Z-graded noncommutative algebraic geometry University of Washington Algebra/Algebraic Geometry Seminar

Transcription

1 Z-graded noncommutative algebraic geometry University of Washington Algebra/Algebraic Geometry Seminar Robert Won Wake Forest University Joint with Jason Gaddis (Miami University) and Cal Spicer (Imperial College London). February 13, / 40

2 Graded k-algebras Throughout, k an algebraically closed field, char k = 0. Γ an abelian (semi)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. If a A γ, deg a = γ and a is homogeneous of degree γ. Most commonly, Γ = N (e.g., polynomials graded by total degree). February 13, 2018 Graded rings and things 2 / 40

3 Modules and morphisms Right A-modules with A-module homomorphisms form a category, mod-a. If mod-a mod-b, we say A and B are Morita equivalent. Remarks For commutative rings, Morita equivalent isomorphic. M n (A) is Morita equivalent to A. Morita invariant properties of A: simple, semisimple, noetherian, artinian, hereditary, etc. February 13, 2018 Graded rings and things 3 / 40

4 Graded modules and morphisms A Γ-graded right A-module M: M = i Γ M i such that M i A j M i+j. Γ-graded right A-modules with graded module homomorphisms form a category gr-(a, Γ) or gr-a. If gr-a gr-b. Then A and B are graded Morita equivalent. February 13, 2018 Graded rings and things 4 / 40

5 What would a commutative algebraist do? A a commutative N-graded k-algebra. The projective scheme Proj A: As a set: homogeneous prime ideals of A not containing the irrelevant ideal A >0. Topological space: the Zariski topology with closed sets for homogeneous ideals a of A. V(a) = {p Proj A a p} Structure sheaf: localize at homogeneous prime ideals. February 13, 2018 Graded rings and things 5 / 40

6 Noncommutative is not commutative Localization is different. Given A commutative and S A multiplicatively closed, a 1 s 1 1 a 2s 1 2 = a 1 a 2 s 1 1 s 1 2. If A noncommutative, can only form AS 1 if S is an Ore set. Definition S is an Ore set if for any a A, s S sa as. February 13, 2018 Graded rings and things 6 / 40

7 Noncommutative is not commutative Forget localization. Who needs it? Even worse, 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 k x, y /(xy qyx) is a noncommutative P 1 but for q n 1 has only three proper homogeneous ideals (namely (x), (y), and (x, y)). February 13, 2018 Graded rings and things 7 / 40

8 Noncommutative is not commutative Forget prime ideals. Can we come up with a space? We re clever! Left prime ideals? Spec(R/[R, R])? Even worse, there may not even be a set. Theorem (Reyes, 2012) Suppose F : Ring Set extends the functor Spec : CommRing Set. Then for n 3, F(M n (C)) =. February 13, 2018 Graded rings and things 8 / 40

9 Sheaves to the rescue All you need is modules Theorem Let X = Proj A for a commutative, f.g. k-algebra A generated in degree 1. (1) Every coherent sheaf on X is isomorphic to M for some f.g. graded A-module M. (2) M = Ñ as sheaves if and only if there is an isomorphism M n = N n. Let gr-a be the category of f.g. A-modules. Take the quotient category qgr-a = gr-a/fdim-a. The above says qgr-a coh(proj A). February 13, 2018 Graded rings and things 9 / 40

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 and 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. Can do geometry with the module category. February 13, 2018 Noncommutative projective schemes 10 / 40

11 From schemes to 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) Let L = O X (1) (which 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). February 13, 2018 Noncommutative projective schemes 11 / 40

12 From schemes to 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). February 13, 2018 Noncommutative projective schemes 12 / 40

13 Noncommutative curves The Hartshorne approach Noncommutative dimension? 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. February 13, 2018 Noncommutative projective schemes 13 / 40

14 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 curves are commutative. February 13, 2018 Noncommutative projective schemes 14 / 40

15 Noncommutative surfaces The right definition of a noncommutative polynomial ring? Definition A a f.g. connected graded k-algebra is Artin-Schelter regular of dimension d if (1) gl.dima = d < (2) GKdim A < and (3) Ext i A(k, A) = δ i,d k. Behaves homologically like a commutative polynomial ring. Commutative AS regular of dimension d k[x 1,..., x d ]. Interesting invariant theory for these rings. Noncommutative P 2 s should be qgr-a for A AS regular of dimension 3. February 13, 2018 Noncommutative projective schemes 15 / 40

