Z-graded noncommutative projective geometry Algebra Seminar

Transcription

1 Z-graded noncommutative projective geometry Algebra Seminar Robert Won University of California, San Diego November 9, / 43

2 Overview 1 Preliminaries Pre-talk catchup Noncommutative things 2 Noncommutative projective schemes 3 Z-graded rings Prior work Generalized Weyl algebras 4 Future direction November 9, / 43

3 What you missed in the pre-talk Throughout, k = k, char(k) = 0 Γ an abelian group A Γ-graded k-algebra A has k-space decomposition A = γ Γ A γ such that A γ A δ A γ+δ A graded right A-module M: M = γ Γ M γ such that M γ A δ M γ+δ Preliminaries November 9, / 43

4 What you missed in the pre-talk A graded by N (or Z), the graded module category gr-a: Objects: finitely generated graded right A-modules Hom sets (degree 0) graded module homomorphisms: hom gr-a (M, N) = {f Hom mod-a (M, N) f (M i ) N i } The shift functor: S i : gr-a gr-a M M i M M 3 M 2 M 1 M 0 M 1 M 1 M 3 M 2 M 1 M 0 M 1 M 2 M 3 M 2 M 1 M 0 M 1 Preliminaries November 9, / 43

5 The Picard group For a category of graded modules (e.g. gr-a) we define the Picard group, Pic(gr-A) to be the group of autoequivalences of gr-a modulo natural isomorphism. (If R is a commutative k-algebra, Pic(mod-R) = Pic(R), the isomorphism classes of invertible unitary R-bimodules) For a noncommutative ring, A, Pic(gr-A) can be nonabelian. Preliminaries November 9, / 43

6 Morita equivalence Two rings R and S are called Morita equivalent if mod-r is equivalent to mod-s (if and only if R-mod equivalent to S-mod). For R and S commutative, then R is Morita equivalent to S if and only if R = S. Example R a ring, then M n (R) (the ring of n n matrices) is Morita equivalent to R. R and S are graded Morita equivalent if gr-r is equivalent to gr-s. Preliminaries November 9, / 43

7 Noncommutative is not commutative Commutative graded ring: R Proj R Noncommutative ring: not enough (prime) ideals The Weyl algebra k x, y /(xy yx 1) is a noncommutative analogue of k[x, y] but is simple. The quantum polynomial ring The N-graded ring k x, y /(xy qyx) is a noncommutative P 1 but for q n 1 has only four homogeneous prime ideals (namely (0), (x), (y), and (x, y)). Preliminaries November 9, / 43

8 Sheaves to the rescue The Beatles (paraphrased): All you need is sheaves. Idea: You can reconstruct the space from the sheaves. Theorem (Rosenberg, Gabriel, Gabber, Brandenburg) Let X, Y be quasi-separated schemes. If qcoh(x) qcoh(y) then X and Y are isomorphic. Preliminaries November 9, / 43

9 Sheaves to the rescue All you need is modules Theorem Let X = Proj R for a commutative, f.g. k-algebra R generated in degree 1. (1) Every coherent sheaf on X is isomorphic to M for some f.g. graded R-module M. (2) M = Ñ as sheaves if and only if there is an isomorphism M n = N n. Let gr-r be the category of f.g. R-modules. Take the quotient category qgr-r = gr-r/fdim-r The above says qgr-r coh(proj R). Preliminaries November 9, / 43

10 Noncommutative projective schemes A (not necessarily commutative) connected graded k-algebra A is A = k A 1 A 2 such that dim k A i < and A is a f.g. k-algebra. Definition (Artin-Zhang, 1994) The noncommutative projective scheme Proj NC A is the triple (qgr-a, A, S) where A is the distinguished object and S is the shift functor. Idea: Use geometry to study Proj NC A = qgr-a to study A. Noncommutative projective schemes November 9, / 43

11 All you need is modules Can attempt to do any geometry that only relies on the module category. If X is a commutative projective k-scheme, for each x X, there is the skyscraper sheaf k(x) coh(x). So the simple objects of coh(x) correspond to points of X. A point module is a graded right module M such that M is cyclic, generated in degree 0, and has dim k M n = 1 for all n. Fact: If A is f.g. connected graded noetherian k-algebra generated in degree 1, then the point modules are simple objects of qgr-a. Points of Proj NC A = simple modules. Noncommutative projective schemes November 9, / 43

