span [τ 0, τ]. That is, the equation τ 0 = x(τ 0 ) + e (τ0 t)b Au(t)dt τ 0 has a solution in τ and u(t). As the system is time-invariant we may always assume that τ 0 = 0 and have to satisfy the equation 0 = x 0 + τ 0 e tb Au(t)dt for every x 0 C n (7.2) Consider the control matrix c(σ) which is the n mn matrix c(σ) = A BA B 2 A... B n- A Assume that rk c(σ) < n then there is a non-zero state s C n such that s τ c(σ) = 0, where s τ denotes the transpose (row column) of s. Because B satisfies the characteristic polynomial χ B (t), B n and all higher powers B m are linear combinations of { n, B, B 2,..., B n }. Hence, s τ B m A = 0 for all m. Writing out the power series expansion of e tb in equation (7.2) this leads to the contradiction that 0 = s τ x 0 for all x 0 C n. Hence, if rk c(σ) < n, then Σ is not completely controllable. Conversely, let rk c(σ) = n and assume that Σ is not completely controllable. That is, the space of all states s(τ, u) = τ 0 e tb Au(t)dt is a proper subspace of C n. But then, there is a non-zero state s C n such that s tr s(τ, u) = 0 for all τ and all functions u(t). Differentiating this with respect to τ we obtain s tr e τb Au(τ) = 0 whence s tr e τb A = 0 (7.3) for any τ as u(τ) can take on any vector. For τ = 0 this gives s tr A = 0. If we differentiate (7.3) with respect to τ we get s tr Be τb A = 0 for all τ and for τ = 0 this gives s tr BA = 0. Iterating this process we show that s tr B m A = 0 for any m, whence s tr [ A BA B 2 A... B n A ] = 0 contradicting the assumption that rk c(σ) = n. That is, we have proved : Proposition 7. A linear time-invariant dynamical system Σ determined by the matrices (A, B, C) is completely controllable if and only if rk c(σ) is maximal. We say that a state x(τ) at time τ is unobservable if Ce (τ t)b x(τ) = 0 for all t. Intuitively this means that the state x(τ) cannot be detected uniquely from the output of the system Σ. Again, if we differentiate this condition a number of times and evaluate at t = τ we obtain the conditions Cx(τ) = CBx(τ) =... = CB n x(τ) = 0. We say that Σ is completely observable if the zero state is the only unobservable state at any time τ. Consider the observation matrix o(σ) of the system Σ which is the pn n matrix o(σ) = [ C tr (CB) tr... (CB n ) tr] tr

246 246 CHAPTER 7. MODULI SPACES. An analogous argument as in the proof of proposition 7. gives us that a linear timeinvariant dynamical system Σ determined by the matrices (A, B, C) is completely observable if and only if rk o(σ) is maximal. For reasons which will become clear in a moment, we call linear time-invariant dynamical systems which are both completely controllable and completely observable Schurian systems. An important problem in system theory is to classify the Schurian systems with the same input/output behavior. We reduce this problem to the study of GL n -orbits in an open subset of a vectorspace. Assume we have two systems Σ and Σ, determined by matrix triples from Sys = M n m (C) M n (C) M p n (C) producing the same output y(t) when given the same input u(t), for all possible input functions u(t). We recall that the output function y for a system Σ = (A, B, C) is determined by y(τ) = Ce B(τ τ0) x(τ 0 ) + τ τ 0 Ce (τ t)b Au(t)dt. Differentiating this a number of times and evaluating at τ = τ 0 as in the proof of proposition 7. equality of input/output for Σ and Σ gives the conditions CB i A = C B i A for all i. As a consequence the systems Σ and Σ have the same Hankel matrix which by definition is the product of the observation matrix with the control matrix of the system : H(Σ) = C CB. CB n [ A BA... B n A ] = CA CBA CBA CB 2 A CB 2n-2 A But then, we have for any v C mn that c(σ)(v) = 0 c(σ )(v) = 0 and we can decompose C pn = V W such that the restriction of c(σ) and c(σ ) to V are the zero map and the restrictions to W give isomorphisms with C n. Hence, there is an invertible matrix g GL n such that c(σ ) = gc(σ) and from the commutative diagram C mn c(σ) C n o(σ) C pn g C mn c(σ ) C n o(σ ) C pn we obtain that also o(σ ) = o(σ)g. Consider the system Σ = (A, B, C ) equivalent with Σ under the base-change matrix g. That is, Σ = g.σ = (ga, gbg, Cg ). Then, [ A, B A,..., B n ] A = gc(σ) = c(σ ) = [ A, B A,..., B n A ] and so A = A. Further, as B i+ A = B i+ A we have by induction on i that the restriction of B on the subspace of B i Im(A ) is equal to the restriction of B on this space. Moreover, as n i=0 B i Im(A ) = C n it follows that B = B. Because o(σ ) = o(σ)g we also have C = C. This finishes the proof of :

247 7.. DYNAMICAL SYSTEMS. 247 Proposition 7.2 Let Σ = (A, B, C) and Σ = (A, B, C ) be two Schurian dynamical systems. The following are equivalent. The input/output behavior of Σ and Σ are equal. 2. The systems Σ and Σ are equivalent, that is, there exists an invertible matrix g GL n such that A = ga, B = gbg and C = Cg. By definition, a dynamical system Σ = (A, B, C) is Schurian if (and only if) the determinant of at least one n n minor of c(σ) and o(σ) is non-zero. That is, the subset Sys s of Schurian dynamical systems is open in Sys and is stable under the GL n -action. Our next job is to classify the orbits under this action. We introduce a combinatorial gadget : the Kalman code. It is an array consisting of (n + ) m boxes each having a position label (i, j) where 0 i n and j m. These boxes are ordered lexicographically that is (i, j ) < (i, j) if and only if either i < i or i = i and j < j. Exactly n of these boxes are painted black subject to the rule that if box (i, j) is black, then so is box (i, j) for all i < i. That is, a Kalman code looks like 0 n m We assign to a completely controllable system Σ = (A, B, C) its Kalman code K(Σ) as follows : let A = [ A A 2... A m ], that is Ai is the i-th column of A. Paint the box (i, j) black if and only if the column vector B i A j is linearly independent of the column vectors B k A l for all (k, l) < (i, j). The painted array K(Σ) is indeed a Kalman code. Assume that box (i, j) is black but box (i, j) white for i < i, then B i A j = (k,l)<(i,j) α kl B k A l but then, B i A j = (k,l)<(i,j) α kl B k+i i A l and all (k + i i, l) < (i, l), a contradiction. Moreover, K(Σ) has exactly n black boxes as there are n linearly independent columns of the control matrix c(σ) when Σ is completely controllable. The Kalman code is a discrete invariant of the orbit O ( Σ) under the action of GL n. This follows from the fact that B i A j is linearly independent of the B k A l for all (k, l) < (i, j) if and only if gb i A j is linearly independent of the gb k A l for any g GL n and the observation that gb k A l = (gbg ) k (ga) l. With V c we will denote the open subset of all completely controllable pairs (A, B) that is, those for which the rank of the n nm matrix [ A BA B 2 A... B n A ] is maximal. We consider the map V = M n m (C) M n (C) ψ Mn (n+)m (C) (A, B) [ A BA B 2 A... B n A B n A ]

248 248 CHAPTER 7. MODULI SPACES. The matrix ψ(a, B) determines a linear map ψ (A,B) : C (n+)m C n and (A, B) is a completely controllable pair if and only if the corresponding linear map ψ (A,B) is surjective. Moreover, there is a linear action of GL n on M n (n+)m (C) by left multiplication and the map ψ is GL n -equivariant. 7.2 Grassmannians. The Kalman code induces a barcode on ψ(a, B), that is the n n minor of ψ(a, B) determined by the columns corresponding to black boxes in the Kalman code. ψ(a, B) By construction this minor is an invertible matrix g GL n. We can choose a canonical point in the orbit O (A, B) : g.(a, B). It does have the characteristic property that the n n minor of its image under ψ, determined by the Kalman code is the identity matrix n. The matrix ψ(g.(a, B)) will be denoted by b(a, B) and is called barcode of the pair (A, B). We claim that the barcode determines the orbit uniquely. The map ψ is injective on the open set V c of completely controllable pairs. Indeed, if [ A BA... B n A ] = [ A B A... B n A ] then A = A, B Im(A) = B Im(A) and hence by induction also B B i Im(A) = B B i Im(A ) for all i n. But then, B = B as both pairs (A, B) and (A, B ) are completely controllable, that is, n i=0 Bi Im(A) = C n = n i=0 B i Im(A ). Hence, the barcode b(a, B) determines the orbit O (A, B) and is a point in the Grassmannian Grass n (m(n+)). We briefly recall the definition of these Grassmannians. Let k l be integers, then the points of the Grassmannian Grass k (l) are in one-to-one correspondence with k-dimensional subspaces of C l. For example, if k = then Grass (l) = P l. We know that projective space can be covered by affine spaces defining a manifold structure on it. Also Grassmannians admit a cover by affine spaces. Let W be a k-dimensional subspace of C l then fixing a basis {w,..., w k } of W determines an k l matrix M having as i-th row the coordinates of w i with respect to the standard basis of C l. Linear independence of the vectors w i means that there is a barcode design I on M w. w k i i 2... i k where I = i < i 2 <... < i k l such that the corresponding k k minor M I of M is invertible. Observe that M can have several such designs. Conversely, given a k l matrix M of rank k determines a k-dimensional subspace of l spanned by the transposed rows. Two k l M and M matrices of rank

249 7.2. GRASSMANNIANS. 249 k determine the same subspace provided there is a basechange matrix g GL k such that gm = M. That is, we can identify Grass k (l) with the orbit space of the linear action of GL k by left multiplication on the open set Mk l max(c) of M k l(c) of matrices of maximal rank. Let I be a barcode design and consider the subset of Grass k (l)(i) of subspaces having a matrix representation M having I as barcode design. Multiplying on the left with M I the GL k -orbit O M has a unique representant N with N I = k. Conversely, any matrix N with N I = k determines a point in Grass k (l)(i). Thus, Grass k (l)(i) depends on k(l k) free parameters (the entries of the negative of the barcode) w. w k i i 2... i k π I and we have an identification Grass k (l)(i) C k(l k). For a different barcode design I the image π I (Grass k (l)(i) Grass k (l)(i )) is an open subset of C k(l k) (one extra nonsingular minor condition) and π I π I is a diffeomorphism on this set. That is, the maps π I provide us with an atlas and determine a manifold structure on Grass k (l). Returning to dynamical systems, the barcode b(a, B) determined by the Kalman code determines a unique point in Grass n (m(n + )). We have V c ψ M max n m(n+) (C) b(.) χ Grass n (m(n + )) where ψ is a GL n -equivariant embedding and χ the orbit map. Observe that the barcode matrix b(a, B) shows that the stabilizer of (A, B) is trivial. Indeed, the minor of g.b(a, B) determined by the Kalman code is equal to g. Moreover, continuity of b implies that the orbit O (A, B) is closed in V c. We claim that ψ is a diffeomorphism to a locally closed submanifold of M n m(n+) (C). To prove this we have to consider the differential of ψ. For all (A, B) W and (X, Y ) T (A,B) (V ) V we have j (B + ɛy ) j (A + ɛx) = B n A + ɛ (B n X + B i Y B n i A). Therefore the differential of ψ in (A, B) V, dψ (A,B) (X, Y ) is equal to [ X BX + Y A B 2 X + BY A + Y BA... B n X + n i=0 Bi Y B n i A ]. Assume dψ (A,B) (X, Y ) is the zero matrix, then X = 0 and substituting in the next term also Y A = 0. Substituting in the third gives Y BA = 0, then in the fourth Y B 2 A = 0 and so on until Y B n A = 0. But then, i=0 Y [ A BA B 2 A... B n A ] = 0. If (A, B) is a completely controllable pair, this implies that Y = 0 and hence shows that dψ (A,B) is injective for all (A, B) V c. By the implicit function theorem, ψ induces a GL n -equivariant diffeomorphism between the open subset V c of completely controllable pairs and a locally closed submanifold of M n (n+)m (C) max. The image of this submanifold under the orbit map χ is again a manifold as all fibers are equal to GL n. This concludes the difficult part of the Kalman theorem :

250 250 CHAPTER 7. MODULI SPACES. Theorem 7.3 The orbit space O c = V c /GL n of equivalence classes of completely controllable pairs is a locally closed submanifold of dimension m.n of the Grassmannian Grass n (m(n + )). In fact V b c O c is a principal GL n -bundle. To prove the dimension statement, consider V c (K) the set of completely controllable pairs (A, B) having Kalman code K and let O c (K) be the image under the orbit map. After identifying V c (K) with its image under ψ, the barcode matrix b(a, B) gives a section O c (K) s V c (K). In fact, GL n O c (K) Vc (K) (g, x) g.s(x) is a GL n -equivariant diffeomorphism because the n n minor determined by K of g.b(a, B) is g. Apply the local product decomposition to the generic Kalman code K g 0 n m obtained by painting the top boxes black from left to right until one has n black boxes. Clearly V c (K g ) is open in V c and one deduces dim O c = dim O c (K g ) = dim V c (K g ) dim GL n = mn + n 2 n 2 = mn. The Kalman theorem implies the existence of an orbit space for completely controllable and Schurian systems. Indeed, let Σ = (A, B, C) completely controllable and let g = g (A,B) GL n be the uniquely determined basechange such that g.(a, B) = b(a, B), then we have a canonical representant (ga, gbg, Cg ) in the orbit O(Σ). As the stabilizer Stab(A, B) is trivial the orbits of (A, B, C) and (A, B, C ) are distinct if C = C. That is the natural projection pr 3 Sys c pr 3 V c. Sys c /GL n b O c descends to define an orbit space which is an M p n (C)- bundle over O c and hence is a manifold. The Schurian systems Sys s form a GL n -stable open subset of Sys c and hence their orbit space is an open submanifold of Sys c /GL n. Theorem 7.4 Let Sys c (resp. Sys s ) the the open subset of Sys = M n m (C) M n (C) M p n (C) Schurian) linear dynamical sys- determined by the completely controllable (resp. tems.. The orbit space for the GL n action on Sys c exists and is a vectorbundle of rank pn over O c. 2. The orbit space for the GL n -action on Sys s exists and is a manifold of dimension mn 2 p.

