Moduli of representations of quivers

Transcription

1 Mouli of representations of quivers Markus Reineke (Wuppertal) Stefano Maggiolo Geometry of mouli, Sophus Lie Conference Center (Norfjorei) June 11 th, 2012 June 15 th, 2011 Contents 1 Why mouli of quivers representations? 1 2 Representations of quivers 2 3 Geometric invariant theory for quiver representation 7 4 Slope stability 10 5 Cohomology of quiver mouli 12 6 Non-commutative algebraic geometry 14 1 Why mouli of quivers representations? First of all, quivers representations are a way to formalize problems in linear algebra, typically very classic. Often these en up with a classification of all solutions, that epens on continuous an iscrete parameters. Mouli spaces are exactly a way to materialize the continuous parameters. Even if one is intereste in completely ifferent mouli problems, for example mouli of vector bunles, quite often can fin that mouli of quivers representations are relate to these problems, for example being moels (not intene in a formal sense), or toy examples, or part of the solution, etc. Another motivation is that mouli of quivers representations are simple enough to be use as testing groun for techniques in mouli theory. Finally, in one irection of non-commutative algebraic geometry, mouli of quivers representations appear as commutative shaows of smooth, noncommutative algebraic geometry. Lecture 1 (1 hour) June 11 th, 2012 s.maggiolo@gmail.com 1

2 Mouli of representations of quivers 2 Representations of quivers 2.1 efinition. A quiver is a finite oriente graph Q; in particular we will enote with Q 0 the set of vertices, with Q 1 the set of eges an we will write α : i j for an ege. The maps s, t : Q 1 Q 0 associate to each ege the source an the target vertex. 2.2 example. We o not rule out the possibility of having loops, or multiple eges between the same source an target, or even multiple loops. We just assume that the sets Q 0 an Q 1 are finite. Intuitively, a representation of a quivers consists of fixing a vector space for every vertex of the quiver an a linear map for every ege. 2.3 efinition. Let k be a fiel an Q a quiver. A k-representation of Q consists of a finite imensional k-vector space V i at each vertex i Q 0 an a k-linear map V α : V i V j for every ege α : i j. 2.4 example. If we have the quiver then stuying the representation of it consists of stuying all maps between vector spaces up to base change in the source or the target. If the quiver is then we are stuying linear self-maps of vector spaces up to a common base change in source an target. In the example we note that we were stuying maps up to base change of the vector spaces. This is formalize in the notion of equivalent representations. 2.5 efinition. A morphism f : V W of representations of quivers is a tuple of k-linear maps f i : V i V j such that all iagrams commute, that is, for every α : i j, the iagram V i V α V j f i W i W α W j f j commutes. Two morphisms f : V W an g : W X can be compose efining (g f ) i = g i f i : V i X i. The efinition of quivers representations an of their morphisms give rise to a k-linear category rep k Q of representations of quivers. This automatically 2

3 Markus Reineke (Wuppertal) gives the notions of isomorphisms an of subobjects. More precisely, we have: two representations V an W are isomorphic if there are morphisms f i : V i V j for every α : i j such that W α = f j V α f 1 i : note how this is exactly what we sai before, since this states that W is isomorphic to V if an only if W α can be obtaine from V α by simultaneous base changes; the representation U is a subrepresentation of V if an only if U i V i for every i Q 0, an V α (U i ) U j for every α : i j Q efinition. The imension vector of Q is the vector im V := (im k V i ) i Q0 N Q example. Suppose we have the quiver then classifying all possible representations up to isomorphisms means classifying all possible matrices m n up to invertible linear combinations of rows an of columns. We en up with only three iscrete invariants: the imensions n an m of V 0 an V 1 an the rank of the matrix. 2.8 example. Suppose we have the quiver then classifying all possible representations up to isomorphisms means classifying all possible square matrices up to conjugacy of elements in GL n. If k = k, then we have Joran s canonical form that gives us continuous invariants (the eigenvalues) an iscrete invariants (the sizes of Joran s blocks). We have seen an example with only iscrete invariants an one with iscrete an continuous ones. We have a theorem that tells us how to iscern the two cases. 2.9 theorem (Gabriel). The classification problem for representations of the quiver Q epens only on iscrete invariants (that is, after fixing the imensions of the vector spaces there are only finitely many isomorphism classes of representations) if an only if the non-oriente graph of Q is a isjoint union of Dynkin iagrams of type A n (n 1), D n (n 4) or E n (6 n 8) example. Suppose you have only one vertex with two loops, that is, you want to classify two enomorphisms up to base change. This is the classic unsolve problem in linear algebra. The strategy coul be to put the first enomorphism in some normal form, an then try to ajust the secon without changing the goo shape of the first. But this turns out to be increibly ifficult, an this is hinte by the fact that the classification problems for this 3