12 Twisted homogeneous coordinate rings We can go from rings to schemes via Proj. R Proj R Can we go back? (Certainly not uniquely: Proj R (d) = Proj R.) X a projective scheme, L a line bundle on X The homogeneous coordinate ring is B(X, L) = k H 0 (X, L n ). n 1 Theorem (Serre) Assume L is ample. Then (1) B = B(X, L) is a f.g. graded noetherian k-algebra. (2) X = Proj B so qgr-b coh(x). Noncommutative projective schemes November 9, / 43

13 Twisted homogeneous coordinate rings X a (commutative) projective scheme, L a line bundle, and σ Aut(X). Define L σ = σ L and L n = L L σ L σn 1. The twisted homogeneous coordinate ring is B(X, L, σ) = k n 1 H 0 (X, L n ). B(X, L, σ) is not necessarily commutative Theorem (Artin-Van den Bergh, 1990) Assume L is σ-ample. Then (1) B = B(X, L, σ) is a f.g. graded noetherian k-algebra. (2) qgr-b coh(x). Noncommutative projective schemes November 9, / 43

14 Noncommutative curves The Hartshorne approach A a k-algebra, V A a k-subspace generating A spanned by {1, a 1,..., a m }. V 0 = k, V n spanned by monomials of length n in the a i. The Gelfand-Kirillov dimension of A. GKdim k[x 1,..., x m ] = m. GKdim A = lim sup log n (dim k V n ) n So noncommutative projective curves should have GKdim 2. Noncommutative projective schemes November 9, / 43

15 Noncommutative curves Theorem (Artin-Stafford, 1995) Let A be a f.g. connected graded domain generated in degree 1 with GKdim(A) = 2. Then there exists a projective curve X, an automorphism σ and invertible sheaf L such that up to a f.d vector space A = B(X, L, σ) As a corollary, qgr-a coh(x). Or noncommutative projective curves are commutative. Noncommutative projective schemes November 9, / 43

16 Noncommutative surfaces The right definition of a noncommutative polynomial ring? Definition A a f.g. connected graded k-algebra is Artin-Schelter regular if (1) gldim A = d < (2) GKdim A < and (3) Ext i A(k, A) = δ i,d k. Behaves homologically like a commutative polynomial ring. k[x 1,..., x m ] is AS-regular of dimension m. Noncommutative P 2 s should be qgr-a for A AS-regular of dimension 3. Noncommutative projective schemes November 9, / 43

17 Noncommutative surfaces Theorem (Artin-Tate-Van den Bergh, 1990) Let A be an AS-regular ring of dimension 3 generated in degree 1. Either (a) A = B(X, L, σ) for X = P 2 or X = P 1 P 1 or (b) A B(E, L, σ) for an elliptic curve E. Or noncommutative P 2 s are either commutative or contain a commutative curve. Other noncommutative surfaces (noetherian connected graded domains of GKdim 3)? Noncommutative P 3 (AS-regular of dimension 4)? Noncommutative projective schemes November 9, / 43

18 Z-graded rings Throughout, graded really meant N-graded Way back to our first example: A 1 = k x, y /(xy yx 1) Ring of differential operators on k[t] x t y d/dt A is Z-graded by deg x = 1, deg y = 1 Simple noetherian domain of GK dim 2 Exists an outer automorphism ω, reversing the grading ω(x) = y ω(y) = x Z-graded rings November 9, / 43

19 Sierra (2009) Sue Sierra, Rings graded equivalent to the Weyl algebra Classified all rings graded Morita equivalent to A 1 Examined the graded module category gr-a 1 : For each λ k \ Z, one simple module M λ For each n Z, two simple modules, X n and Y n X Y For each n, exists a nonsplit extension of X n by Y n and a nonsplit extension of Y n by X n Z-graded rings November 9, / 43

20 Sierra (2009) Computed Pic(gr-A 1 ) Shift functor, S: Autoequivalence ω: Z-graded rings November 9, / 43

21 Sierra (2009) Theorem (Sierra) There exist ι n, autoequivalences of gr-a 1, permuting X n and Y n and fixing all other simple modules. Also ι i ι j = ι j ι i and ι 2 n = Id gr-a Theorem (Sierra) Pic(gr-A 1 ) = (Z/2Z) (Z) D = Zfin D. Z-graded rings November 9, / 43