251 7.3. MIXED SEMI-INVARIANTS Mixed semi-invariants. The space Sys of all linear systems determined by the numbers (m, n, p) can be identified with the representation space of the quiver situation rep α Q with α = (m, n, p) and Q the quiver m p n Instead of considering GL(α)-orbits we only consider the GL n -action. Up till now we have only considered GL(α)-invariant functions to classify (closed) orbits. Observe that a completely controllable system Σ = (A, B) does not determine a closed GL n -orbit in V = M n m M n as the action of the scalar matrix g = ɛ n gives the system (ɛa, B) and hence (0 n m, B) is a (not completely controllable) system belonging to the orbit closure O(Σ). Still, we were able to construct a nice orbit space for such systems because the orbit O(Σ) is closed in the open subvariety V s. We will give an interpretation of the orbit map in invariant-theoretic ( language. There is a natural embedding Grass k (l) l P N where N = + given by k) sending a point to the N-tuple of all determinants of the k k minors determined by the different bar-code designs I w. w k i i 2... i k 2 3 w i... w ik det w ki... w kik Composing the orbit map b with this embedding, a system Σ = (A, B) is send to the N-tuple of determinants det b I (A, B). For a different point g.(a, B) in the orbit O(Σ) we have that det b I (g.(a, B)) = det(g)det b I (A, B) That is, these functions are semi-invariants for GL n. In general, if V is a GL n - module, a polynomial function f on V is said to be a semi-invariant if for all v V we have f(g.v) = χ(g)f(v) for some character GL n χ C and we recall that every character of GL n is of the form det k for some k Z. Equivalently, f is an invariant polynomial for the restricted action of the special linear group SL n = { g GL n det(g) = } on V. In chapter, we ran into semi-invariants in the description of the orbit space for the GL n -action on rep α M = M n M n C n C n using Hilbert stairs. Recall that a Hilbert stair σ, that is, the lower triangular part of a square of n n array of boxes filled with go-stones subject to the rules each row contains exactly one stone, and each column contains at most one stone of each color. determines a sequence W (σ) = {, w 2,..., w n } of monomials in the noncommuting variables x and y, placing at the top of the stairs and descending the chair following the rule that every go-stone has a top word T which we may assume we have

252 252 CHAPTER 7. MODULI SPACES. constructed before and a side word S and they are related as indicated below T T T S xt yt For a quadruple (X, Y, u, v) rep α M we replace every occurrence of x in the word w i (x, y) by X and every occurrence of y by Y to obtain an n n matrix w i = w i (X, Y ) M n (C) and by left multiplication on u a column vector w i.v. The evaluation of σ on (X, Y, u, v) is the determinant of the n n matrix σ(x, Y, u, v) = det u w 2.u w 3.u... w n.u These functions were used to separate the orbits of cyclic quadruples. As for every monomial w(x, y) and every g GL n we have that w(gx, gy g )gu = gw(x, Y )u we see that the functions σ(x, Y, u, v) are again semi-invariants for the action of GL n, or equivalently, SL n -invariants on rep α M. In this section we will determine all such mixed semi-invariants for GL n acting on the vectorspace W = M n... M }{{ n V } n... V n V }{{} n... Vn }{{} k m p made up of k matrix-components M n on which GL n act by simultaneous conjugation, m vector-components V n on which GL n -acts by left-multiplication and p covector-components Vn on which GL n act via the contragradient action. That is, W is the representation space of the quiver situation m k p n where we restrict the usual GL(α)-action to the GL n -component. We will determine the generating semi-invariant polynomials, that is, the SL n -invariant functions on W. In chapter 3 we worked out a similar problem in great detail, here we merely sketch the main steps. In section 6 we will generalize these calculations to determine the GL(α)-semi-invariants on an arbitrary quiver situation rep α Q. As always, we first determine the multilinear SL n -invariants, that is the SL n - invariant linear maps M n... M }{{ n V } n... V n V }{{} n... Vn }{{} i j z f C By the identification M n = V n Vn we have to determine the SL n -invariant linear maps Vn i+j Vn i+z f C

253 7.3. MIXED SEMI-INVARIANTS. 253 The description of such invariants is given by classical invariant theory, see for example [32, II.5,Thm. 2.5.A]. Theorem 7.5 The multilinear SL n -invariants f of the situation above are linear combinations of invariants of one of the following two types. For (i,..., i n, h,..., h n,..., t,..., t n, s,..., s r ) a permutation of the i + j vector indices and (u,..., u r ) a permutation of the i + z covector indices, consider the SL n -invariant [v i,..., v in ] [v h,..., v hn ]... [v t,..., v tn ] φ u (v s )... φ ur (v sr ) where the brackets are the determinantal invariants [v a,..., v an ] = det [ v a v a2... v an ] 2. For (i,..., i n, h,..., h n,..., t,..., t n, s,..., s r ) a permutation of the i + z covector indices and (u,..., u r ) a permutation of the i + j vector indices, consider the SL n -invariant [φ i,..., φ in ] [φ h,..., φ hn ]... [φ t,..., φ tn ] φ u (v s )... φ ur (v sr ) where the brackets are the determinantal invariants [φ a,..., φ an ] = det φ a. φ an Observe that we do not have at the same time brackets of vectors and of covectors, due to the relation φ (v )... φ (v n ) [v,..., v n ] [φ,..., φ n ] = det.. φ n (v )... φ n (v n ) Our next job is to give a matrix-interpretation of these basic invariants. consider the case of a bracket of vectors (the case of covectors is similar) Let us [v i,..., v in ] If all the indices {i,..., i n } are original vector-indices (and so do not come from the matrix-terms) we save this term and go to the next factor. Otherwise, if say i is one of the matrix indices, A i = φ i v i, then the covector φ i must be paired up in a scalar product φ i (v u ) with a vector v u. Again, two cases can occur. If u is a vector index, we have that φ i (v u )[v i,..., v in ] = [A i v u, v i2,..., v in ] = [v i, v i2,..., v in ] Otherwise, we can keep on matching the matrix indices and get an expression φ i (v u ) φ u (v u2 ) φ u2 (v u3 )... until we finally hit again a vector index, say u l, but then we have the expression φ i (v u ) φ u (v z )... φ ul (v ul ) [v i,..., v in ] = [Mv ul, v i2,..., v in ] where M = A i A u... A ul. One repeats the same argument for all vectors in the brackets. As for the remaining scalar product terms, we have a similar procedure of

254 254 CHAPTER 7. MODULI SPACES. matching up the matrix indices and one verifies that in doing so one obtains factors of the type φ(m v) and tr(m) where M is a monomial in the matrices. As we mentioned, the case of covectorbrackets is similar except that in matching the matrix indices with a covector φ, one obtains a monomial in the transposed matrices. Having found these interpretations of the basic SL n -invariant linear terms, we can proceed by polarization and restitution processes as explained in chapter 3, to finish the proof of the next result, due to C. Procesi [24, Thm 2.]. Theorem 7.6 The SL n -invariants of W = rep α Q where Q is the quiver m k p n are generated by the following four types of functions, where we write a typical element in W as (A,..., A }{{ k, v },..., v m, φ }{{},..., φ p ) }{{} k m p with the A i the matrices corresponding to the loops, the v j making up the rows of the n m matrix and the φ j the columns of the p n matrix. tr(m) where M is a monomial in the matrices A i, scalar products φ j (Mv i ) where M is a monomial in the matrices A i, vector-brackets [M v i, M 2 v i2,..., M n v in ] where the M j are monomials in the matrices A i, covector-brackets [M φ τ i,..., M n φ τ i n ] where the M j are monomials in the matrices A i, 7.4 General subrepresentations. Throughout this section we fix a quiver Q on k vertices {v,..., v k } and dimension vectors α = (a,..., a k ) and β = (b,..., b k ). We want to describe morphisms between representations V rep α Q and W rep β Q. That is, we consider the closed subvariety Hom Q (α, β) M a b... M ak b k rep α Q rep β Q consisting of the triples (φ, V, W ) where φ = (φ,..., φ k ) is a morphism of quiverrepresentations V W. Projecting to the two last components we have an onto morphism between affine varieties Hom Q (α, β) h rep α Q rep β Q In chapter 4.2 we have proved that the dimension of fibers is an uppersemicontinuous function. That is, for every natural number d, the set {Φ Hom Q (α, β) dim Φ h (h(φ)) d} is a Zariski open subset of Hom Q (α, β). As the target space rep α Q rep β Q is irreducible, it contains a non-empty open subset hom min where the dimension of

255 7.4. GENERAL SUBREPRESENTATIONS. 255 the fibers attains a minimal value. This minimal fiber dimension will be denoted by hom(α, β). Similarly, we could have defined an affine variety Ext Q (α, β) where the fiber over a point (V, W ) rep α Q rep β Q is given by the extensions Ext CQ (V, W ). If χ Q is the Euler-form of Q we recall that for all V rep α Q and W rep β Q we have dim C Hom CQ (V, W ) dim C Ext Q (V, W ) = χ Q(α, β) Hence, there is also an open set ext min of rep α Q rep β Q where the dimension of Ext (V, W ) attains a minimum. This minimal value we denote by ext(α, β). As hom min ext min is a non-empty open subset we have the numerical equality hom(α, β) ext(α, β) = χ Q (α, β). In particular, if hom(α, α+β) > 0, there will be an open subset where the morphism V φ W is a monomorphism. Hence, there will be an open subset of repα+β Q consisting of representations containing a subrepresentation of dimension vector α. We say that α is a general subrepresentation of α + β and denote this with α α + β. We want to characterize this property. To do this, we introduce the quiver-grassmannians Grass α (α + β) = k Grass ai (a i + b i ) i= which is a projective manifold. Consider the following diagram of morphisms of reduced varieties rep α+β Q s pr rep α+β α Q rep α+β Q Grass α (α + β) with the following properties p pr 2 Grass α (α + β) rep α+β Q Grass α (α+β) is the trivial vectorbundle with fiber rep α+β Q over the projective smooth variety Grass α (α + β) with structural morphism pr 2. rep α+β α Q is the subvariety of rep α+β Q Grass α (α+β) consisting of couples (W, V ) where V is a subrepresentation of W (observe that this is for fixed W a linear condition). Because GL(α+β) acts transitively on the Grassmannian Grass α (α + β) (by multiplication on the right) we see that repα α+β Q is a subvectorbundle over Grass α (α + β) with structural morphism p. In particular, Q is a reduced variety. rep α+β α The morphism s is a projective morphism, that is, can be factored via the natural projection rep α+β Q P N f rep α+β α Q s π 2 rep α+β Q

256 256 CHAPTER 7. MODULI SPACES. where f is the composition of the inclusion rep α+β α Q rep α+β Q Grass α (α + β) with the natural inclusion of Grassmannians in projective spaces recalled in the previous section Grass α (α + β) k i= with Pni the Segre embedding k i= Pni P N. In particular, s is proper by [9, Thm. II.4.9], that is, maps closed subsets to closed subsets. We are interested in the scheme-theoretic fibers of s. If W rep α+β Q lies in the image of s, we denote the fiber s (W ) by Grass α (W ). Its geometric points are couples (W, V ) where V is an α-dimensional subrepresentation of W. Whereas Grass α (W ) is a projective scheme, it is in general neither smooth, nor irreducible nor even reduced. Therefore, in order to compute the tangent space in a point (W, V ) of Grass α (W ) we have to clarify the functor it represents on the category commalg of commutative C-algebras. Let C be a commutative C-algebra, a representation R of the quiver Q over C consists of a collection R i = P i of projective C-modules of finite rank and a collection of C-module morphisms for every arrow a in Q a j R i j = P j Ra P i = R i The dimension vector of the representation R is given by the k-tuple (rk C R,..., rk C R k ). A subrepresentation S of R is determined by a collection of projective sub-summands (and not merely sub-modules) S i R i. In particular, for W rep α+β Q we define the representation W C of Q over the commutative ring C by { (W C ) i = C C W i (W C ) a = id C C W a With these definitions, we can now define the functor represented by Grass α (W ) as the functor assigning to a commutative C-algebra C the set of all subrepresentations of dimension vector α of the representation W C. Lemma 7.7 Let x = (W, V ) be a geometric point of Grass α (W ), then Proof. T x Grass α (W ) = Hom CQ (V, W V ) The tangent space in x = (W, V ) are the C[ɛ]-points of Grass α (W ) lying ψ W V over (W, V ). To start, let V be a homomorphism of representations of Q and consider a C-linear lift of this map ψ : V W. Consider the C-linear subspace of W C[ɛ] = C[ɛ] W spanned by the sets {v + ɛ ψ(v) v V } and ɛ V This determines a C[ɛ]-subrepresentation of dimension vector α of W C[ɛ] lying over (W, V ) and is independent of the chosen linear lift ψ. Conversely, if S is a C[ɛ]-subrepresentation of W C[ɛ] lying over (W, V ), then S ɛs = V W. But then, a C-linear complement of ɛs is spanned by elements of the form v + ɛψ(v) where ψ(v) W and ɛ ψ is determined modulo an element W V of ɛ V. But then, we have a C-linear map ψ : V and as S is a C[ɛ]- subrepresentation, ψ must be a homomorphism of representations of Q. We can now give a characterization for general α-dimensional subrepresentations, proved by A. Schofield in citeschofield.