4 Mouli of representations of quivers Lecture 2 (1 hour) June 11 th, 2012 quiver has (im V) continuous parameters. This can be shown assuming that the first can be iagonalize with pairwise ifferent eigenvalues; then permutations an iagonal matrices can act on this enomorphism without estroying the nice properties. But then one can prove that the secon enomorphisms can be put in a form with ones on the lower iagonal, so the total number of parameters is n + (n 2 (n 1)) = n The example is har also because of the high number of parameters. It turns out that this is the situation for almost all quivers example. The classification problem for the quiver of the previous example can be embee into the classification problem for almost all quivers. For example, the problem for the quiver (where the number in parenthesis is the multiplicity of the ege, that is, the number of times it appears in Q 1 ) (2) can be solve, while (3) cannot because if the two spaces have the same imension, we can change bases so that one map is the ientity; then the other two maps give precisely the same situation as before example. Let us consier the quiver... with n source vertices. If n = 1, we are in the case A 2, so the classification problem can be solve; so it is with n = 2 or n = 3, where we have A 3 or D 4 ; with four we have D 4, that can also be solve with some aitional work. But in the case with five spaces below, we can ecie to put the same vector space V as the sources an V 2 as the target; we can then set the maps to be, respectively, (i, i), (i, 0), (0, i), (i, ϕ), (i, ψ). Then the two maps ϕ an ψ gives the embeing of the quiver with one vertex an two loops example. Consier the quiver... it is of type A n, so by Gabriel s Theorem there are only iscrete invariants. Inee, the firsts are the imensions of the spaces; then of course there are the ranks of the maps, but also the ranks of all the compositions. One can 4

5 Markus Reineke (Wuppertal) prove that these are all the iscrete invariants one nees, even if proving this with rows an columns operations is not easy even in the case of n = The path algebra 2.14 theorem. The category rep k Q is equivalent to the category mo kq of moules over the algebra kq (the path algebra of Q) of finite imension efinition. Let Q be a quiver; kq as a vector space is spanne by paths in the quiver Q (in orer to have a unital algebra, there must be also the paths of length zero starting from each vertex); the multiplication of two paths is 0 if the paths cannot be compose, whereas it is the concatenation if the first path ens where the secon starts. This makes kq an associative algebra with unit (given by the sum of all length zero paths). This hints to another reason to consier only finite graph: otherwise we woul have formal problems efining the unit of kq example. If Q is a vertex with a loop, then kq = k[t]. If we have two loops we o not get polynomials in two variables but the free algebra in two generators, k X, Y. If Q is... then kq is the algebra of lower triangular matrices. How oes one associate a left moule over kq to a representation of Q? As a k-vector space, the moule M is the irect sum of all vector spaces involve, that is, M := i Q 0 V i. Since the path algebra is generate by the path of length 0 or 1, we can efine the action of kq on M just specifying the actions of these paths. A path of length zero starting from the i-th component acts by selecting the i-th component, that is (v 1,..., v k ) (0,..., 0, v i, 0,..., v i ) ; instea, α : i j acts on M mapping the component V i to V j via V α an putting 0 everywhere else, that is (v 1,..., v i,..., v j,..., v k ) (0,..., 0,..., 0, α(v i ), 0,..., 0). This efines the functor on objects, an one woul have to check that this efinition extens to morphisms, an that it is an equivalence. To prouce an inverse, one uses the actions of the length zero paths to split the moule in the V i components, an then uses the length one paths to extract the linear maps. 5