16 Noncommutative surfaces Theorem (Artin, Tate, and 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 (and hence qgr-a coh(p 2 ) or qgr-a coh(p 1 P 1 )), (b) A B(E, L, σ) for an elliptic curve E (and hence qgr-b coh(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)? February 13, 2018 Noncommutative projective schemes 16 / 40

17 The first Weyl algebra Throughout, graded really meant N-graded. Recall the first Weyl algebra: A 1 = k x, y /(xy yx 1). Simple noetherian domain of GK dim 2. A 1 does not admit a nontrivial N-grading. February 13, 2018 Z-graded rings 17 / 40

18 The first Weyl algebra 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 (A 1 ) 0 = k[xy] k Z-graded module category gr-a? February 13, 2018 Z-graded rings 18 / 40

19 The first Weyl algebra Sue Sierra, Rings graded equivalent to the Weyl algebra (2009) 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 February 13, 2018 Z-graded rings 19 / 40

20 The first Weyl algebra Computed Pic(gr-A 1 ), the Picard group (group of autoequivalences) of gr-a 1 Shift functor, S: Exists an outer automorphism ω, reversing the grading ω(x) = y ω(y) = x February 13, 2018 Z-graded rings 20 / 40

21 The first Weyl algebra Theorem (Sierra 2009) 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 2009) Pic(gr-A 1 ) = (Z/2Z) (Z) D = Zfin D. February 13, 2018 Z-graded rings 21 / 40

22 The first Weyl algebra Paul Smith, A quotient stack related to the Weyl algebra (2011) Proves that gr-a 1 coh(χ) 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 ) coh(χ) February 13, 2018 Z-graded rings 22 / 40

23 The first Weyl algebra 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 n Z] 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 2011) There is an equivalence of categories gr-a 1 gr-(c, Z fin ). February 13, 2018 Z-graded rings 23 / 40

24 Z-graded rings Theorem (Artin and 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 ) coh(χ). Question (Smith) What happens in positive characteristic? February 13, 2018 Z-graded rings 24 / 40

25 Generalized Weyl algebras (GWAs) Introduced by V. Bavula Generalized Weyl algebras and their representations (1993) R a ring; σ Aut(R); a Z(R) The generalized Weyl algebra R(σ, a) with base ring R R(σ, a) = R x, y ( ) xy = a yx = σ(a) rx = xσ(r), r R dy = yσ 1 (r), r R Theorem (Bell and Rogalski 2016) Every simple Z-graded domain of GKdim 2 is graded Morita equivalent to a GWA with R = k[z] or k[z, z 1 ]. February 13, 2018 Z-graded rings 25 / 40

26 Generalized Weyl algebras Let: R = k[z] and σ(z) = z + 1, or R = k[z, z 1 ] and σ(z) = ξz for ξ k R(σ, a) = R x, y ( xy = a yx = σ 1 ) (a) xz = σ(z)x yz = σ 1 (z)y Two roots α, β of a are congruent if σ i (α) = β for some i Z. Example (The first Weyl algebra) Take R = k[z], σ(z) = z + 1, and a = z R(σ, z) = k[z] x, y ( ) k x, y = xy = z yx = z 1 (xy yx 1) = A 1. xz = (z + 1)x yz = (z 1)y February 13, 2018 Z-graded rings 26 / 40

27 Generalized Weyl algebras Properties of A = R(σ, 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 = 2, f has congruent roots but no multiple roots, f has a multiple root A is Z-graded by deg x = 1, deg y = 1, deg r = 0 for r R. February 13, 2018 Z-graded rings 27 / 40

28 Simple rings of GK dimension 2 A = R(σ, f ) is simple if and only if f has no congruent roots. The simple modules of gr-a: σ 2 (α 2 ) σ 1 (α 2 ) α 2 σ(α 2 ) σ 2 (α 2 ) σ 2 (α 1 ) σ 1 (α 1 ) α 1 σ(α 1 ) σ 2 (α 1 ) Parametrized by MaxSpec R (mostly). Along the σ-orbit of each root α i of f, two modules analogous to X n and Y n. If the multiplicity of α i is n i, there are two length n i -irreducible modules whose factors are all X or all Y. February 13, 2018 Z-graded rings 28 / 40