257 7.4. GENERAL SUBREPRESENTATIONS. 257 Theorem 7.8 The following are equivalent. α α + β. 2. Every representation W rep α+β Q has a subrepresentation V of dimension α. 3. ext(α, β) = 0. Proof. Assume., then the image of the proper map s : rep α+β α Q rep α+β Q contains a Zariski open subset. As properness implies that the image of s must also be a closed subset of rep α+β Q it follows that Im s = rep α+β Q, that is 2. holds. Conversely, 2. clearly implies. so they are equivalent. We compute the dimension of the vectorbundle repα α+β Q over Grass α (α + β). Using that the dimension of a Grassmannians Grass k (l) is k(l k) we know that the base has dimension k i= a ib i. Now, fix a point V W in Grass α (α + β), then the fiber over it determines all possible ways in which this inclusion is a subrepresentation of quivers. That is, for every arrow in Q of the form a j i we need to have a commuting diagram V i V j W i W j Here, the vertical maps are fixed. If we turn V rep α Q, this gives us the a i a j entries of the upper horizontal map as degrees of freedom, leaving only freedom for the lower horizontal map determined by a linear map Wi V Wj i, that is, having b i (a j + b j ) degrees of freedom. Hence, the dimension of the vectorspace-fibers is (a i a j + b i (a j + b j )) j a i giving the total dimension of the reduced variety rep α+β α Q. But then, dim rep α+β α Q dim rep α+β Q = = k a i b i + (a i a j + b i (a j + b j )) a j i (a i + b i )(a j + b j ) a i= i= j j i i k a i b i a i b j = χ Q (α, β) a s repα+β Assume that 2. holds, then the proper map repα α+β Q is onto and as both varieties are reduced, the general fiber is a reduced variety of dimension χ Q (α, β), whence the general fiber contains points such that their tangentspaces have dimension χ Q (α, β). By the foregoing lemma we can compute the dimension of this tangentspace as dim Hom CQ (V, W V ). But then, as χ Q (α, β) = dim C Hom CQ (V, W V ) dim C Ext CQ(V, W V )

258 258 CHAPTER 7. MODULI SPACES. it follows that Ext (V, W V ) = 0 for some representation V of dimension vector α and W V of dimension vector β. But then, ext(α, β) = 0, that is, 3. holds. Conversely, assume that ext(α, β) = 0. Then, for a general point W rep α+β Q in the image of s and for a general point in its fiber (W, V ) repα α+β Q we have dim C Ext CQ (V, W V ) = 0 whence dim C Hom CQ (V, W V ) = χ Q(α, β). But then, the general fiber of s has dimension χ Q (α, β) and as this is the difference in dimension between the two irreducible varieties, the map is generically onto. Finally, properness of s then implies that it is onto, giving 2. and finishing the proof. 7.5 Schofield s criterium. In all moduli space problems we will encounter, it will be crucial to determine the dimension vectors of general subrepresentations, or by the foregoing section, to compute ext(α, β). An inductive algorithm to do this was discovered by A. Schofield [27]. Recall that α β iff a general representation W rep β Q contains a subrepresentation S W of dimension vector α. Similarly, we denote β γ if and only if a general representation W rep β Q has a quotient-representation W T of dimension vector γ. As before, Q will be a quiver on k-vertices {v,..., v k } and we denote dimension vectors α = (a,..., a k ), β = (b,..., b k ) and γ = (c,..., c k ). We will first determine the rank of a general homomorphism V W between representations V repα Q and W rep β Q. We denote Hom(α, β) = k i=m bi a i and Hom(V, β) = Hom(α, β) = Hom(α, W ) for any representations V and W as above. With these conventions we have Lemma 7.9 There is an open subset Hom m (α, β) rep α Q rep β Q and a dimension vector γ def = rk hom(α, β) such that for all (V, W ) Hom min (α, β) Proof. dim C Hom CQ (V, W ) is minimal, and {φ Hom CQ (V, W ) rk φ = γ} is a non-empty Zariski open subset of Hom CQ (V, W ). Consider the subvariety Hom Q (α, β) of the trivial vectorbundle Hom Q (α, β) Hom(α, β) rep α Q rep β Q Φ pr rep α Q rep β Q of triples (φ, V, W ) such that V φ W is a morphism of representations of Q. The fiber Φ (V, W ) = Hom CQ (V, W ). As the fiber dimension is upper semi-continuous, there is an open subset Hom min (α, β) of rep α Q rep β Q consisting of points (V, W ) where dim C Hom CQ (V, W ) is minimal. For given dimension vector δ = (d,..., d k ) we consider the subset Hom Q (α, β, δ) = {(φ, V, W ) Hom Q (α, β) rk φ = δ} Hom Q (α, β) This is a constructible subset of Hom Q (α, β) and hence there is a dimension vector γ such that Hom Q (α, β, γ) Φ (Hom min (α, β)) is constructible and dense in Φ (Hom min (α, β)). But then, Φ(Hom Q (α, β, γ) Φ (Hom min (α, β)))

259 7.5. SCHOFIELD S CRITERIUM. 259 is constructible and dense in Hom min (V, W ). Therefore it contains an open subset Hom m (V, W ) satisfying the requirements of the lemma. Lemma 7.0 Assume we have short exact sequences of representations of Q { = 0 S V X 0 2 = 0 Y W T 0 then there is a natural onto map Ext CQ(V, W ) Ext CQ (S, T ) Proof. We will see in chapter 9 that gldim CQ, whence applying derived functors to the given sequences we obtain the following part of the natural longexact sequences RHom(W, ) RHom(T, ).. RHom(V, 2 )... Ext(V, W ) Ext(V, T ) 0 RHom(S, 2 )... Ext(S, W ) Ext(S, T ) 0 from which the statement follows. 0 0 Theorem 7. Let γ = rk hom(α, β) (with notations as in lemma 7.9), then. α γ α γ β β γ 2. ext(α, β) = χ Q (α γ, β γ) = ext(α γ, β γ) Proof. The first statement is obvious from the definitions, for if γ = rk hom(α, β), then a general representation of dimension α will have a quotient-representation of dimension γ (and hence a subrepresentation of dimension α γ) and a general representation of dimension β will have a subrepresentation of dimension γ (and hence a quotient-representation of dimension β γ. The strategy of the proof of the second statement is to compute the dimension of the subvariety of Hom(α, β) rep α rep β rep γ defined by H factor = {(φ, V, W, X) V φ X = Im φ W factors as representations }

260 260 CHAPTER 7. MODULI SPACES. in two different ways. Consider the intersection of the open set Hom m (α, β) determined by lemma 7.9 with the open set of couples (V, W ) such that dim Ext(V, W ) = ext(α, β) and let (V, W ) lie in this intersection. In the previous section we have proved that dim Grass γ (W ) = χ Q (γ, β γ) Let H be the subbundle of the trivial vectorbundle over Grass γ (W ) H Hom(α, W ) Grass γ (W ) Grass γ (W ) consisting of triples (φ, W, U) with φ : i C ai W a linear map such that Im(φ) is contained in the subrepresentation U W of dimension γ. That is, the fiber over (W, U) is Hom(α, U) and therefore has dimension k i= a ic i. With H full we consider the open subvariety of H of triples (φ, W, U) such that Im φ = U. We have k dim H full = a i c i + χ Q (γ, β γ) But then, H factor is the subbundle of the trivial vectorbundle over H full i= H factor rep α Q H full π H full consisting of quadruples (V, φ, W, X) such that V φ W is a morphism of representations, with image the subrepresentation X of dimension γ. The fiber of π over a triple (φ, W, X) is determined by the property that for each arrow a j i the following diagram must be commutative, where we decompose the vertex spaces V i = X i K i for K = Ker φ X i K i A B 5 C D Xj K j h ci 0 i X i A Xj where A is fixed, giving the condition B = 0 and hence the fiber has dimension equal to (a i c i )(a j c j ) + c i (a j c j ) = a i (a j c j ) a j i a j i a j i h cj This gives our first formula for the dimension of H factor H factor = k a i c i + χ Q (γ, β γ) + i= 0 i a i (a j c j ) a j i

261 7.5. SCHOFIELD S CRITERIUM. 26 On the other hand, we can consider the natural map H factor Φ rep α Q defined by sending a quadruple (V, φ, W, X) to V. the fiber in V is given by all quadruples (V, φ, W, X) such that V φ W is a morphism of representations with Im φ = X a representation of dimension vector γ, or equivalently Φ (V ) = {V φ W rk φ = γ} Now, recall our restriction on the couple (V, W ) giving at the beginning of the proof. There is an open subset max of rep α Q of such V and by construction max Im Φ, Φ (max) is open and dense in H factor and the fiber Φ (V ) is open and dense in Hom CQ (V, W ). This provides us with the second formula for the dimension of H factor dim H factor = dim rep α Q + hom(α, W ) = a i a j + hom(α, β) a Equating both formulas we obtain the equality which is equivalent to χ Q (γ, β γ) + k a i c i a i c j = hom(α, β) a i= χ Q (γ, β γ) + χ Q (α, γ) χ Q (α, β) = ext(α, β) Now, for our (V, W ) we have that Ext(V, W ) = ext(α, β) and we have exact sequences of representations 0 S V X 0 0 X W T 0 and using lemma 7.0 this gives a surjection Ext(V, W ) Ext(S, T ). On the other hand we always have from the homological interpretation of the Euler form the first inequality dim C Ext(S, T ) χ Q (α γ, β γ) = χ Q (γ, β γ) χ Q (α, β) + χ Q (α, γ) = ext(α, β) As the last term is dim C Ext(V, W ), this implies that the above surjection must be an isomorphism and that dim C Ext(S, T ) = χ Q (α γ, β γ) whence dim C Hom(S, T ) = 0 But this implies that hom(α γ, β γ) = 0 and therefore ext(α γ, β γ) = χ Q (α γ, β γ). Finally, ext(α γ, β γ) = dim Ext(S, T ) = dim Ext(V, W ) = ext(α, β) j i j i finishing the proof. Theorem 7.2 For all dimension vectors α and β we have ext(α, β) = max α α β β χ Q (α, β ) = max β β χ Q (α, β ) = max α α χ Q (α, β)