6 Mouli of representations of quivers 2.17 example. As sai before, the path algebra of is the algebra of lower triangular matrixes, 2 2, where the element in position (i, j) correspons to the path from i to j; if we have a representation f : V W, the corresponing moule M is V W, an the action of ( ) a 0 b c on ( v w ) is just the multiplication of matrices, provie that every time we use the (2, 1) element we apply f to v: ( ) a 0 b c ( ) ( v := w a v b f (v) + c w The equivalence between rep k Q an mo kq is very important, because it tells us that all general results for moules over rings hols also for representations of quivers. For example, the Joran-Höler Theorem, translates to the fact that all representations V of Q amit a filtration V = V 0 V 1 V n = 0 such that V i 1 /V i is simple, that is, it oes not have subrepresentations. Another example is the Krull-Schmit Theorem, that states that V can be written as n i=1 U i, where U i are inecomposable representations, that is, cannot be written as X Y for X, Y = 0. Moreover, V is inecomposable if an only if En(V) is a local ring. Also, all the machinery of homological algebra can be use, because it is available for the moule category that has enough projective efinition. For every i Q 0, let P i be a representation of Q (possibly infinite imensional), efine letting (P i ) j be the vector space generate by paths from i to j (note that the vector space (P i ) j has infinite imension when there are cycles between i an j); efine also, for α : j k, (P i ) α composing a path from i to j with the arrow from j to k. Since kq, can be viewe as a moule over kq, it is also a representation. One can prove that it ecompose as the sum of all P i, an this proves that the P i are projective. In particular, we obtain Hom(P i, V) = V i. There is a stanar projective resolution of a representation V. The first step is i Q 0 P i k V i V, where the tensor prouct means taking as many copies of P i as the imension of V i. The sum is a finitely generate sum of projective, so it is projective. The surprising part is that the kernel of this map is automatically projective an can be escribe as α : i j P j V i. Summing up, the stanar projective resolution of V is 0 α : i j P j V i ). i Q 0 P i k V i V 0. As a corollary, the Ext groups of inex at least 2 are trivial in rep k Q (in a similar way to what happens in the category of vector bunles on a smooth 6

7 Markus Reineke (Wuppertal) projective variety). Another corollary is that χ(m, N) = im Hom(M, N) im Ext 1 (M, N) = im M, im N, where, e := i Q0 i e i i j i e j is a non-symmetric bilinear form, calle the Euler form because it gives the homological Euler characteristic (the alternating sum of Ext groups). 3 Geometric invariant theory for quiver representation From now on we will work over an algebraically close fiel k of characteristic 0. We also fix a imension vector N Q 0 (since there is just one vector space of imension i, we can think we have fixe vector spaces V i for each i Q 0 ). We consier the affine k-space R (Q) := α : i j Hom k (V i, V k ). This is a space of parameters for all quiver representations of Q. The group G := i Q0 GL(V i ) acts on R (Q) via (g i ) i (V α ) α := (g j V α g 1 i ) α : i j. This is a linear action of a reuctive algebraic group on an affine space. Recall that we saw that if V an W were isomorphic representations, then we coul write W exactly as g V for some g G. Hence, the orbits of R (Q) relative to G correspon to the isomorphism classes of representations of Q with im =. If we have a representation V, we write [V] for its isomorphism class an O V for the orbit of V by G. An orbit is always locally close, so we can compute its coimension: Lecture 3 (1 hour) June 12 th, 2012 coim R (Q) O(V) = im R (Q) im O V = = im R (Q) im G + im Stab G (V) = = im R (Q) im G + im Aut Q (V) = = i j 2 i + im Aut Q(V) = α : i j i Q 0 = im Aut Q (V),. This is because an element g stabilizing V is an automorphism of V in the category of quivers representations. Note how we starte from a geometric invariant (the coimension) an we coul compute it using only some categorical objects: the automorphisms an the Euler form. We also note that coim R (Q) O V 1,. We can use this fact to prove half of Gabriel s Theorem. Assume that for all imension vectors, there exist only finitely many isomorphism classes of representations of imension vector. So, for every, we have finitely many orbits in the space of orbits. But orbits are locally close, therefore they form a stratification of the space of orbits, an this imply that there must be a ense orbit. That is, for every, there exists a V R (Q) such that O V R (Q) is ense, hence of coimension 0. Thus, for every,, 1 by the computation of the coimension of an orbit. The quaratic form q() :=, is then positive efinite, an by the theory of quaratic forms, Q must be a isjoint union of Dynkin iagrams. Of course, from the point of view of mouli of representations, the quivers that are union of Dynkin iagram are the most boring, because the associate 7