22 Smith (2011) Paul Smith, A quotient stack related to the Weyl algebra Proves that Gr-A 1 Qcohχ χ is a quotient stack whose coarse moduli space is the affine line Spec k[z], and whose stacky structure consists of stacky points BZ 2 supported at each integer point Gr-A 1 Gr(C, Z fin ) Qcohχ Z-graded rings November 9, / 43

23 Smith (2011) Z fin the group of finite subsets of Z, operation XOR Constructs a Z fin graded ring C := hom(a 1, ι J A 1 ) = J Z fin = k[z z n] where deg x n = {n} k[x n n Z] (x 2 n + n = x 2 m + m m, n Z) C is commutative, integrally closed, non-noetherian PID Theorem (Smith) There is an equivalence of categories Gr-A 1 Gr(C, Z fin ). Z-graded rings November 9, / 43

24 Z-graded rings Theorem (Artin-Stafford, 1995) Let A be a f.g. connected N-graded domain generated in degree 1 with GKdim(A) = 2. Then there exists a projective curve X such that qgr-a coh(x). Theorem (Smith, 2011) A 1 is a f.g. Z-graded domain with GKdim(A 1 ) = 2. There exists a commutative ring C and quotient stack χ such that Gr-A 1 Gr(C, Z fin ) Qcoh(χ). Z-graded rings November 9, / 43

25 Generalized Weyl algebras (GWAs) Introduced by V. Bavula Generalized Weyl algebras and their representations (1993) D a ring; σ Aut(D); a Z(D) The generalized Weyl algebra D(σ, a) with base ring D D(σ, a) = D x, y ( ) xy = a yx = σ(a) dx = xσ(d), d D dy = yσ 1 (d), d D Theorem (Bell-Rogalski, 2015) Every simple Z-graded domain of GKdim 2 is graded Morita equivalent to a GWA. Z-graded rings November 9, / 43

26 The main object D = k[z]; σ(z) = z 1; a = f (z) A(f ) = k x, y, z ( ) xy = f (z) yx = f (z 1) xz = (z + 1)x yz = (z 1)y Two roots α, β of f (z) are congruent if α β Z Example (The first Weyl algebra) Take f (z) = z D(σ, a) = k[z] x, y ( ) k x, y = xy = z yx = z 1 (xy yx 1) = A 1. zx = x(z 1) zy = y(z + 1) Z-graded rings November 9, / 43

27 The main object Properties of A(f ): Noetherian domain Krull dimension 1 Simple if and only if no congruent roots 1, f has neither multiple nor congruent roots gl.dim. A(f ) = 2, f has congruent roots but no multiple roots, f has a multiple root We can give A(f ) a Z grading by deg x = 1, deg y = 1, deg z = 0 Z-graded rings November 9, / 43

28 Questions and strategy For these generalized Weyl algebras A(f ): What does gr-a(f ) look like? What is Pic(gr-A(f ))? Can we construct a commutative Γ-graded ring C such that gr(c, Γ) gr-a(f )? Strategy: First quadratic f Distinct, non-congruent roots Congruent roots Double root Z-graded rings November 9, / 43

29 Distinct, non-congruent roots Let f (z) = z(z + α) for some α k \ Z. A = A(f ) = ( xy = z(z + α) k x, y, z ) yx = (z 1)(z + α 1) xz = (z + 1)x yz = (z 1)y We still have A is simple, K.dim(A) = gl.dim(a) = 1 Still exists an outer automorphism ω reversing the grading ω(x) = y ω(y) = x ω(z) = 1 + α z Z-graded rings November 9, / 43

30 Distinct, non-congruent roots Graded simple modules: α 2 α 1 α α One simple graded module M λ for each λ k \ (Z Z + α) For each n Z two simple modules X 0 n and Y 0 n For each n Z two simple modules X α n and Y α n A nonsplit extension of X 0 n by Y 0 n and vice versa A nonsplit extension of X α n by Y α n and vice versa Z-graded rings November 9, / 43

31 Distinct, non-congruent roots Theorem (W) There exist numerically trivial autoequivalences, ι (n, ) permuting X 0 n and Y 0 n and fixing all other simple modules. Similarly, there exist ι (,n) permuting X α n and Y α n. Theorem (W) Pic(gr-A) = (Z/2Z) (Z) D Z-graded rings November 9, / 43