262 262 CHAPTER 7. MODULI SPACES. Proof. Let V and W be representation of dimension vector α and β such that dim Ext(V, W ) = ext(α, β). Let S V be a subrepresentation of dimension α and W T a quotient representation of dimension vector β. Then, we have ext(α, β) = dim C Ext(V, W ) dim C Ext(S, T ) χ Q (α, β ) where the first inequality is lemma 7.0 and the second follows from the interpretation of the Euler form. Therefore, ext(α, β) is greater or equal than all the terms in the statement of the theorem. The foregoing theorem asserts the first equality, as for rk hom(α, β) = γ we do have that ext(α, β) = χ Q (α γ, β γ). In the proof of the above theorem, we have found for sufficiently general V and W an exact sequence of representations 0 S V W T 0 where S is of dimension α γ and T of dimension β γ. Moreover, we have a commuting diagram of surjections Ext(V, W ) Ext(V, T )... Ext(S, W ) Ext(S, T ) and the dashed map is an isomorphism, hence so are all the epimorphisms. Therefore, we have { ext(α, β γ) dim Ext(V, T ) = dim Ext(V, W ) = ext(α, β) ext(α γ, β) dim Ext(S, W ) = dim Ext(V, W ) = ext(α, β) Further, let T be a sufficiently general representation of dimension β γ, then it follows from Ext(V, T ) Ext(S, T ) that ext(α γ, β γ) dim Ext(S, T ) dim Ext(V, T ) = ext(α, β γ) but the left term is equal to ext(α, β) by the above theorem. But then, we have ext(α, β) = ext(α, β γ). Now, we may assume by induction that the theorem holds for β γ. That is, there exists β γ β such that ext(α, β γ) = χ Q (α, β ). Whence, β β and ext(α, β) = χq (α, β ) and the middle equality of the theorem holds. By a dual argument so does the last. This gives us the following inductive procedure to find all the dimension vectors of general subrepresentations. Take a dimension vector α and assume by induction we know for all β < α the set of general subrepresentations β β. Then, β α if and only if 0 = ext(β, α β) = max β β χ Q (β, α β) where the first equality is the main result of the foregoing section and the last is the result above. 7.6 θ-semistable representations. Let Q be a quiver on k vertices {v,..., v k } and fix a dimension vector α. So far, we have considered the algebraic quotient map rep α Q iss α Q

263 7.6. θ-semistable REPRESENTATIONS. 263 classifying closed GL(α)-orbits in rep α Q, that is, isomorphism classes of semisimple representations of dimension α. We have seen that the invariant polynomial maps are generated by traces along oriented cycles in the quiver. Hence, if Q has no oriented cycles, the quotient variety iss α Q is reduced to one point corresponding to the semi-simple S a... S a k k where S i is the trivial one-dimensional simple concentrated in vertex v i. Still, in these cases one can often classify nice families of representations. For example, consider the quiver situation x y z Then, rep α Q = C 3 and the action of GL(α) = C C is given by (λ, µ).(x, y, z) = ( λ µ x, λ µ y, λ µ z). The only closed GL(α)-orbit in C3 is (0, 0, 0) as the one-parameter subgroup λ(t) = (t, ) has the property lim λ(t).(x, y, z) = (0, 0, 0) t 0 so (0, 0, 0) O(x, y, z) for any representation (x, y, z). Still, if we trow away the zero-representation, then we have a nice quotient map C 3 {(0, 0, 0)} π P 2 (x, y, z) [x : y : z] and as O(x, y, z) = C (x, y, z) we see that every GL(α)-orbit is closed in this complement C 3 {(0, 0, 0)}. We will generalize such settings to arbitrary quivers. A character of GL(α) is an algebraic group morphism χ : GL(α) C. They are fully determined by an integral k-tuple θ = (t,..., t k ) Z k where GL(α) χ θ C (g,..., g k ) det(g ) t.....det(g k ) t k For a fixed θ we can extend the GL(α)-action to the space rep α C by GL(α) rep α Q C rep α Q C g.(v, c) = (g.v, χ θ (g)c) The coordinate ring C[rep α Q C] = C[rep α ][t] can be given a Z-gradation by defining deg(t) = and deg(f) = 0 for all f C[rep α Q]. The induced action of GL(α) on C[rep α Q C] preserves this gradation. Therefore, the ring of invariant polynomial maps C[rep α Q C] GL(α) = C[rep α Q][t] GL(α) is also graded with homogeneous part of degree zero the ring of invariants C[rep α ] GL(α). An invariant of degree n, say ft n with f C[rep α Q] has the characteristic property that f(g.v ) = χ n θ (g)f(v ) that is, f is a semi-invariant of weight χ n θ. That is, the graded decomposition of the invariant ring is C[rep α Q C] GL(α) = R 0 R... with R i = C[rep α Q] GL(α),χn θ With these notations, the moduli space of semi-stable quiver representations of dimension α was introduced by A. King in [2] to be the variety M ss α (Q, θ) = P roj C[rep α Q C] GL(α) = P roj n=0 C[rep α Q] GL(α),χn θ

264 264 CHAPTER 7. MODULI SPACES. Recall that for a positively graded affine commutative C-algebra R = i=0 R i, the geometric points of P roj R correspond to graded-maximal ideals m not containing the positive part R + = i= R i. Intersecting m with the part of degree zero R 0 determines a point of Spec R 0, the affine variety with coordinate ring R 0 and gives rise to a structural morphism P roj R Spec R 0. The Zariski closed subsets of P roj R are of the form V(I) = {m P roj R I m} for a homogeneous ideal I R. Also recall that P roj R can be covered by affine varieties of the form X(f) with f a homogeneous element in R +. The coordinate ring of this affine variety is the part of degree zero of the graded localization R g f. We refer to [9, II.2] for more details. Example 7.3 Consider again the quiver-situation x y z and character θ = (, ), then the three coordinate functions x, y and z of C[rep α Q] are semiinvariants of weight χ θ. It is then clear that the invariant ring is equal to C[rep α Q C] GL(α) = C[xt, yt, zt] where the three generators all have degree one. That is, as desired, M ss α (Q, θ) = P roj C[xt, yt, zt] = P 2 We will now investigate which orbits in rep α Q are parameterized by the moduli space Mα ss (Q, θ). We say that a representation V rep α Q is χ θ -semistable if and only if there is a semi-invariant f C[rep α Q] GL(α),χnθ for some n such that f(v ) 0. The subset of rep α Q consisting of all χ θ -semistable representations will be denoted by rep ss α (Q, θ). Observe that rep ss α (Q, θ) is Zariski open (but it may be empty for certain (α, θ)). We can lift a representation V rep α Q to points V c = (V, c) rep α Q C and use GL(α)-invariant theory on this larger GL(α)- module V(f) Vc V 0 V(t) Let c 0 and assume that the orbit closure O(V c ) does not intersect V(t) = rep α Q {0}. As both are GL(α)-stable closed subsets of rep α Q C we know from the separation property of invariant theory, see 4.6, that this is equivalent to the existence of a GL(α)-invariant function g C[rep α Q C] GL(α) such that g(o(v c )) 0 but g(v(t)) = 0. We have seen that the invariant ring is graded,

265 7.6. θ-semistable REPRESENTATIONS. 265 hence we may assume g to be homogeneous, that is, of the form g = ft n for some n. But then, f is a semi-invariant on rep α Q of weight χ n θ and we see that V must be χ θ -semistable. Moreover, we must have that θ(α) = k i= t ia i = 0, for the one-dimensional central torus of GL(α) µ(t) = (t a,..., t ak ) GL(α) acts trivially on rep α Q but acts on C via multiplication with k i= hence if t aiti θ(α) 0 then O(V c ) V(t). More generally, we have from the strong form of the Hilbert criterium proved in 4.4 that O(V c ) V(t) = if and only if for every one-parameter subgroup λ(t) of GL(α) we must have that lim λ(t).v c / V(t). We t 0 can also formulate this in terms of the GL(α)-action on rep α Q. The composition of a one-parameter subgroup λ(t) of GL(α) with the character C λ(t) GL(α) χθ C is an algebraic group morphism and is therefore of the form t t m for some m Z and we denote this integer by θ(λ) = m. Assume that λ(t) is a one-parameter subgroup such that lim λ(t).v = V exists in rep α Q, then as t 0 λ(t).(v, c) = (λ(t).v, t m c) we must have that θ(λ) 0 for the orbitclosure O(V c ) not to intersect V(t). That is, we have the following characterization of χ θ -semistable representations. Proposition 7.4 The following are equivalent. V rep α Q is χ θ -semistable. 2. For c 0, we have O(V c ) V(t) =. 3. For every one-parameter subgroup λ(t) of GL(α) we have lim t 0 λ(t).v c / V(t) = rep α Q {0}. 4. For every one-parameter subgroup λ(t) of GL(α) such that lim t 0 λ(t).v exists in rep α Q we have θ(λ) 0. Moreover, these cases can only occur if θ(α) = 0. Assume that g = ft n is a homogeneous invariant function for the GL(α)-action on rep α Q C and consider the affine open GL(α)-stable subset X(g). The construction of the algebraic quotient in 4.6 and the fact that invariant rings here are graded asserts that the closed GL(α)-orbits in X(g) are classified by the points of the graded localization at g which is of the form (C[rep α Q C] GL(α) ) g = R f [h, h ] for some homogeneous invariant h and where R f is the coordinate ring of the affine open subset X(f) in Mα ss (Q, θ) determined by the semi-invariant f of weight χ n θ. As the moduli space is covered by such open subsets we have Proposition 7.5 The moduli space of θ-semistable representations of rep α Q M ss α (Q, θ) classifies closed GL(α)-orbits in the open subset rep ss α (Q, θ) of all χ θ -semistable representations of Q of dimension vector α.

266 266 CHAPTER 7. MODULI SPACES. Example 7.6 In the foregoing example rep ss α (Q, θ) = C 3 {(0, 0, 0)} as for all these points one of the semi-invariant coordinate functions is non-zero. For θ = (, ) the lifted GL(α) = C C - action to rep α Q C = C 4 is given by (λ, µ).(x, y, z, t) = ( µ λ x, µ λ y, µ λ z, λ µ t) We have seen that the ring of invariants is C[xt, yt, zt]. Consider the affine open set X(xt) of C 4, then the closed orbits in X(xt) are classified by C[xt, yt, zt] g xt = C[ y x, z x ][xt, xt ] and the part of degree zero C[ y x, z x ] is the coordinate ring of the open set X(x) in P2. In 4.5 we were able to classify closed GL n -orbits in rep n A with semi-simple representations. We will now give a representation theoretic interpretation of closed GL(α)-orbits in rep ss α (Q, θ). Again, the starting point is that one-parameter subgroups λ(t) of GL(α) correspond to filtrations of representations. Let us go through the motions one more time. For λ : C GL(α) a one-parameter subgroup and V rep α Q we can decompose for every vertex v i the vertex-space in weight spaces where λ(t) acts on the weight space V (n) i allows us to define a filtration V i = n Z V (n) i V ( n) i as multiplication by t n. This decomposition = m n V (m) i For every arrow a j, i λ(t) acts on the components of the arrow maps V (n) i V m,n a V (m) j by multiplication with t m n. That is, a limit lim V a exists if and only if Va m,n = 0 t 0 for all m < n, that is, if V a induces linear maps V ( n) i V a V ( n) j Hence, a limiting representation exists if and only if the vertex-filtration spaces V ( n) i determine a subrepresentation V n V for all n. That is, a one-parameter subgroup λ such that lim λ(t).v exists determines a decreasing filtration of V by subrepresentations t... V n V n+... Further, the limiting representation is then the associated graded representation lim λ(t).v = n Z t 0 V n V n+ where of course only finitely many of these quotients can be nonzero. For the given character θ = (t,..., t k ) and a representation W rep β Q we denote θ(w ) = t b t k b k where β = (b,..., b k ) Assume that θ(v ) = 0, then with the above notations, we have an interpretation of θ(λ) as k θ(λ) = t i ndim C V (n) i = nθ( V n ) = θ(v n ) V i= n+ n Z n Z n Z

267 7.6. θ-semistable REPRESENTATIONS. 267 Definition 7.7 A representation V rep α Q is said to be θ-semistable if θ(v ) = 0 and for all subrepresentations W θ(w ) 0. V we have θ-stable if V is θ-semistable and if the only subrepresentations W such that θ(w ) = 0 are V and 0. Proposition 7.8 For V rep α Q the following are equivalent. V is χ θ -semistable. 2. V is θ-semistable. V Proof.. 2. : Let W be a subrepresentation of V and let λ be the one-parameter subgroup associated to the filtration V W 0, then lim λ(t).v exists t 0 whence by proposition we have θ(λ) 0, but we have θ(λ) = θ(v ) + θ(w ) = θ(w ) 2.. : Let λ be a one-parameter subgroup of GL(α) such that lim t 0 λ(t).v exists and consider the induced filtration by subrepresentations V n defined above. By assumption all θ(v n ) 0, whence θ(λ) = n Z θ(v n ) 0 and again proposition finishes the proof. Lemma 7.9 Let V rep α Q and W rep β Q be both θ-semistable and V f W a morphism of representations. Then, Ker f, Im f and Coker f are θ-semistable representations. Proof. Consider the two short exact sequences of representations of Q { 0 Ker f V Im f 0 0 Im f W Coker f 0 As θ( ) is additive, we have 0 = θ(v ) = θ(ker f)+θ(im f) and as both are subrepresentations of θ-semistable representations V resp. W, the right-hand terms are 0 whence are zero. But then, from the second sequence also θ(coker f) = 0. Being submodules of θ-semistable representations, Ker f and Im f also satisfy θ(s) 0 for all their subrepresentations U. Finally, a subrepresentation T Coker f can be lifted to a subrepresentation T W and θ(t ) 0 follows from the short exact sequence 0 Im f T T 0. That is, the full subcategory rep ss (Q, θ) of rep Q consisting of all θ-semistable representations is an Abelian subcategory and clearly the simple objects in rep ss (Q, θ) are precisely the θ-stable representations. As this Abelian subcategory has the necessary finiteness conditions, one can prove a version of the Jordan-Hölder theorem. That is, every θ-semistable representation V has a finite filtration V = V 0 V... V z = 0 of subrepresentation such that every factor Vi V i+ is θ-stable. Moreover, the unordered set of these θ-stable factors are uniquely determined by V.