8 Mouli of representations of quivers mouli space is a union of points. In all other cases, the mouli space must be the orbit space of R (Q) with respect to G. The only problem is that, to be sure that such a quotient exists as a geometric object, we nee to use geometric invariant theory. 3.1 Summary of GIT We will assume the setting of our mouli problem, so we will consier a linear action of a reuctive group G on a vector space X. The basic iea is to fin a variety whose functions are the invariant functions k[x] G. Hilbert prove that this ring is finitely generate, so it qualifies to be the coorinate ring of some variety X /G := Spec k[x] G. Since k[x] G k[x], taking the ual we have a morphism π : X X /G. It satisfies several properties that are natural for quotients: universal property: for every morphism ϕ : X Y that is G-invariant, there exists exactly one way to split this morphism through π; surjectivity; each fiber of π contains a unique close G-orbit (if one orbit is in the closure of another, we cannot expect that a morphism is able to istinguish between them); close k-points of X /G correspons to close k-orbits of X. This was a summary of pre-mumfor GIT. Now, what we woul like is for π to actually separate all orbits. As sai before, the only way to o this is to restrict our space X to X st, the stable locus, that is, the locus of points x X such that Stab G (x) is finite. If we restrict to X st, then the map π actually has one orbit per fiber. Lecture 4 (1 hour) June 13 th, example. Suppose that G m acts on A 2 as t (x, y) := (tx, ty). The orbits are all the rays (not close) an the origin (close), so A 2 /G m = {0}. Instea, the aim is to have A 2 /G m = P 1. Recall that to o Mumfor-style GIT we nee also a line bunle on the space X with a G-linearization. This blows own to the choice of a character χ : G G m, an we can efine the set of χ-semi-invariant functions as k[x] G,χ := { f : X k g G, x X, f (gx) = χ(g) f (x)}. The problem is that if we multiply two χ-semi-invariant functions we get a χ 2 -semi-invariant function, an so k[x] G,χ is not a ring. What we can o is to take all χ n -semi-invariant functions, for every n, an construct a grae ring k[x] G χ := n 0 k[x] G,χn. Our hope is that the Proj of this grae ring is a sensible quotient. We nee some other efinition: X χ sst is the locus of points x X such that f (x) = 0 for some f k[x] G χ of positive egree; insie Xχ sst we efine X χ st as the locus of points with the aitional conition of having a finite stabilizer. Both the semi-stable locus an the stable locus are open. 8

9 Markus Reineke (Wuppertal) From now on, we will consier not X but X χ sst, so if we say close orbit it will mean close in X χ sst. The situation is the following open X χ st X χ sst X π open X χ st /G X χ sst /G X /G π open projective π Proj k[x] G χ Proj k[x] G χ Spec k[x] G where the morphism X χ sst /G X /G is projective because the latter s ring is the egree 0 of the former s ring. If the action of G on the stable locus is not only with finite stabilizers but also free, then X χ st /G is smooth an the map π over it is a G-principal bunle. 3.2 example. In the case of A 2 with the action of G m by ilations, we can choose the trivial character χ = i, an we obtain k[a 2 ] G m χ = k[x, Y], where X an Y are χ-semi-invariant functions of egree 1. It s Proj is of course P 1. We can check also that the semi-stable locus is A 2 \ {0}. 3.3 example. Look at the actions of G m on A 2 by t (x, y) := (tx, ty 1 ), where the generic orbit is an hyperbola, an we have three special orbit: the axes an the origin. Choosing the ientity character, one sees that the stable locus consists of all the hyperbolas an the puncture x axis; choosing instea the character t t 1, the semi-stable locus comprises all hyperbolas an the puncture y-axis. This shows that the semi-stable locus is not intrinsic: we have to make a choice, an this choice is summe up in the choice of the character. 3.4 example. We can efine grassmannians using GIT by acting with GL k on M n k (with k n) an using the eterminant as the character. The quotient is the grassmannian G(k, n). 3.2 Application to the action of G on R (Q) Recall that we associate to the class of a representation [V] the orbit O V. The problem is that if we apply GIT irectly, we obtain stabilizers of positive imension, hence the stable locus woul be empty by efinition. But there is a one imensional family of elements of G acting trivially on R (Q): the elements that rescale the bases of all vector spaces by the same amount. Therefore we can consier PG as G moulo the scalars. Since Stab G (V) = Aut(V), we have that Stab PG (V) = Aut(V)/G m. Now, the only ecent characters χ : PG G m are the ones associating to the matrices g i the prouct of their eterminants to some power ϑ i : χ((g i ) i ) = i et(g i ) ϑ i. But, to make this well efine for PG, there is the aitional conition that i ϑ i = 0. We will write χ ϑ for the character χ. 9