29 Simple rings of GK dimension 2 Theorem(s) (W 2018 and W, Spicer) Let A be a simple Z-graded domain of GK dimension 2. Then there exist autoequivalences permuting X n and Y n and fixing all other simple modules. Corollary (W, Spicer) n i=1 Z fin = Z fin Pic(gr-A). February 13, 2018 Z-graded rings 29 / 40

30 Simple rings of GK dimension 2 Theorem Let A be a G-graded ring and let Γ Pic(gr-(S, G)). For each γ Γ, choose F γ. As long as F γ, F δ, and F γ+δ satisfy some technical condition for each γ, δ Γ, then R = γ Γ Hom gr-(s,g) (S, F γ S) is an associative Γ-graded ring. Further, if γ Γ F γs is a projective generator of gr-(s, G), then gr-(r, Γ) gr-(s, G). Idea: Find a subgroup Γ Pic gr-(s, G) large enough so that γ Γ F γs generates but small enough that R is nice. February 13, 2018 Z-graded rings 30 / 40

31 Simple rings of GK dimension 2 Theorem (W, Spicer) Let A be a simple Z-graded domain of GK dimension 2. Then there exists a commutative Z fin -graded ring B such that gr-a gr-(b, Z fin ). Corollary (W, Spicer) [ ] Spec B gr-a coh. Spec kz fin Question What are the properties of the commutative rings B and the quotient stacks? February 13, 2018 Z-graded rings 31 / 40

32 Non-simple rings? A = R(σ, f ) is simple if and only if no congruent roots. Let R = k[z], σ(z) = z + 1, f = z(z m) for some m Z. gr-a has simple modules with finite k-dimension. Theorem (W) qgr-k[z](σ, z(z m)) gr-k[z](σ, z 2 ). Conjecture For a non-simple GWA A, there is a simple GWA A such that qgr-a gr-a. February 13, 2018 Z-graded rings 32 / 40

33 Non-simple rings? Fact (Sierra) There is a Morita context between k[z](σ, z(z m)) and k[z](σ, z 2 ). Question How is the Morita context related to the quotient functor gr-k[z](σ, z(z m)) qgr-k[z](σ, z(z m)) gr-k[z](σ, z 2 )? Question What about other GWAs? Different base rings? Higher degree GWAs? February 13, 2018 Z-graded rings 33 / 40

34 Simple rings in higher dimension Theorem (Bell and Rogalski 2016) Every simple Z-graded domain of GKdim 2 is graded Morita equivalent to a GWA with R = k[z] or k[z, z 1 ]. Q: What if A is a simple Z-graded ring with GKdim A > 2? Restrict to Ore domains A ( commutative or noetherian). Localize at homogeneous elements: graded quotient ring Q gr (A). Fact: Q gr (A) = D[t, t 1 ; σ] for a division ring D and automorphism σ : D D. If Q gr (A) = K[t, t 1 ; σ] for a field K, A is birationally commutative. Restrict to birationally commutative A. February 13, 2018 Z-graded rings 34 / 40

35 Simple rings in higher dimension Theorem (Bell and Rogalski 2016) Every simple Z-graded domain of GKdim 2 is graded Morita equivalent to a GWA with R = k[z] or k[z, z 1 ]. Q: What if A is a simple Z-graded ring with GKdim A > 2? (A: Should be hard.) Restrict to Ore domains A ( commutative or noetherian). Localize at homogeneous elements: graded quotient ring Q gr (A). Fact: Q gr (A) = D[t, t 1 ; σ] for a division ring D and automorphism σ : D D. If Q gr (A) = K[t, t 1 ; σ] for a field K, A is birationally commutative. Restrict to birationally commutative A. February 13, 2018 Z-graded rings 34 / 40

36 Simple rings in higher dimension Bell and Rogalski s simple Z-graded rings. R a commutative noetherian k-algebra. X = Spec R. σ : R R an automorphism, induces σ : X X. February 13, 2018 Z-graded rings 35 / 40