268 268 CHAPTER 7. MODULI SPACES. Theorem 7.20 For a θ-semistable representation V rep α Q the following are equivalent. The orbit O(V ) is closed in rep ss α (Q, α). 2. V W e... W e l l with every W i a θ-stable representation. That is, the geometric points of the moduli space Mα ss (Q, θ) are in natural oneto-one correspondence with isomorphism classes of α-dimensional representations which are direct sums of θ-stable subrepresentations. The quotient map rep ss α (Q, θ) M ss α (Q, θ) maps a θ-semistable representation V to the direct sum of its Jordan-Hölder factors in the Abelian category rep ss (Q, θ). Proof. Assume that O(V ) is closed in rep ss α (Q, θ) and consider the θ-semistable representation W = gr ss V, the direct sum of the Jordan-Hölder factors in rep ss (Q, θ). As W is the associated graded representation of a filtration on V, there is a one-parameter subgroup λ of GL(α) such that lim λ(t).v W, that is t 0 O(W ) O(V ) = O(V ), whence W V and 2. holds. Conversely, assume that V is as in 2. and let O(W ) be a closed orbit contained in O(V ) (one of minimal dimension). By the Hilbert criterium there is a oneparameter subgroup λ in GL(α) such that lim λ(t).v W. Hence, there is a finite t 0 filtration of V with associated graded θ-semistable representation W. As none of the θ-stable components of V admits a proper quotient which is θ-semistable (being a direct summand of W ), this shows that V W and so O(V ) = O(W ) is closed. The other statements are clear from this. The striking similarities between θ-stable representations and simple representations will become more transparent in chapter 3 when we discuss universal localizations. It will turn out that θ-stable representations become simple representations of a certain universal localization of the path algebra CQ. Example 7.2 Consider the modular group P SL 2 (Z) Z 2 Z 3, the free product of the cyclic groups of order two and three with generators σ resp. τ. Let S be an n-dimensional simple representation of P SL 2 (Z). Let ξ be a 3-rd root of unity, then restricting S to these finite Abelian subgroups we have ( S Z2 S a S a 2 S Z3 T b T b 2 ξ T b 3 ξ 2 where S x resp. T x are the one-dimensional representations on which σ resp. τ acts via multiplication with x. Observe that a + a 2 = b + b 2 + b 3 = n and we associate to S a representation V of the quiver situation b a b 2 a 2 b 3 with V i = S a i i and V 2j = T b j j a ij b j a i is the composition of and where the linear map corresponding to an arrow V aij : S a i i S Z2 = V Z3 T b j j

269 7.6. θ-semistable REPRESENTATIONS. 269 of the canonical injections and projections. If α = (a, a 2, b, b 2, b 3 ) then we take as θ = (,, +, +, +). Observe that i,j V aij : C n C n is a linear isomorphism. If W V is a subrepresentation, then θ(w ) 0. Indeed, if the dimension vector of W is β = (c, c 2, d, d 2, d 3 ) and assume that θ(w ) < 0, then k = c +c 2 > l = d +d 2 +d 3, but then the restriction of V aij to W gives a linear map C k C l having a kernel which is impossible. Hence, V is a θ-semistable representation of the quiver. In fact, V is even θ-stable, for consider a subrepresentation W V with dimension vector β as before and θ(w ) = 0, that is, c + c 2 = d + d 2 + d 3 = m, then the isomorphism i,j V aij W and the decomposition into eigenspaces of C m with respect to the Z 2 and Z 3 -action, makes C m into an m-dimensional representation of P SL 2 (Z) which is a subrepresentation of S. S being simple then implies that W = V or W = 0, whence V is θ-stable. The underlying reason is that the group algebra CP SL 2 (Z) is a universal localization of the path algebra CQ of the above quiver. Remains to determine the situations (α, θ) such that the corresponding moduli space Mα ss (Q, θ) is non-empty, or equivalently, such that the Zariski open subset rep ss α (Q, θ) rep α Q is non-empty. Theorem 7.22 Let α be a dimension vector such that θ(α) = 0. Then,. rep ss α (Q, α) is a non-empty Zariski open subset of rep α Q if and only if for every β α we have that θ(β) The θ-stable representations rep s α(q, α) are a non-empty Zariski open subset of rep α Q if and only if for every 0 β α we have that θ(β) > 0 Observe that the Schofield criterium gives an inductive procedure to calculate these conditions. Sometimes we can bypass the troublesome inductive step using our description of dimension vectors of simple representations. Example 7.23 It is possible to determine the weight space decomposition vectors α = (a, a 2, b, b 2, b 3 ) of simple n = a + a 2 = b + b 2 + b 3 -dimensional representations of the modular group P SL 2 (Z) by first computing the dimension vectors β = (c, c 2, d, d 2, d 3 ) of general subrepresentations of α and then to check whether for all of these c + c 2 < d + d 2 + d 3. An alternative method is to compute local quiver settings and use the description of semisimple dimension vectors. With S ij we denote the simple -dimensional representation of P SL 2 (Z) determined by S ij Z2 = S i and S ij Z3 Let n = x +...+x 6 and we aim to study the local structure of rep n CP SL 2 (Z) in a neighborhood of the semi-simple n-dimensional representation V ξ = S x S x 2 2 S x 3 3 S x 4 2 S x 5 22 S x 6 23 To determine the structure of Q ξ we have to compute dim Ext (S ij, S kl ). To do this we view the S ij as representations of the quiver Q in the example above. For example S 2 is the representation 0 of dimension vector (, 0; 0,, 0).For representations of Q, the dimensions of Hom and Ext-groups are determined by the bilinear form χ Q =

270 270 CHAPTER 7. MODULI SPACES. If V rep α Q and W rep β Q where α = (a, a 2 ; b, b 2, b 3 ) with a + a 2 = b + b 2 + b 3 = k and β = (c, c 2 ; d, d 2, d 3 ) with c + c 2 = d + d 2 + d 3 = l we have dim Hom(V, W ) dim Ext (V, W ) = χ Q (α, β) = kl (a c + a 2 c 2 + b d + b 2 d 2 + b 3 d 3 ) Because the translation from P SL 2 (Z)-representations to Q-representations is full and faithful and as Hom(S ij, S kl ) = C δ ikδ jl we have that ( dim Ext if i k and j l (S ij, S kl ) = 0 otherwise But then, the local quiver setting (Q ξ, α ξ) is x 4 x 3 x 2 x 5 x 6 x We want to determine whether the irreducible component of rep n CP SL 2 (Z) containing V ξ contains simple P SL 2 (Z)-representations, or equivalently, whether α ξ is the dimension vector of a simple representation of Q ξ, that is, χ Q ξ (α ξ, ɛ j ) 0 and χ Q ξ (ɛ j, α ξ ) for all j 6 The Euler-form of Q ξ is determined by the matrix where we number the vertices cyclicly χ Q = ξ leading to the following set of inequalities 8 >< x x 5 + x 6 x 2 x 4 + x 6 >: x 3 x 4 + x 5 8 >< x 4 x 2 + x 3 x 5 x + x 3 >: x 6 x + x 2 Finally, observe that V ξ corresponds to a Q-representation of dimension vector (x + x 2 + x 3, x 4 + x 5 + x 6 ; x + x 4, x 2 + x 5, x 3 + x 6 ). If we write this dimension vector as (a, a 2 ; b, b 2, b 3 ) then the inequalities are equivalent to the conditions a i b j for all i 2 and j 3 which gives us the desired restriction on the quintuples a a 2 at least when a i 3 and b j 2. The remaining cases are handled similarly. Observe that we can use a similar strategy to determine the restrictions on simple representations of any free product Z p Z q of two cyclic groups. b b 2 b 3

271 7.7. SEMI-INVARIANTS OF QUIVERS. 27 The graded algebra C[rep α C] GL(α) of all semi-invariants on rep α Q of weight χ n θ for some n 0 has as degree zero part the ring of polynomial invariants C[rep α Q] GL(α). This embedding determines a proper morphism M ss α (Q, θ) π issα Q which is onto whenever rep ss α (Q, α) is non-empty. In particular, if Q is a quiver without oriented cycles, then the moduli space of θ-semistable representations of dimension vector α, Mα ss (Q, θ), is a projective variety. 7.7 Semi-invariants of quivers. Because the moduli space Mα ss (Q, θ) is defined to be the projective scheme of the graded algebra of semi-invariants of weight χ n θ for some n M ss α (Q, α) = P roj n=0 C[rep α Q] GL(α),χn θ we need some control on the semi-invariants of quivers. A generating set of semiinvariants was described by A. Schofield and M. Van den Bergh in [28]. The strategy of proof should be clear by now. First, we describe a large set of semi-invariants, apart from the invariant polynomials which we know to be generated by traces of oriented cycles in the quiver we expect determinantal semi-invariants as in the case of mixed GL n -semi-invariants of section 3. Then we use classical invariant theory to describe all multilinear semi-invariants of GL(α), or equivalently, all multilinear invariants of SL(α) = SL a... SL ak and describe them in terms of these determinantal semi-invariants. Finally, we show by polarization and restitution that these semi-invariants do indeed generate all semi-invariants. Let Q be a quiver on k vertices {v,..., v k }. We introduce the additive C- category add Q generated by the quiver. For every vertex v i we introduce an indecomposable object which we denote by i. An arbitrary object in add Q is then a sum of those e... k e k Morphisms in the category add Q are defined by the rules Hom( i Hom( i, j ) = j, i ) = i where the right hand sides are the C-vectorspaces spanned by all oriented paths from v i to v j in the quiver Q, including the idempotent (trivial) path e i when i = j. Clearly, for any k-tuples of positive integers α = (u,..., u k ) and β = (v,..., v k ) i Hom( u... k u k v,... k v k ) is defined in the usual way in the additive category add Q, that is by the matrices

272 272 CHAPTER 7. MODULI SPACES. where composition arises via matrix multiplication M v u (. M vk u ( k )... M v u k ( k ).... )... M vk u k ( k ) Now, fix a dimension vector α = (a,..., a k ) and a morphism in add Q u... k u k φ v... k v k For any representation V rep α Q we can replace each occurrence of an arrow a j of i Q in φ by the a j a i -matrix V a. This way we obtain a rectangular matrix V (φ) M P k i= aivi P k aiui(c) i= If we are in a situation such that a i v i = a i u i, then we can define a semiinvariant polynomial function on rep α Q by P α,φ (V ) = det V (φ) We call such semi-invariants determinantal semi-invariants. One verifies that P φ,α is a semi-invariant of weight χ θ where θ = (u v,..., u k v k ). We will show that such determinantal semi-invariant together with traces along oriented cycles generate all semi-invariants. Because semi-invariants for the GL(α)-action on rep α Q are the same as invariants for the restricted action of SL(α) = SL a... SL ak, we will describe the multilinear SL(α)-invariants from classical invariant theory. Because rep α Q = M aj a i (C) a j i = a j i C ai C aj we have to consider multilinear SL(α)-invariants of C ai C aj = [ C ai j i i i i C ai ] Hence, any multilinear SL(α)-invariant can be written as f = k i= f i where f i is a SL ai -invariant of i C ai i C ai In section 3 we have recalled Weyl s result describing all multilinear SL m -invariants on B C m C C m. By polarization and restitution it follows from this that the linear SL m -invariants are determined by the following three sets

273 7.7. SEMI-INVARIANTS OF QUIVERS. 273 traces, that is, for each pair (b, c) we have C m C m = M m (C) T r C. brackets, that is, for each m-tuple (b,..., b m ) we have an invariant bj C m C defined by v b... v bm det [ ] v b... v bm cobrackets, that is, for each m-tuple (c,..., c m ) we have an invariant ci C m C defined by φ c... φ cm det φ c. φ cm Multilinear SL m -invariants of B C m C C m are then spanned by invariants constructed from the following data. Take three disjoint index-sets I, J and K and consider surjective maps { B µ I K C ν J K subject to the following conditions { # µ (k) = = # ν (k) for all k K. # µ (i) = m = # ν (j) for all i I and j J. To this data γ = (µ, ν, I, J, K) we can associate a multilinear SL m -invariant f γ ( B v b C φ c ) defined by φ ν (k)(v µ (k)) det [ φ ] c v b... v bm det. k K i I j J where µ (i) = {b,..., b m } and ν (j) = {c,..., c m }. Observe that f γ is determined only up to a sign by the data γ. But then, we also have a spanning set for the multilinear SL(α)-invariants on rep α Q = v [ v C av C av ] v determined by quintuples Γ = (µ, ν, I, J, K) where we have disjoint index-sets partitioned over the vertices v {v,..., v k } of Q I = v I v J = v J v K = v K v together with surjective maps from the set of all arrows A of Q { A µ I K A ν J K φ cm where we have for every arrow w a that v { µ(a) I v K v ν(a) J w K w