10 Mouli of representations of quivers Applying the GIT theory we get open projective M ϑ st (Q) M ϑ sst (Q) M ssimp (Q) R (Q) χ ϑ st /PG R (Q) χ ϑ sst /PG R (Q) /PG As before, the last term is affine, an the first is smooth because the stabilizer is Aut(V)/G m, that is a connecte, zero-imensional group insie En(V)/G m : stabilizers are trivial. We use the inex semi-simple for the affine quotient. This is because one coul prove that the close orbits correspon to representations V that can be ecompose in a sum of irreucible representations, that is, V is semi-simple. That is, M ssimp (Q) parametrizes semi-simple representation of Q (with im = ). If Q has no oriente cycles, the only irreucible representations are the S i. 1 This implies that the only close orbit in R (Q) is {0} an M ssimp (Q) = {pt}: it is a situation similar to the one of A 2, where we ha only one close orbit (the origin), that maske all other interesting non-close orbit. In the case Q has oriente cycles, a theorem of Le Bruyn-Procesi states that k[r (Q)] G is generate by the traces of the compositions of the maps along the oriente cycles. More precisely, by functions of the form V Tr(V α1 V αn V α2 V α1 ) for a cycle α 1 α n α 1. The problem is that the theorem is non-constructive, hence we on t know how many of such trace functions we nee to take to generate the whole ring (an which are the relations amongst the generators). 3.5 example. Consier the quiver Q with one vertex an two loops, then M ssimp (Q) = A 5 when = 2, an the generators of the ring are the traces of A, B, AB, A 2, B 2. If = 3, we have 10 traces with one relation. If we have = 3 with three loops, one nees to take 48 traces an there are 365 relations. Lecture 5 (1 hour) June 14 th, Slope stability We have an interpretation for the quotient M ssimp (Q) (that is, it parametrizes semi-simple representations), but we still have to iscover interpretations for the two spaces M ϑ sst (Q) an M ϑ st (Q). There is a nice interpretations analogous to the one foun in mouli spaces of vector spaces, using slope stability. We efine the rank of a imension vector as the sum of all imensions of the vector spaces, an we choose an arbitrary egree function eg: Z Q 0 Q, which is simply a linear function. In the setting of vector bunle on curves we have a natural egree function, here we o not. Given a egree function, we can efine the slope function as µ : N Q 0 \ {0} Q 1 Maybe the P i? 10

