The graded module category of a generalized Weyl algebra

Transcription

1 The graded module category of a generalized Weyl algebra Final Defense Robert Won Advised by: Daniel Rogalski May 2, / 39

2 Overview 1 Graded rings and things 2 Noncommutative is not commutative 3 Noncommutative projective schemes 4 Z-graded rings The first Weyl algebra Generalized Weyl algebras 5 Future direction May 2, / 39

3 Graded k-algebras Throughout, k an algebraically closed field, char k = 0. A k-algebra is a ring A with 1 which is a k-vector space such that λ(ab) = (λa)b = a(λb) for λ k and a, b A. Examples The polynomial ring, k[x] The ring of n n matrices, M n (k) The complex numbers, C = R 1 + R i Graded rings and things May 2, / 39

4 Graded k-algebras 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. Graded rings and things May 2, / 39

5 Graded k-algebras Examples Any ring A with trivial grading k[x, y] graded by N k[x, x 1 ] graded by Z If you like physics: superalgebras graded by Z/2Z Or combinatorics: symmetric and alternating polynomials graded by Z/2Z Remark Usually, graded ring means N-graded. Graded rings and things May 2, / 39

6 Graded modules and morphisms A graded right A-module M: M = i Z M i such that M i A j M i+j. Finitely generated graded right A-modules with graded module homomorphisms form a category gr-a. Example N-graded ring k[x, y] Homogeneous ideals (x), (x, x + y), (x 2 xy) are graded submodules Nonhomogeneous ideals (x + 3), (x 2 + y) are not Graded rings and things May 2, / 39

7 Graded modules and morphisms Idea: Study A by studying its module category, mod-a (analogous to studying a group by its representations). 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. Graded rings and things May 2, / 39

8 Proj of a graded k-algebra A a commutative N-graded k-algebra The scheme Proj A As a set: homogeneous primes ideals of A not containing 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 Noncommutative is not commutative May 2, / 39

9 Proj of a graded k-algebra Interplay beteween algebra and geometry algebra graded k-algebra A homogeneous prime ideals I J factor rings homogeneous elements of A geometry space Proj A points V(I) V(J) subschemes functions on Proj A Noncommutative is not commutative May 2, / 39

10 Noncommutative is not commutative Noncommutative rings are ubiquitous. Examples Ring A, nonabelian group G, the group algebra A[G] M n (A), n n matrices Differential operators on k[t], generated by t and d/dt is isomorphic to the first Weyl algebra A 1 = k x, y /(xy yx 1). If you like physics, position and momentum operators don t commute in quantum mechanics. Noncommutative is not commutative May 2, / 39

11 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. Noncommutative is not commutative May 2, / 39

12 Noncommutative is not commutative 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 The N-graded ring k x, y /(xy qyx) is a noncommutative P 1 but for q n 1 has only three homogeneous ideals (namely (x), (y), and (x, y)). Noncommutative is not commutative May 2, / 39

13 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. Noncommutative is not commutative May 2, / 39

14 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). Noncommutative is not commutative May 2, / 39

15 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. Noncommutative projective schemes May 2, / 39

16 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 May 2, / 39

17 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 such that qgr-a coh(x) Or noncommutative projective curves are commutative. Noncommutative projective schemes May 2, / 39

18 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) gl.dima = d < (2) GKdim A < and (3) Ext i A(k, A) = δ i,d k. Interesting invariant theory for these rings. 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 May 2, / 39

19 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) qgr-a coh(x) for X = P 2 or X = P 1 P 1, or (b) A B and qgr-b coh(e) 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 May 2, / 39

20 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 is not N-graded Z-graded rings May 2, / 39

21 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 Exists an outer automorphism ω, reversing the grading ω(x) = y ω(y) = x Z-graded rings May 2, / 39

22 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 May 2, / 39

23 Sierra (2009) Computed Pic(gr-A 1 ), the Picard group (group of autoequivalences) of gr-a 1 Shift functor, S: Autoequivalence ω: Z-graded rings May 2, / 39

24 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 May 2, / 39

25 Smith (2011) Paul Smith, A quotient stack related to the Weyl algebra Proves that Gr-A 1 Qcoh(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 ) Qcoh(χ) Z-graded rings May 2, / 39

26 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 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) There is an equivalence of categories Gr-A 1 Gr(C, Z fin ). Z-graded rings May 2, / 39

27 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 ) coh(χ). Z-graded rings May 2, / 39

28 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 May 2, / 39

29 Generalized Weyl algebras For us, 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 May 2, / 39

30 Generalized Weyl algebras 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 May 2, / 39

31 Questions Focus on quadratic f (z) = z(z + α). For these GWAs 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 )? Z-graded rings May 2, / 39

32 Results If α k \ Z (distinct roots): α 2 α 1 α α If α = 0 (double root): If α N + (congruent roots): Z-graded rings May 2, / 39

33 Results Theorem(s) (W) In all cases, there exist numerically trivial autoequivalences ι n permuting X n and Y n and fixing all other simple modules. Corollary (W) For quadratic f, Pic(gr-A(f )) = Z fin D. Z-graded rings May 2, / 39

34 Results Let α k \ Z. Define a Z fin Z fin graded ring C: ( C = hom A A, ι(j,j )A ) (J,J ) Z fin Z fin = k[a n, b n n Z] (a 2 n + n = a 2 m + m, a 2 n = b 2 n + α 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 May 2, / 39

35 Results Let α = 0. Define a Z fin -graded ring B: B = J Z fin hom A (A, ι J A) = k[z][b 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 May 2, / 39

36 Results In congruent root (α N + ), we have finite-dimensional modules. 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 May 2, / 39

37 The upshot In all cases, There exist numerically trivial autoequivalences permuting X and Y and fixing all other simples. Pic(gr-A(f )) = Z fin D. There exists a Z fin -graded commutative ring R such that qgr-a(f ) gr(r, Z fin ). Z-graded rings May 2, / 39

38 Questions Z fin -grading on B gives an action of Spec kz fin on Spec B [ ] Spec B χ = Spec kz fin Other Z-graded domains of GK dimension 2? Other GWAs? U(sl(2)), Down-up algebras, quantum Weyl algebra, Simple Z-graded domains Construction of B: {F γ γ Γ} Pic(gr-A) B = γ Γ Hom qgr-a (A, F γ A) Opposite view: O a quasicoherent sheaf on χ: Hom Qcoh(χ) (O, S n O) n Z Future direction May 2, / 39

39 Thank you! May 2, / 39