37 Simple rings in higher dimension Bell and Rogalski s simple Z-graded rings. R a commutative noetherian k-algebra. X = Spec R. σ : R R an automorphism, induces σ : X X. Let R[t, t 1 ; σ] denote the skew-laurent ring with tr = σ(r)t and t 1 r = σ 1 (r)t 1 for all r R. February 13, 2018 Z-graded rings 35 / 40

38 Simple rings in higher dimension Bell and Rogalski s simple Z-graded rings. R a commutative noetherian k-algebra. X = Spec R. σ : R R an automorphism, induces σ : X X. Let R[t, t 1 ; σ] denote the skew-laurent ring with tr = σ(r)t and t 1 r = σ 1 (r)t 1 for all r R. Z a σ-lonely closed subset of X: σ i (Z) Z =. February 13, 2018 Z-graded rings 35 / 40

39 Simple rings in higher dimension Bell and Rogalski s simple Z-graded rings. R a commutative noetherian k-algebra. X = Spec R. σ : R R an automorphism, induces σ : X X. Let R[t, t 1 ; σ] denote the skew-laurent ring with tr = σ(r)t and t 1 r = σ 1 (r)t 1 for all r R. Z a σ-lonely closed subset of X: σ i (Z) Z =. H, J ideals of R such that Spec R/H, Spec R/J Z. February 13, 2018 Z-graded rings 35 / 40

40 Simple rings in higher dimension Bell and Rogalski s simple Z-graded rings. R a commutative noetherian k-algebra. X = Spec R. σ : R R an automorphism, induces σ : X X. Let R[t, t 1 ; σ] denote the skew-laurent ring with tr = σ(r)t and t 1 r = σ 1 (r)t 1 for all r R. Z a σ-lonely closed subset of X: σ i (Z) Z =. H, J ideals of R such that Spec R/H, Spec R/J Z. Define the Z-graded ring B(Z, H, J) R[t, t 1 ; σ] February 13, 2018 Z-graded rings 35 / 40

41 Simple rings in higher dimension Bell and Rogalski s simple Z-graded rings. R a commutative noetherian k-algebra. X = Spec R. σ : R R an automorphism, induces σ : X X. Let R[t, t 1 ; σ] denote the skew-laurent ring with tr = σ(r)t and t 1 r = σ 1 (r)t 1 for all r R. Z a σ-lonely closed subset of X: σ i (Z) Z =. H, J ideals of R such that Spec R/H, Spec R/J Z. Define the Z-graded ring B(Z, H, J) R[t, t 1 ; σ] R }{{} deg 0 February 13, 2018 Z-graded rings 35 / 40

42 Simple rings in higher dimension Bell and Rogalski s simple Z-graded rings. R a commutative noetherian k-algebra. X = Spec R. σ : R R an automorphism, induces σ : X X. Let R[t, t 1 ; σ] denote the skew-laurent ring with tr = σ(r)t and t 1 r = σ 1 (r)t 1 for all r R. Z a σ-lonely closed subset of X: σ i (Z) Z =. H, J ideals of R such that Spec R/H, Spec R/J Z. Define the Z-graded ring B(Z, H, J) R[t, t 1 ; σ] }{{} R Jt }{{} deg 0 deg 1 February 13, 2018 Z-graded rings 35 / 40

43 Simple rings in higher dimension Bell and Rogalski s simple Z-graded rings. R a commutative noetherian k-algebra. X = Spec R. σ : R R an automorphism, induces σ : X X. Let R[t, t 1 ; σ] denote the skew-laurent ring with tr = σ(r)t and t 1 r = σ 1 (r)t 1 for all r R. Z a σ-lonely closed subset of X: σ i (Z) Z =. H, J ideals of R such that Spec R/H, Spec R/J Z. Define the Z-graded ring B(Z, H, J) R[t, t 1 ; σ] }{{} R Jt }{{} deg 0 deg 1 Jσ(J)t 2 }{{} deg 2 February 13, 2018 Z-graded rings 35 / 40