11 Markus Reineke (Wuppertal) by µ() = eg / rk. We have to exclue 0 because its rank is zero. 4.1 efinition. A quiver representation V is calle semi-stable if µ(im U) µ(im V) for every U V nonzero. It is calle stable if the same hols with a strict inequality. V is calle poly-stable if V is a irect sum of stable representations with the same slope. Insie rep k Q one can fin the subcategory of semi-stable representations of slope µ, an it is an abelian subcategory, meaning that, for example, kernels an cokernels are insie the subcategory. Moreover, insie it, the stable objects are precisely the irreucible one with respect to the subcategory. Given a imension vector, we can efine ϑ i := eg() eg(e i ) rk() (where e i is the imension vector (0,..., 0, 1, 0,..., 0)). Using this ϑ, we have ϑ i i = 0 as requeste to let the action escen to PG. This is a natural choice for any given egree function eg an imension vector. 4.2 theorem (King). A representation V is in R χ ϑ sst (Q) if an only if V is semistable; V R χ ϑ st (Q) if an only if V is stable; O V R χ ϑ sst (Q) is close if an only if V is poly-stable. Note that on the right sies we use the notion of stability, that is relative to a egree function. On the left, this choice is hien in the ϑ, that is constructe to aapt to the chosen egree function. Therefore, from now on we rop ϑ from the notation. 4.3 corollary. The space M sst (Q) parametrizes isomorphism classes of poly-stable representations with im = ; the (smooth) space M st (Q) parametrizes isomorphism classes of stable representations with im =. We have no irect way to say that a space M st (Q) is nonempty: there are some criterions, but they are highly recursive criterions, hence not very effective. Let us assume to have M st (Q) nonempty, then we coul ask what is its imension. Of course, im M st(q) = im R (Q) im PG, that we alreay seen being equal to 1,. A necessary conition for the mouli space to be nonempty, is then that, 1. This in particular shows that for Dynkin quivers, being the Euler form positive efinite, the imension is 0 (or the mouli space is empty). What happens when we change the egree function? For example, let Q be the quiver (m) an let the imension vector be := ( 1, 2 ). The egree is a linear function, hence of the form eg() := a a 2 2 with a i Z. If we plot the coefficients a i in a Z 2, then we can ientify a locus, (a 1 = a 2 ) on which R χ ϑ sst (Q) = {0}, hence M sst with (a 2 = {pt}. If we choose a egree function above the iagonal, namely > a 1 ), then R χ ϑ sst (Q) =, except for 1 2 = 0, for which the 11

12 Mouli of representations of quivers mouli space is anyway a single point. In the other irections instea we have more interesting results. Here a representation with maps f i is (semi-)stable if an only if for every nonzero U V we have im i=1 m f i(u) im U im im W V. This happens when the linear maps f i are in some sense orthogonal to each other, in particular it is a genericity conition that is open. This inee is the so calle Kronecker mouli space, stuie by Drezet. But even in this case it is not easy to ecie whether the stable locus is nonempty. Note that for this quiver, a subrepresentation is a choice of vector spaces U V V an U W W such that f i UV U W but U W must contain also the sum of all images: U W f i (U V ). 4.4 example. We look at mouli spaces of small imension. For example, we choose the quiver with the imension vector specifie in the iagram. A representation of this is a configuration of 4 vectors in a 2-imensional vector space, or passing to the projective spaces, 4 points in P 1. The mouli space is inee parametrize by the cross-ratio: for example, with eg(s 1,..., s 4, t) = e, M sst(q) = P 1, an M st(q) = P1 \ {0, 1, }. 4.5 example. In the case of five maps with a common target, the stable or semi-stable mouli space is P 2 blown up in four points. If the quiver an imension vector is then the mouli spaces (both stable an semi-stable) are P 2 blown up in three points. 4.6 example. The quiver an imension vector 1 (m) k has mouli spaces G(m, k); 2 (3) 3 amits a G m action with 13 fixe points, an for both the stable an semi-stable mouli spaces coincie. 4.7 conjecture. If M st (Q) = Msst (Q), then it amits an affine pairing. 5 Cohomology of quiver mouli Lecture 6 (1 hour) June 14 th, 2012 We will assume that Q has no oriente cycles, an that is eg-coprime, that is, for every nonzero e <, µ(e) = µ() (in other wors, gc( i ) = 1 an 12