32 Distinct, non-congruent roots Define a Z fin Z fin graded ring C: C = k[a n, b n n Z] modulo the relations a 2 n + n = a 2 m + m and a 2 n = b 2 n + α for all m, n Z with deg a n = ({n}, ) and deg b n = (, {n}) Theorem (W) There is an equivalence of categories gr(c, Z fin Z fin ) gr-a. Z-graded rings November 9, / 43

33 Multiple root Let f (z) = z 2. A = A(f ) = k x, y, z ( xy = z 2 yx = (z 1) 2 ) xz = (z + 1)x yz = (z 1)y K.dim(A) = 1 Exists an outer automorphism ω reversing the grading ω(x) = y ω(y) = x ω(z) = 1 z Now gl.dim(a) = Z-graded rings November 9, / 43

34 Multiple root The graded simple modules For each λ k \ Z, M λ For each n Z, X n and Y n X Y Nonsplit extensions of X n by Y n and vice versa. Also self-extensions of X n by X n and Y n and Y n. Z-graded rings November 9, / 43

35 Multiple root Theorem (W) There exist numerically trivial autoequivalences, ι n permuting X n and Y n and fixing all other simple modules. Theorem (W) Pic(gr-A) = (Z/2Z) (Z) D Z-graded rings November 9, / 43

36 Multiple root Define a Z fin graded ring B: B = J Z fin hom A (A, ι J A) = k[z][a n n Z] (b 2 n = (z + n) 2 n Z). Theorem (W) B is a reduced, non-noetherian, non-domain of Kdim 1 with uncountably many prime ideals. Theorem (W) There is an equivalence of categories gr(b, Z fin ) gr-a. Z-graded rings November 9, / 43

37 Two congruent roots Let f (z) = z(z + m) for some m N. A = A(f ) k x, y, z = ( ) xy = z(z + m) yx = (z 1)(z + m 1) xz = (z + 1)x yz = (z 1)y What do we lose? A no longer simple There exist new finite-dimensional simple modules Now gl.dim(a) = 2 What remains? Still have K.dim(A) = 1 Still exists an outer automorphism ω reversing the grading ω(x) = y ω(y) = x ω(z) = 1 + m z Z-graded rings November 9, / 43

38 Two congruent roots The graded simple modules For each λ k \ Z, M λ For each n Z, X n, Y n, and Z n X Z Y Z n is finite dimensional proj. dim Z n = 2 Leads to more finite length modules than in previous cases Z-graded rings November 9, / 43

39 Two congruent roots Theorem (W) If F is an autoequivalence of gr-a then there exists a = ±1 and b Z such that {F(X n ), F(Y n )} = {X an + b, Y an + b } and F(Z n ) = Z an + b. Theorem (W) There exist numerically trivial autoequivalences, ι n permuting X n and Y n and fixing all other simple modules. Z-graded rings November 9, / 43

40 Two congruent roots In this case, finite dimensional modules Z n. Consider the quotient category qgr-a = gr-a/fdim-a Theorem (W) There is an equivalence of categories gr(b, Z fin ) qgr-a. Z-graded rings November 9, / 43

41 The upshot In all cases, There exist numerically trivial autoequivalences permuting X and Y and fixing all other simples. Pic(gr-A(f )) = (Z 2 ) Z D. There exists a Z fin -graded commutative ring R such that qgr-a(f ) gr(r, Z fin ). Z-graded rings November 9, / 43

42 Questions Z fin -grading on B gives an action of Spec kz fin on Spec B [ ] Spec B χ = Spec kz fin What are the properties of χ? Other Z-graded domains of GK dimension 2? GWAs defined by non-quadratic f? Other base rings D? U(sl(2)) Quantum Weyl algebra Simple Z-graded domains Future direction November 9, / 43

43 Questions Theorem (Smith, 2011) A 1 a Z-graded domain of GK dim 2. Stack χ such that Gr-A 1 Gr(C, Z fin ) Qcoh(χ). For A a Z-graded domain of GK dim 2, is there always a stack? Construction of B: Γ Pic(gr-A) B = Hom qgr-a (A, FA) F Γ Take Γ = S = Z then B = A. Opposite view: O a quasicoherent sheaf on χ: Hom Qcoh(χ) (O, S n O) n Z Future direction November 9, / 43