274 274 CHAPTER 7. MODULI SPACES. and these maps µ and ν are subject to the numerical restrictions { # µ (k) = = # ν (k) for all k K. # µ (i) = a v = # ν (j) for all i I v and all j J v. Such a quintuple Γ = (µ, ν, I, J, K) determines for every vertex v a quintuple γ v = (µ v = µ { a }, v ν v = ν { a v }, I v, J v, K v ) satisfying the necessary numerical restrictions to define the SL av -invariant f γv described before. Then, the multilinear SL(α)-invariant on rep α Q determined by Γ is defined to be f γ = v f γv and we have to show that these semi-invariants lie in the linear span of the determinantal semi-invariants. First, consider the case where the index set K is empty. If we denote the total number of arrows in Q by n, then the numerical restrictions imposed give us two expressions for n a v.# I v = n = a v.# J v v v Every arrow a w determines v a pair of indices µ(a) I v and ν(a) J w. To the quintuple Γ we assign a map Φ Γ in add Q I... k I k Φ Γ J... k J k I which decomposes as a block-matrix in blocks M v,w Hom( v v, w which the (i, j) entry is given by the sum of arrows µ(a)=i ν(a)=j w a v J w ) of For a representation V rep α Q, V (Φ Γ ) is an n n matrix and the determinant defines the determinantal semi-invariant P Φα,Γ which we claim to be equal to the basic invariant f Γ possibly up to a sign. We introduce a new quiver situation. Let Q be the quiver with vertices the elements of I J and with arrows the set A of arrows of Q, but this time w take the starting point of the arrow in a Q to be µ(a) I and the terminating vertex to be ν(a) J. That is, Q is a bipartite quiver µ(a) a ν(a) I J

275 7.7. SEMI-INVARIANTS OF QUIVERS. 275 On Q we have the quintuple Γ = (µ, ν, I, J, K ) where K =, I = i I I i = i I {i} J = j J J j = j J {j} and µ = µ, ν = ν. We define an additive functor add Q s add Q by i s v j s w s a a for all i I v and all j J w. The functor s induces a functor rep Q s rep Q defined by V s V s. If V rep α Q then s(v ) rep α Q where { α c i = a v if i I v = (c,..., c p, d,..., d q ) with }{{}}{{} d j = a w if j J w # I # J That is, the characteristic feature of Q is that every vertex i I is the source of exactly c i arrows (follows from the numerical condition on µ) and that every vertex j J is the sink of exactly d j arrows in Q. That is, locally Q has the following form c c or d d There are induced maps rep α Q s rep α Q GL(α) s GL(α ) where the latter follows from functoriality by considering GL(α) as the automorphism group of the trivial representation in rep α Q. These maps are compatible with the actions as one checks that s(g.v ) = s(g).s(v ). Also s induces a map on the coordinate rings C[rep α Q] s C[repα Q ] by s(f) = f s. In particular, for the determinantal semi-invariants we have s(p α,φ ) = P α,s(φ ) and from the compatibility of the action it follows that when f is a semi-invariant the GL(α ) action on rep α Q with character χ, then s(f) is a semi-invariant for the GL(α)-action on rep α Q with character s(χ) = χ s. In particular we have that s(p α,φ Γ ) = P α,s(φγ ) = P α,φγ and s(f Γ ) = f Γ Hence in order to prove our claim, we may replace the triple (Q, α, Γ) by the triple (Q, α, Γ ). We will do this and forget the dashes from here on. In order to verify that f Γ = ±P α,φγ it suffices to check this equality on the image of W = C ci C dj in C ci C dj a i j a i j One verifies that both f Γ and P α,φγ are GL(α)-semi-invariants on W of weight χ θ where θ = (,...,,,..., ) }{{}}{{} # I # J Using the characteristic local form of Q = Q, we see that W is isomorphic to the GL(α)- module W (C ci... C }{{ ci ) (C } dj... C dj ) M }{{} ci (C) M dj (C) i I c i j J d i I j J j

276 276 CHAPTER 7. MODULI SPACES. and the i factors of GL(α) act by inverse right-multiplication on the component M ci (and trivially on all others) and the j factors act by left-multiplication on the component M dj (and trivially on the others). That is, GL(α) acts on W with an open orbit, say that of the element w = ( c,..., cp, d,..., dq ) W One verifies immediately from the definitions that that both f Γ and P α,φγ evaluate to ± in w. Hence, indeed, f Γ can be expressed as a determinantal semi-invariant. Remains to consider the case when K is non-empty. For k K two situations can occur µ (k) = a and ν (k) = b are distinct, then k corresponds to replacing the arrows a and b by their concatenation a k b µ (k) = a = ν (k) then a is a loop in Q and k corresponds to taking the trace of a. This time we construct a new quiver Q with vertices {w,..., w n } corresponding to the set A of arrows in Q. The arrows in Q will correspond to elements of K, that is if k K we have the arrow (or loop) in Q with notations as before a k k b or a a We consider the connected components of Q. They are of the following three types (oriented cycle) : To an oriented cycle C in Q corresponds an oriented cycle C C in the original quiver Q. We associate to it the trace tr(c C ) of this cycle. (open paths) : An open path P in Q corresponds to an oriented path P P in Q which may be a cycle. To P we associate the corresponding path P P in Q. (isolated points) : They correspond to arrows in Q. We will now construct a new quiver Q having the same vertex set {v,..., v k } as Q but with arrows corresponding to the set of paths P P described above. The starting and ending vertex of the arrow corresponding to P P are of course the starting and ending vertex of the path P P in Q. Again, we define an additive functor add Q s add Q by the rules k v s v and s P P j i j P P i If the path P P maps is the concatenation of the arrows a d... a in Q, we define the { µ (P P ) = µ(a ) ν (P P ) = ν(a d) whence { {P P } µ I {P P } ν J

277 7.7. SEMI-INVARIANTS OF QUIVERS. 277 that is, a quintuple Γ = (µ, ν, I, J, K = ) for the quiver Q. One then verifies that f Γ = s(f Γ ) C tr(c C) = s(p α,φγ ) tr(c C) C = P α,s(φγ ) tr(c C) C finishing the proof of the fact that multilinear semi-invariants lie in the linear span of determinantal semi-invariants (and traces of oriented cycles). The arguments above can be reformulated in a more combinatorial form which is often useful in constructing semi-invariants of a specific weight, as is necessary in the study of the moduli spaces Mα ss (Q, θ). Let Q be a quiver on the vertices {v,..., v k }, fix a dimension vector α = (a,..., a k ) and a character χ θ where θ = (t,..., t k ) such that θ(α) = 0. We will call a bipartite quiver Q r j l i L R on left vertex-set L = {l,..., l p } and right vertex-set R = {r,..., r q } and a dimension vector β = (c,..., c p ; d,..., d q ) to be of type (Q, α, θ) if the following conditions are met All left and right vertices correspond to vertices of Q, that is, there are maps { L l {v,..., v k } R r {v,..., v k } possibly occurring with multiplicities, that is there is a map L R m N + such that c i = m(l i )a z if l(l i ) = v z and d j = m(r j )a z if r(r j ) = v z. There can only be an arrow l r i j if for v k = l(l i ) and v l = r(r i ) there is an oriented path v k v l in Q allowing the trivial path and loops if v k = v l. Every left vertex l i is the source of exactly c i arrows in Q and every rightvertex r j is the sink of precisely d j arrows in Q. Consider the u u matrix where u = i c i = j d j (both numbers are equal to the total number of arrows in Q ) where the i-th row contains the entries of the i-th arrow in Q with respect to the obvious left and right bases. Observe

278 278 CHAPTER 7. MODULI SPACES. that this is a GL(β) semi-invariant on rep β Q with weight determined by the integral k + l-tuple (,..., ;,..., ). If we fix for every arrow a from l i to r j in Q an m(r j ) m(l i ) matrix p a of linear combinations of paths in Q from l(l i ) to r(r j ), we obtain a morphism rep α Q rep β Q sending a representation V rep α Q to the representation W of Q defined by W a = p a (V ). Composing this map with the above semi-invariant we obtain a GL(α) semi-invariant of rep α Q with weight determined by the k-tuple θ = (t,..., t k ) where t i = m(r j ) m(l j ). j r (v i) j l (v i) We call such semi-invariants standard determinantal. Summarizing the arguments of this section we have proved after applying polarization and restitution processes Theorem 7.24 The semi-invariants of the GL(α)-action on rep α Q are generated by traces of oriented cycles and by standard determinantal semi-invariants. 7.8 Brauer-Severi varieties. In this section we will reconsider the Brauer-Severi scheme BS n (A) of an algebra, introduced in chapter 2. In the generic case, that is when A is the free algebra C x,..., x m, we show that it is a moduli space of a certain quiver situation. This then allows us to give the étale local description of BS n (A) whenever A is a Cayleysmooth algebra. Again, this local description will be a moduli space. The generic Brauer-Severi scheme of degree n for m-generators, BSn m (gen) is defined as follows. Consider the free algebra on m generators C x,..., x m and consider the GL n -action on rep n C x,..., x m C n = Mn m C n given by g.(a,..., A m, v) = (ga g,..., ga m g, gv) and consider the open subset Brauer s (gen) consisting of those points (A,..., A m, v) where v is a cyclic vector, that is, there is no proper subspace of C n containing v and invariant under left multiplication by the matrices A i. The GL n -stabilizer is trivial in every point of Brauer s (gen) whence we can define the orbit space BS m n (gen) = Brauer s (gen)/gl n Consider the following quiver situation m n on two vertices {v, v 2 } such that there are m loops in v 2 and consider the dimension vector α = (, n). Then, clearly rep α Q = C n M m n rep n C x,..., x m C n where the isomorphism is as GL n -module. On rep α Q we consider the action of the larger group GL(α) = C GL n acting as (λ, g).(v, A,..., A m ) = (gvλ, ga g,..., ga m g ) Consider the character χ θ where θ = ( n, ), then θ(α) = 0 and consider the open subset of θ-semistable representations in rep α Q.

279 7.8. BRAUER-SEVERI VARIETIES. 279 Lemma 7.25 The following are equivalent for V = (v, A,..., A m ) rep α Q. V is θ-semistable. 2. V is θ-stable. 3. V Brauer s (gen). Consequently, M ss α (Q, α) BS m n (gen) Proof.. 2. : If V is θ-semistable it must contain a largest θ-stable subrepresentation W (the first term in the Jordan-Hölder filtration for θ-semistables). In particular, if the dimension vector of W is β = (a, b) < (, n), then θ(β) = 0 which is impossible unless β = α whence W = V is θ-stable : Observe that v 0, for otherwise V would contain a subrepresentation of dimension vector β = (, 0) but θ(β) = n is impossible. Assume that v is noncyclic and let U C n be a proper subspace say of dimension l < n containing v and stable under left multiplication by the A i, then V has a subrepresentation of dimension vector β = (, l) and again θ(β ) = l n < 0 is impossible. 3.. : By cyclicity of v, the only proper subrepresentations of V have dimension vector β = (0, l) for some 0 < l n, but they satisfy θ(β) > 0, whence V is θ-(semi)stable. As for the last statement, recall that geometric points of Mα ss (Q, α) classify isomorphism classes of direct sums of θ-stable representations. As there are no proper θ-stable subrepresentations, Mα ss (Q, α) classifies the GL(α)-orbits in Brauer s (gen). Finally, as in chapter, there is a one-to-one orbits between the GL n -orbits as described in the definition of the Brauer-Severi variety and the GL(α)-orbits on rep α Q. By definition, Mα ss (Q, θ) = P roj n=0 C[rep α Q] GL(α),χnθ and we can either use the results of section 3 or the previous section to show that these semi-invariants f are generated by brackets, that is, f(v ) = det [ w (A,..., A m )v... w n (A,..., A m )v ] where the w i are words in the noncommuting variables x,..., x m. As in section I.3 we can restrict these n-tuples of words {w,..., w n } to sequences arising from multicolored Hilbert n-stairs. That is, the lower triangular part of a square n n array n this time filled with colored stones i rules each row contains exactly one stone n each column contains at most one stone of each color where i m subject to the two coloring

280 280 CHAPTER 7. MODULI SPACES. The relevant sequences W (σ) = {, w 2,..., w n } of words are then constructed by placing the identity element at the top of the stair, and descend according to the rule Every go-stone has a top word T which we may assume we have constructed before and a side word S and they are related as indicated below T T S i x i T In a similar way to the argument in chapter we can cover M ss α (Q, α) = BS m n (gen) by open sets determined by Hilbert stairs and find representatives of the orbits in σ-standard form, that is replacing every i-colored stone in σ by a at the same spot in A i and fill the remaining spots in the same column of A i by zeroes 0... i i i 0 0 n j n A i =... 0 j As this fixes (n )n entries of the mn 2 +n entries of V, one recovers the following result of M. Van den Bergh [3] Theorem 7.26 The generic Brauer-Severi variety BS m n (gen) of degree n in m generators is a smooth variety which can be covered by affine open subsets each isomorphic to C (m )n2 +n. For an arbitrary affine C-algebra A, one defines the Brauer stable points to be the open subset of rep n A C n Brauer s n(a) = {(φ, v) rep n A C n φ(a)v = C n } As Brauer stable points have trivial stabilizer in GL n all orbits are closed and we can define the Brauer-Severi variety of A of degree n to be the orbit space BS n (A) = Brauer s n(a)/gl n We claim that Quillen-smooth algebras have smooth Brauer-Severi varieties. Indeed, as the quotient morphism Brauer s n(a) BSn (A) is a principal GL n -fibration, the base is smooth whenever the total space is smooth. The total space is an open subvariety of rep n A C n which is smooth whenever A is Quillen-smooth.