13 Markus Reineke (Wuppertal) egree is generic, avoiing finitely many hyperplanes). For example, in the case (m), the only reasonable egree function is eg( 1, 2 ) = 1, an we also require that gc( 1, 2 ) = 1. These requirements implies that M (Q) := M st (Q) = Msst (Q), because the equality in the semi-stability conition cannot be reache uner these conitions. In particular, M (Q) is smooth because the stable locus is smooth, an is projective because the semi-stable locus is projective. 5.1 Harer-Narasimhan stratification of the unstable locus 5.1 lemma (Harer-Narasimhan). For all representations V, there exists a unique filtration 0 = V 0 V 1 V s = V such that all quotients V i /V i 1 are semi-stable an the slopes are ecreasing, that is, µ(v 1 /V 0 ) > µ(v 2 /V 1 ) > > µ(v s /V s 1 ). The surprising fact about this lemma is that the filtration respecting these properties is unique. Moreover, with some care it can be one in such a way that it also respect functoriality (that is, in such a way that it is preserve by morphisms). Thanks to the uniqueness we can efine the following. 5.2 efinition. The Harer-Narasimhan type of a representation V is the list (im V 1 /V 0,..., im V s /V s 1 ) (N Q 0) s. Obviously, if ( 1,..., s ) is a HN type, then i = (this implies that after fixing the imension vector, there are only finitely many HN types for that imension vector). Also, by efinition, µ( 1 ) > > µ( s ). Moreover, if V is semistable, then the HN filtration is 0 V. The HN type is therefore (), an this conition is equivalent to semi-stability. Because of the finiteness conition, we can stratify the space R (Q) by the HN type: R (Q) = =( 1,..., s ), i = S = =( 1,..., s ), i = {V R (Q) HN type of V is }. It is a finite stratification, an S () correspons to the χ ϑ -semi-stable locus, that is open in R (Q). Moreover, since R (Q) is a vector space, we know that it is the only open stratum. It woul be great if the stratum S ( 1,..., s ) was relate to s k=1 Rχ ϑ sst (Q), k that is, the bounary strata were constructe starting from mouli of smaller representations. What happens is slightly more complicate, that is we have S ( 1,..., s ) = G P B, where B is a vector bunle over s k=1 Rχ ϑ sst (Q). The group P k is a parabolic subgroup in G with Levi factor s k=1 G k (that is, we are taking upper triangular block matrices with iagonal blocks equal to G k, insie G that is a prouct of GL groups). 13

14 Mouli of representations of quivers Note that we are not stratifying the mouli space irectly, we are stratifying the big parameter space R (Q). This oes not give irectly ways to ecompose the mouli space in a useful way to compute invariants. Consier K 0 (Var /k) that is the set of all isomorphism classes of quasiprojective varieties X up to relations of the form [X] = [A] + [U] whenever A X is close an U = X \ A. We put an operation on this set, that is [X] [Y] := [X Y], that makes it into a ring. We can efine operators on this ring together with the formal sum. To keep things simple, we consier k = C; then for example, we can efine χ : K 0 (Var /C) Q sening [X] to χ c (X), the Euler characteristic with compact support. This is known to respect the relation for which we quotiente, so it is well efine. Another operator we can efine on K 0 (Var /C) is the virtual Poincaré polynomial, that is a ring homomorphism K 0 (Var /C) Q[q] sening [X] with X smooth an projective to i 0 h i (X, Q)q i. It can be prove that there is only one ring homomorphism with this behaiour for smooth projective varieties. Another possibility is to count points over finite fiels, that respect the relation an is multiplicative with respect to the cartesian prouct. What is the class of R (Q) in K 0 (Var /k)? Since it is an affine space, it is just [L] im R (Q), the class of a line with the appropriate exponent. But we have the HN filtration on R (Q), so [R (Q)] = [R χ ϑ sst (Q)] + [S ], proper HN type an we can express the classes in the sum using the fornula saw before: [R (Q)] = [R χ ϑ sst (Q)] + proper HN type In the en we get the motivic HN recursion: [R sst (Q)] = [R (Q)] [G ] [G ] [L] im B s k=1 [G ] [ s k=1 Rχ ϑ sst (Q)] [L] im B k. [P ] [R sst(q) ]. [G k] This is a completely formal relation that we can use to compute the motive of the mouli space M (Q), in terms of things that are either motives of smaller imensional mouli of representations, or known such as affine spaces. If is eg-coprime, then, for example, [R sst (Q)]/[G ] an [M (Q)]/[G m ] have the same virtual Poincaré polynomial. 6 Non-commutative algebraic geometry Lecture 7 (1 hour) June 15 th, 2012 There are two approaches when you ecie to o non-commutative algebraic geometry: the small scale approach is when you actually take noncommutative algebras, but such that they are of finite rank over their center (so that you can hope that many properties extens); the large scale approach is when you instea take very non-commutative algebras, for example the 14