44 Simple rings in higher dimension Bell and Rogalski s simple Z-graded rings. R a commutative noetherian k-algebra. X = Spec R. σ : R R an automorphism, induces σ : X X. Let R[t, t 1 ; σ] denote the skew-laurent ring with tr = σ(r)t and t 1 r = σ 1 (r)t 1 for all r R. Z a σ-lonely closed subset of X: σ i (Z) Z =. H, J ideals of R such that Spec R/H, Spec R/J Z. Define the Z-graded ring B(Z, H, J) R[t, t 1 ; σ] σ 1 (H)t 1 }{{}}{{} R Jt Jσ(J)t }{{} 2 }{{} deg 1 deg 0 deg 1 deg 2 February 13, 2018 Z-graded rings 35 / 40

45 Simple rings in higher dimension Bell and Rogalski s simple Z-graded rings. R a commutative noetherian k-algebra. X = Spec R. σ : R R an automorphism, induces σ : X X. Let R[t, t 1 ; σ] denote the skew-laurent ring with tr = σ(r)t and t 1 r = σ 1 (r)t 1 for all r R. Z a σ-lonely closed subset of X: σ i (Z) Z =. H, J ideals of R such that Spec R/H, Spec R/J Z. Define the Z-graded ring B(Z, H, J) σ 1 (H)σ 2 (H)t 2 σ 1 (H)t 1 }{{}}{{}}{{} R Jt Jσ(J)t }{{} 2 }{{} deg 2 deg 1 deg 0 deg 1 deg 2 R[t, t 1 ; σ] February 13, 2018 Z-graded rings 35 / 40

46 Simple rings in higher dimension Bell and Rogalski s simple Z-graded rings. R a commutative noetherian k-algebra. X = Spec R. σ : R R an automorphism, induces σ : X X. Let R[t, t 1 ; σ] denote the skew-laurent ring with tr = σ(r)t and t 1 r = σ 1 (r)t 1 for all r R. Z a σ-lonely closed subset of X: σ i (Z) Z =. H, J ideals of R such that Spec R/H, Spec R/J Z. Define the Z-graded ring B(Z, H, J) σ 1 (H)σ 2 (H)t 2 σ 1 (H)t 1 }{{}}{{}}{{} R Jt Jσ(J)t }{{} 2 }{{} deg 2 deg 1 deg 0 deg 1 deg 2 R[t, t 1 ; σ] Idea: twisted extended Rees algebra : Jt Jt Jσ(J)t t February 13, 2018 Z-graded rings 35 / 40

47 Simple rings in higher dimension Example R = k[z], σ(z) = z + 1, H = (z), J = R. Then B(Z, H, J) is and ((z 1)(z 2)) t 2 (z 1)t 1 k[z] k[z]t k[z]t 2 via x t and y (z 1)t 1. A 1 = k x, y /(xy yx 1) = B(Z, H, J) Other generalized Weyl algebras: J = R, H principal. February 13, 2018 Z-graded rings 36 / 40

48 Simple rings in higher dimension Properties of B = B(Z, H, J): B is simple if and only if R is σ-simple: no nontrivial two-sided ideals I such that σ(i) = I. B is noetherian if {σ i (Z)} is critically dense in X: for any closed Y X, Y σ i (Z) = for all but finitely many i. Theorem (Bell and Rogalski 2016) Let A be a simple, finitely generated, Z-graded k-algebra which is a birationally commutative Ore domain. Then for some B(Z, H, J). gr-a gr-b(z, H, J) February 13, 2018 Z-graded rings 37 / 40

49 Simple rings in higher dimension Theorem (W, Gaddis) Let B = B(Z, H, J). The graded simple right B-modules are described as follows. For each maximal ideal m of R such that H m, or J m there are graded simple modules: M B m = mb + B 1 B and its shifts M m n for each n Z, M + B m = σ 1 (m)b + B 1 B and its shifts M+ m n for each n Z. For each maximal ideal m of R such that σ i (H) m and σ i (J) m for every i, M m = B is a simple module. mb February 13, 2018 Z-graded rings 38 / 40

50 Simple rings in higher dimension Question What is the Krull dimension/global dimension/gk dimension of B(Z, H, J)? Question (Bell-Rogalski) Is every ideal of B(Z, H, J) two-generated? (Conjectured to be true for every simple noetherian ring.) February 13, 2018 Z-graded rings 39 / 40

51 Thank you! February 13, / 40