281 7.8. BRAUER-SEVERI VARIETIES. 28 Proposition 7.27 If A is Quillen-smooth, then for every n we have that the Brauer-Severi variety of A at degree n is smooth. Next, we bring in the approximation at level n. Observe that for every affine C-algebra A we have a GL n -equivariant isomorphism rep n A rep tr n A@ n More generally, we can define for every Cayley-Hamilton algebra A of degree n the trace preserving Brauer-Severi variety to be the orbit space of the Brauer stable points in rep tr A n Cn. We denote this variety with BSn tr (A). Again, the same argument applies Proposition 7.28 If A is Cayley-smooth of degree n, then the trace preserving Brauer-Severi variety BS tr n (A) is smooth. We have seen that the moduli spaces are projective fiber bundles over the variety determined by the invariants, M ss α (Q, θ) issα Q Similarly, the (trace preserving) Brauer-Severi variety is a projective fiber bundle over the quotient variety of rep n A, that is, there is a proper map BS n (A) π issn A and we would like to study the fibers of this map. Recall that when A is an order in a central simple algebra of degree n, then the general fiber will be isomorphic to the projective space P n embedded in a higher dimensional P N. Over non-azumaya points we expect this P n to degenerate to more complex projective varieties which we would like to describe. To perform this study we need to control the étale local structure of the fiber bundle π in a neighborhood of ξ iss n A. Again, it is helpful to consider first the generic case, that is when A = C x,..., x m or T m n. In this case, we have seen that the following two fiber bundles are isomorphic BS m n (gen) issn T m n and M ss α (Q, θ) issα Q where α = (, n), θ = ( n, ) and the quiver m n has Euler form χ Q = [ ] 0 m A semi-simple α-dimensional representation V ζ of Q has representation type (, 0) (0, d ) e... (0, d k ) e k with d i e i = n and hence corresponds uniquely to a point ξ iss n T m n of representation type τ = (e, d ;... ; e k, d k ). The étale local structure of rep α Q and of iss α Q near ζ is determined by the local quiver Q ζ on k + -vertices, say {v 0, v,..., v k } with dimension vector α ζ = (, e,..., e k ) and where Q ζ has the following local form for i

282 282 CHAPTER 7. MODULI SPACES. every triple (v 0, v i, v j ) as can be verified from the Euler-form a j e j dj a ij d i a ji e i a i where a ij = (m )d i d j = a ji and a i = (m )d 2 i +, a j = (m )d 2 j +. The dashed part of Q ζ is the same as the local quiver Q ξ describing the étale local structure of iss n T m n near ξ. Hence, we see that the fibration BSn m (gen) issn T m n is étale isomorphic in a neighborhood of ξ to the fibration of the moduli space M ss α ζ (Q ζ, θ ζ ) issαζ Q ζ iss αξ Q ξ in a neighborhood of the trivial representation and where θ ζ = ( n, d,..., d k ). Another application of the Luna slice results gives the following Theorem 7.29 Let A be a Cayley-smooth algebra of degree n. Let ξ iss tr n correspond to the trace preserving n-dimensional semi-simple representation A M ξ = S e... S e k k where the S i are distinct simple representations of dimension d i and occurring with multiplicity e i. Then, the projective fibration BS tr n (A) π iss tr n A is étale isomorphic in a neighborhood of ξ to the fibration of the moduli space M ss α ζ (Q ζ, θ ζ ) issαζ Q ζ iss αξ Q ξ in a neighborhood of the trivial representation. Here, Q ξ is the local marked quiver describing the étale local structure of rep tr n A near ξ, where Q ζ is the extended marked quiver situation, which locally for every triple (v 0, v i, v j ) has the following shape where the dashed region is the local marked quiver Q ξ describing

283 7.8. BRAUER-SEVERI VARIETIES. 283 Ext tr A (M ξ, M ξ ) and where α ζ = (, e,..., e k ) and θ ζ = ( n, d,..., d k ). m j e j dj a ij d i m i e i u j a ji u i In the next chapter we will use this local description to describe the fibers of the Brauer-Severi fibration.

284 284 CHAPTER 7. MODULI SPACES.

285 Chapter 8 Nullcones. When A is a Quillen-smooth algebra we have found a local description of the variety rep n A of its n-dimensional representations, of the variety iss n A os isomorphism classes of semi-simple n-dimensional representations and of the Brauer-Severi variety BS n A. In this chapter we will develop tools to study the fibers of the structural morphisms BS n A rep n A ψ π issn A In particular, we will be able to compute the number and dimensions of their irreducible components allowing to determine the flat locus of these morphisms. The basic observation is that these fibers are nullcones of certain quiver settings, that is, quiver representations on which all polynomial invariants evaluate to zero. What we will do is to give a representation theoretic interpretation of a stratification by vectorbundles over flag varieties of these nullcones, due to W. Hesselink [0]. The strategy is easy to explain in the generic case, that is, of m-tuples of n n matrices. Here, (A,..., A m ) lies in the nullcone if and only if by applying permutation Jordan-moves simultaneously to the components A i, they all become strictly upper triangular matrices. Sometimes, we can do better and bring all the non-zero entries of the A i together in a smaller upper right-hand side corner, such as C = for 4 4 matrices. For a given m-tuple it is easy to determine the smallest corner which can be obtained by only applying permutation moves. Remains the problem whether we can simultaneously conjugate the m-tuple to produce another tuple (A..., A m) in the orbit which can be permuted to a strictly smaller corner. If this is not possible, we will say that the corner type C is optimal for (A,..., A m ). To verify this it is clear that the border region of the corner, such as B = will be relevant. We will assign a new quiver setting to the border region. Observe that there is a parabolic subgroup P of GL n preserving the corner C and its action on the border region is coming from its Levi subgroup L which is a product of GL l s. 285

286 286 CHAPTER 8. NULLCONES. In the example, P = and L = and the corresponding quiver setting is easily seen to be m m 2 The crucial observation is now, to assign a character to GL(α) = L such that an m-tuple (A,..., A m ) has optimal corner type C if and only if the representation in rep n Q = B it determines by looking only at the border entries is θ-semistable. By Schofield s criterium we have a combinatorial way to verify whether there are such θ- semistable representations. The corresponding stratum in Hesselinks s stratification is S = GL n.u where U is the collection of all such m-tuples. We have the following size-reduction of the problem : there is a natural one-to-one correspondence between GL n -orbits in S, and P -orbits in U Moreover, as U is determined by θ-semistables, there is a moduli space of quiver representation M ss α (Q, θ) ar the heart of the stratum. 8. Cornering matrices. In this section we will outline the basic idea of the Hesselink stratification of the nullcone [0] in the generic case, that is, the action of GL n by simultaneous conjugation on m-tuples of matrices M m n = M n... M n. With Null m n we denote the nullcone of this action Null m n = {x = (A,..., A m ) M m n 0 = (0,..., 0) O(x)} By the Hilbert criterium we know that x = (A,..., A m ) belongs to the nullcone if and only if there is a one-parameter subgroup C λ GLn such that lim λ(t).(a,..., A m ) = (0,..., 0). t 0 We recall from chapter 4 that any one-parameter subgroup of GL n is conjugated to one determined by an integral n-tuple (r,..., r n ) Z n by t r 0 λ(t) =... 0 t rn Moreover, permuting the basis if necessary, we can conjugate this λ to one where the n-tuple if dominant, that is, r r 2... r n. By applying permutation Jordan-moves, that is, by simultaneously interchanging certain rows and columns in all A i, we may therefore assume that the limit-formula holds for a dominant one-parameter subgroup λ of the maximal torus c 0 T n C... C }{{ = { }... c i C } GL n n 0 c n

287 8.. CORNERING MATRICES. 287 of GL n. Computing its action on a n n matrix A we obtain t r 0 a... a n t r 0 t r r a... t r rn a n =.. 0 t rn a n... a nn 0 r rn t rn r a n... t rn rn a nn But then, using dominance r i r j for i j, we see that the limit is only defined if a ij = 0 for i j, that is, when A is a strictly upper triangular matrix. We have proved our first cornering result. Lemma 8. Any m-tuple x = (A,..., A m ) Nulln m has a point in its orbit O(x) under simultaneous conjugation (A,..., A m) with all A i strictly upper triangular matrices. In fact permutation Jordan-moves suffice. For specific m-tuples x = (A,..., A m ) it might be possible to improve on this result. That is, we want to determine the smallest corner C in the upper right hand corner of the matrix, such that all the component matrices A i can be conjugated simultaneously to matrices A i having only non-zero entries in the corner C C = and no strictly smaller corner C can be found with this property. Our first task will be to compile a list of the relevant corners and to define an order relation on this set. Consider the weight space decomposition of Mn m for the action by simultaneous conjugation of the maximal torus T n, M m n = i,j n M m n (π i π j ) = i,j n C m π i π j where c = diag(c,..., c n ) T m acts on any element of Mn m (π i π j ) by multiplication with c i c j, that is, the eigenspace Mn m (π i π j ) is the space of the (i, j)-entries of the m-matrices. We call W = {π i π j i, j n} the set of T n -weights of M m n. Let x = (A,..., A m ) Null m n and consider the subset E x W consisting of the elements π i π j such that for at least one of the matrix components A k the (i, j)-entry is non-zero. Repeating the argument above, we see that if λ is a one-parameter subgroup of T n determined by the integral n-tuple (r,..., r n ) Z n such that lim λ(t).x = 0 we have π i π j E x we have r i r j Conversely, let E W be a subset of weights, we want to determine the subset {s = (s,..., s n ) R n s i s j π i π j E } and determine a point in this set, minimal with respect to the usual norm s = s s2 n

288 288 CHAPTER 8. NULLCONES. Let s = (s,..., s n ) attain such a minimum. We can partition the entries of s in a disjoint union of strings {p i, p i +,..., p i + k i } with k i N and subject to the condition that all the numbers p ij def = p i + j with 0 j k i occur as components of s, possibly with a multiplicity that we denote by a ij. We call a string string i = {p i, p i +,..., p i + k i } of s balanced if and only if k i s j = a ij (p i + j) = 0 s k string i j=0 In particular, all balanced strings consists entirely of rational numbers. We have Lemma 8.2 Let E W, then the subset of R n determined by R n E = { (r,..., r n ) r i r j π i π j E} has a unique point s E = (s,..., s n ) of minimal norm s E. This point is determined by the characteristic feature that all its strings are balanced. In particular, s E Q n. Proof. Let s be a minimal point for the norm in R n E and consider a string of s and denote with S the indices k {,..., n} such that s k string. Let π i π j E, then if only one of i or j belongs to S we have a strictly positive number a ij s i s j = + r ij with r ij > 0 Take ɛ 0 > 0 smaller than all r ij and consider the n-tuple s ɛ = s + ɛ(δ S,..., δ ns ) with δ ks = if k S and 0 otherwise with ɛ ɛ 0. Then, s ɛ R n E for if π i π j E and i and j both belong to S or both do not belong to S then (s ɛ ) i (s ɛ ) j = s i s j and if one of i or j belong to S, then (s ɛ ) i (s ɛ ) j = + r ij ± ɛ by the choice of ɛ 0. However, the norm of s ɛ is s ɛ = s +2ɛ s k + ɛ 2 #S k S Hence, if the string would not be balanced, k S s k 0 and we can choose ɛ small enough such that s ɛ < s, contradicting minimality of s. For given n we can compile a list S n of all dominant n-tuples (s,..., s n ) (that is, s i s j whenever i j) having all its strings balanced, as follows. List all Young-diagrams Y n = {Y,...} having n boxes. For every diagram Y l fill the boxes with strictly positive integers subject to the rules. the total sum is equal to n 2. no two rows are filled identically 3. at most one row has length