15 Markus Reineke (Wuppertal) free algebra A := k x 1,..., x n. Here we will follow the secon approach. 6.1 example. We efine A m := Specm k[x 1,..., x m ], where Specm is, settheoretically, the set of maximal, two-sie ieals. The non-commutative analogous is NA m := Specm k x 1,..., x m. Insie it, we can fin the cofinite part, that is Specm cofin k x 1,..., x m = {m im k k x 1,..., x m /m < }. of course in the commutative case this is equivalent to Specm. We attach to these spaces a topology that is constructe in the same way as the Zariski topology. The nice thing is that we can construct the cofinite part using quivers. The following is a reinterpretation of a theorem initially state in a ifferent language. 6.2 theorem (Artin, 1969). We have Specm cofin k x 1,..., x m = 1 M simp (m loops). In the theorem, M simp (Q) is the open part of M ssimp (Q) corresponing to R st (Q). For example, in the case of a single vertex with m loops, we have that the simple representations are the ones where there is not a common invariant subspace for all the linear maps. In particular, for m = 1 there are no simple representations. The corresponence in the theorem sens the class of V = (ϕ 1,..., ϕ m ) to its annihilator, Ann(V) = {P k x 1,..., x m P(ϕ 1,..., ϕ m ) = 0}. We can now reformulate the theorem of Le Bruyn-Procesi an see the generators insie the non-commutative ring: we can see Specm cofin k x 1,..., x m insie Tr C x 1,..., x m, the set of trace-like functions, that is of linear functions t : C x 1,..., x m C such that t(pq) = t(qp). This embeing is realize sening the representation V to the map P Tr P(ϕ 1,..., ϕ m ). Differently from the usual algebraic geometry where the affine space is one of the simplest object, in the non-commutative case the affine space is alreay very wil. For example, a notorious har problem in quiver representations is the rationality of M ssimp (Q): this problem can be reuce to the case where Q consists of one vertex with m loops, an inee it is not even known if the non-commutative affine space is rational. 6.3 example. The scheme Hilb (A m ) is the space parametrizing zero imensional subschemes of A m of length. These are in turn in a corresponence with ieals I k[x 1,..., x m ] such that im k k[x 1,..., x m ]/I =. In the noncommutative case, the analogous is N Hilb (m), the space of left ieals I in k x 1,..., x m such that im k k x 1,..., x m /I =. This quotient is no longer a ring but only a vector space (because we aske for left ieals), but we actually just care about the imensions. In reality, the left ieal are choosen because they are easier to figure out with respect to the more natural efinition with two-sie ieals. 15

16 Mouli of representations of quivers We efine a quiver Q with a map between two vector spaces of imension 1 an, with the target having also m loops. The imension vector is = (1, ) an we choose the egree to be eg(a, b) = a. Choosing a representation is the same as choosing a vector v in the secon space plus m linear maps. It is (semi-)stable if an only if v is a cyclic vector with respect to the other maps, that is, applying all possible polynomials of ϕ 1,..., ϕ m to v, we can arrive to any vector. 6.4 lemma. The mouli space M st (Q) is isomorphic to N Hilb(m). We associate to a point [(v, ϕ 1,..., ϕ m )] the annihilator {P P(ϕ 1,..., ϕ m )(v) = 0}. In the other irection, we associate to a left ieal I the point [(1 A/I, x 1,..., x m )], where x i are the generators of the ieal. The avantage in the first case was that we got natural coorinates in M simp (Q) using the embeing in the trace-like function using the Le Bruyn- Procesi Theorem. In this case, the avantage is that we can apply HN recursion to get a recursive formula for the Betti numbers of N Hilb (m). We can construct a two variables generating function putting together all Betti numbers of N Hilb (m) : F(q, t) = q (m 1)( 2 ) im H i (N Hilb (m), Q)q i/2 t. 0 i The twist q i/2 comes from the fact that these spaces have no o cohomology. 6.5 theorem. The function F(q, t) is etermine by the following algebraic q-ifferent functional equation: F(t, q) = 1 1 F(q, qt)f(q, q 2 t) F(q, q m 1 t). Using this equation one can prove things like a formula for the Euler characteristic: ( ) e(n Hilb (m) 1 m ) =. (m 1) + 1 Let us consier n 0 e(hilb n (A i ))t n ; when i {2, 3}, we can rewrite this as i 1(1 t i ) j where j = 1 for i = 2 an j = i for i = 3. We can o the same for N Hilb (m), where F(1, t) = i 1 (1 t i i DT(m) ) i, where DT (m) i is the Donalson-Thomas invariant of the Hilbert scheme, for which it exists a formula using the formula for the Euler characteristic. 16