289 8.. CORNERING MATRICES. 289 This gives a list T n = {T,...} of tableaux. For every tableau T l T n, for each of its rows (a, a 2,..., a k ) find a solution p to the linear equation a x + a 2 (x + ) a k (x + k) = 0 and define the a i -tuple of rational numbers (p,..., p, p +,..., p +,... p + k,..., p + k) }{{}}{{}}{{} a a k a 2 Repeating this process for every row of T l we obtain an n-tuple, which we then order. 2 For example, for n = 5 and the tableaux the linear equations are { 2x + x + = 0 giving p = 3 x + x + = 0 giving p 2 = 2 The corresponding 5-tuple is therefore s = ( 2 3, 2, 3, 3, 2 ). The list S n will be the combinatorial object underlying the relevant corners and the stratification of the nullcone. Example 8.3 (S n for small n) For n = 2, we have giving ( 2, 2 ) and 2 giving (0, 0). For n = 3 we have five types tableau s s 2 s 3 s 2 S 3 = S 4 has eleven types tableau s s 2 s 3 s 4 s 2 S 4 = Observe that we ordered the elements in S n according to s. The reader is invited to verify that S 5 has 28 different types. To every s = (s,..., s n ) S n we associate the following data

290 290 CHAPTER 8. NULLCONES. the corner C s is the subspace of Mn m consisting of those m tuples of n n matrices with zero entries except perhaps at position (i, j) where s i s j. A partial ordering is defined on these corners by the rule C s < C s s < s the parabolic subgroup P s which is the subgroup of GL n consisting of matrices with zero entries except perhaps at entry (i, j) when s i s j 0. the Levi subgroup L s which is the subgroup of GL n consisting of matrices with zero entries except perhaps at entry (i, j) when s i s j = 0. Observe that L s = GL aij where the a ij are the multiplicities of p i + j. Example 8.4 Using the sequence of types in the previous example, we have that the relevant corners and subgroup for 3 3 matrices are C s P s L s For 4 4 matrices the relevant corners are Returning to the corner-type of an m-tuple x = (A,..., A m ) Nulln m, we have seen that E x W determines a unique s Ex Q n which up to permuting the entries an element s of S n. As permuting the entries of s translates into permuting rows and columns in M n (C) we have Theorem 8.5 Every x = (A,..., A m ) Nulln m can be brought by permutation Jordan-moves to an m-tuple x = (A,..., A m) C s. Here, s is the dominant reordering of s Ex with E x W the subset π i π j determined by the non-zero entries at place (i, j) of one of the components A k. The permutation of rows and columns is determined by the dominant reordering. The m-tuple s (or s Ex ) determines a one-parameter subgroup λ s of T n where λ corresponds to the unique n-tuple of integers (r,..., r n ) N + s Z n with gcd(r i ) = For any one-parameter subgroup µ of T n determined by an integral n-tuple µ = (a,..., a n ) Z n and any x = (A,..., A n ) Null m n we define the integer m(x, µ) = min {a i a j x contains a non-zero entry in M m n (π i π j ) } From the definition of R n E it follows that the minimal value s E and λ se s Ex = λ sex m(x, λ sex ) and s = λ s m(x, λ s ) We can now state to what extend λ s is an optimal one-parameter subgroup of T n. is

291 8.2. OPTIMAL CORNERS. 29 Theorem 8.6 Let x = (A,..., A m ) Nulln m and let µ be a one-parameter subgroup contained in T n such that lim λ(t).x = 0, then t 0 λ sex m(x, λ sex ) µ m(x, µ) µ m(x,µ) Rn E x The proof follows immediately from the observation that and the minimality of s Ex. Phrased differently, there is no simultaneous reordering of rows and columns that admit an m-tuple x = (A,..., A m ) C s for a corner C s < C s. 8.2 Optimal corners. In the foregoing section we have transformed an m-tuple x = (A,..., A m ) Null m n by interchanging rows and columns to an m-tuple in corner-form C s. However, it is still possible that another point in the orbit O(x) say y = g.x = (B,..., B m ) can be transformed by interchanging rows and columns in a smaller corner. Example 8.7 Consider one 3 3 nilpotent matrix of the form 2 x = 4 0 a b with ab Then, E x = {π π 2, π π 3 } and the corresponding s = s Ex corner type = ( 2 3, 3, ) so x is clearly of 3 C s = However, x is a nilpotent matrix of rank and by the Jordan-normalform we can conjugate it in standard form, that is, there is some g GL 3 such that 2 y = g.x = gxg = For this y we have E y = {π π 2 } and the corresponding s Ey = ( 2,, 0), which can be brought 2 into standard dominant form s = ( 2, 0, ) by interchanging the two last entries. Hence, by 2 interchanging the last two rows and columns, y is indeed of corner type and we have that C s < C s. C s = Trivial as this example seems, we needed the Jordan-normalform to produce it. As there are no known canonical forms for m tuples of n n matrices, it is a difficult problem to determine the optimal corner type in general. Definition 8.8 We say that x = (A,..., A m ) Nulln m is of optimal corner type C s if after reordering rows and columns, x is of corner type C s and there is no point y = g.x in the orbit which is of corner type C s with C s < C s. Using the results of the foregoing chapter we can give an elegant solution to the problem of determining the optimal corner type of an m-tuple in Null m n. We assume that x = (A,..., A m ) is brought into corner type C s with s = (s,..., s n ) S n. We will associate a quiver-representation to x. As we are interested in checking whether we can transform x to a smaller corner-type, it is intuitively clear that the border region of C s will be important.

292 292 CHAPTER 8. NULLCONES. the border B s is the subspace of C s consisting of those m-tuples of n n matrices with zero entries except perhaps at entries (i, j) where s i s j =. Example 8.9 For 3 3 matrices we have the following corner-types C s having border-regions B s and associated Levi-subgroups L s C s B s L s For 4 4 matrices the relevant data are as follows C s = B s = L s = C s = B s = L s = From these examples, it is clear that the action of the Levi-subgroup L s on the border B s is a quiver-setting. In general, let s S n be determined by the tableau T s, the associated quiver-setting (Q s, α s ) is Q s is the quiver having as many connected components as there are rows in the tableau T s. If the i-th row in T s is (a i0, a i,..., a iki ) then the corresponding string of entries in s is of the form {p i,..., p }{{} i, p i +,..., p i +,..., p }{{} i + k i,..., p i + k i } }{{} a i0 a iki a i and the i-th component of Q s is defined to be the quiver Q i on k i + vertices having m arrows between the consecutive vertices, that is Q i is m m m... m 0 2 k i

293 8.2. OPTIMAL CORNERS. 293 the dimension vector α i for the i-th component quiver Q i is equal to the i-th row of the tableau T s, that is α i = (a i0, a i,..., a iki ) and the total dimension vector α s is the collection of these component dimension vectors. With these notations we have Proposition 8.0 The action of the Levi-subgroup L s = i,j GL a ij on the border B s coincides with the base-change action of GL(α s ) on the representation space rep αs Q s. The isomorphism B s repαs Q s is given by sending an m-tuple of border B s -matrices (A,..., A m ) to the representation in rep αs Q s where the j-th arrow between the vertices v a and v a+ of the i-th component quiver Q i is given by the relevant block in the matrix A j. Example 8. Let us give a couple of examples for 4 4 matrices tableau L s B s (Q s, α s) 2 m m m m 2 2 m Finally, we associate to s S n a character χ s of the Levi-subgroup L s = GL(α s ) χ s the character GL(α s ) C is determined by the integral n-tuple θ s = (t,..., t n ) Z n where if entry k corresponds to the j-th vertex of the i-th component of Q s we have def t k = n ij = d.(p i + j) where d is the least common multiple of the numerators of the p i s for all i. Equivalently, the n ij are the integers appearing in the description of the one-parameter subgroup λ s = (r,..., r n ) grouped together according to the ordering of vertices in the quiver Q s. Recall that the character χ s is then defined to be n χ s (g...., g n ) = det(g i ) ti or in terms of GL(α s ) it sends an element g ij GL(α s ) to i,j det(g ij) nij. i=

294 294 CHAPTER 8. NULLCONES. Example 8.2 For the n = 4 examples above we obtain the following characters (indicated as top-labels of the vertices) tableau t t 2 t 3 t 4 (Q s, α s, θ s) m m m m m 0 Observe that θ s(α s) = 0. Using these conventions we can now state the main result of this section, giving a solution to the problem of optimal corners. Theorem 8.3 Let x = (A,..., A m ) Nulln m be of corner type C s. Then, x is of optimal corner type C s if and only if under the natural maps C s Bs repαs Q s (the first map forgets the non-border entries) x is mapped to a θ s -semistable representation in rep αs Q s. 8.3 The Hesselink stratification. We have seen that every orbit in Nulln m has a representative x = (A,..., A m ) with all A i strictly upper triangular matrices. That is, if N M n is the subspace of strictly upper triangular matrices, then the action map determines a surjection GL n N m ac Null m n Recall that the standard Borel subgroup B is the subgroup of GL n consisting of all upper triangular matrices and consider the action of B on GL n Mn m determined by b.(g, x) = (gb, b.x) Then, B-orbits in GL n N m are mapped under the action map ac to the same point in the nullcone Null m n. Consider the morphisms GL n M m n π GLn /B M m n which sends a point (g, x) to (gb, g.x). The quotient GL n /B is called a flag variety and is a projective manifold. Its points are easily seen to correspond to complete flags F : 0 F F 2... F n = C n with dim C F i = i of subspaces of C n. For example, if n = 2 then GL 2 /B P. Consider the fiber π of a point (g, (B,..., B m )) GL n /B Mn m. These are the points { g h = b B (h, (A,..., A m )) such that ba i b = g B i g for all i m.

295 8.3. THE HESSELINK STRATIFICATION. 295 Therefore, the fibers of π are precisely the B-orbits in GL n Mn m. That is, there exists a quotient variety for the B-action on GL n Mn m which is the trivial vectorbundle of rank mn 2 T = GL n /B Mn m p GLn /B over the flag variety GL n /B. We will denote with GL n B N m the image of the subvariety GL n N m of GL n Mn m under this quotient map. That is, we have a commuting diagram GL n N m GLn Mn m GL n B N m GLn /B Mn m Hence, V = GL n B N m is a sub-bundle of rank m. n(n ) 2 of the trivial bundle T over the flag variety. Note however that V itself is not trivial as the action of GL n does not map N m to itself. Theorem 8.4 Let U be the open subvariety of m-tuples of strictly upper triangular matrices N m consisting of those tuples such that one of the component matrices has rank n. the action map ac induces a commuting diagram GL n B U GLn.U GL n B N m ac Null m n where the upper map is an isomorphism of GL n -varieties if we define the action on fiber bundles to be given by left multiplication in the first component. Therefore, there is a natural one-to-one correspondence between GL n -orbits in GL n.u and B-orbits in U. Further, ac is a desingularization of the nullcone and Nulln m is irreducible of dimension n(n ) (m + ). 2 Proof. Let A N be a strictly upper triangular matrix of rank n and g GL n such that gag N, then g B as one verifies by first bringing A into Jordannormal form J n (0). This implies that over a point x = (A,..., A m ) U the fiber of the action map GL n N m ac Null m n has dimension n(n ) 2 = dim B. Over all other points the fiber has at least dimension n(n ) 2.But then, by the dimension formula we have dim Nulln m = dim GL n + dim N m n(n ) dim B = (m + ) 2 Over GL n.u this map is an isomorphism of GL n -varieties. Irreducibility of Nulln m follows from surjectivity of ac as C[Nulln m ] C[GL n ] C[N m ] and the latter is a domain. These facts imply that the induced action map GL n B N m ac Null m n is birational and as the former is a smooth variety (being a vectorbundle over the flag manifold), this is a desingularization.

296 296 CHAPTER 8. NULLCONES. Example 8.5 Let n = 2 and m =. We have seen in chapter 3 that Null2 is a cone in 3-space» with the singular top the orbit of the zero-matrix and the open complement the orbit of In this case the flag variety is P and the fiber bundle GL 2 B N has rank one. The action map can be depicted as above and is a GL 2 -isomorphism over the complement of the fiber of the top. The foregoing theorem gives us a reduction in the complexity, both in the dimension of the acting group as in the dimension of the space acted upon, of the study of GL n -orbits in the nullcone Null m n, to B-orbits in N m. at least on the stratum GL n.u described before. The aim of the Hesselink stratification of the nullcone is to extend this reduction also to the complement. Let s S n and let C s be the vectorspace of all m-tuples in Mn m which are of corner-type C s. We have seen that there is a Zariski open subset (but, possibly empty) U s of C s consisting of m-tuples of optimal corner type C s. Observe that the action of conjugation of GL n on Mn m induces an action of the associated parabolic subgroup P s on C s. The Hesselink stratum S s associated to s is the subvariety GL n.u s where U s is the open subset of C s consisting of the optimal C s -type tuples The results of the foregoing section allow us to prove, similar to the foregoing result, the following reduction of complexity result from GL n -orbits in the Hesselink stratum S s to P s -orbits in optimal corner tuples U s. Theorem 8.6 With notations as before we have a commuting diagram GL n Ps U s S s GL n Ps C s ac S s where ac is the action map, S s is the Zariski closure of S s in Nulln m and the upper map is an isomorphism of GL n -varieties. Here, GL n /P s is the flag variety associated to the parabolic subgroup P s and is a projective manifold. The variety