Graded modules over generalized Weyl algebras Advancement to Candidacy Robert Won Advised by: Dan Rogalski December 4, 2014 1 / 41
Overview 1 Preliminaries Graded rings and modules Noncommutative things 2 Prior work Sierra (2009) Smith (2011) Generalized Weyl algebras 3 Results Non-congruent roots Congruent roots Multiple root 4 Future direction December 4, 2014 2 / 41
Graded k-algebras k an algebraically closed field of characteristic 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 If a A γ, deg a = γ and a is homogeneous of degree γ Examples Any ring R with trivial grading k[x] graded by Z k[x, x 1 ] graded by Z Preliminaries December 4, 2014 3 / 41
Graded modules and morphisms A graded right A-module M: M = i Z M i such that M i A j M i+j A graded module homomorphism, f : M N: f (M i ) N i A graded module homomorphism of degree d, f : M N: f (M i ) N i+d Preliminaries December 4, 2014 4 / 41
Graded modules and morphisms A graded submodule N = N M i i Z If N M a graded submodule M/N is a graded factor module with M/N = (M/N) i = (M i + N)/N. i Z i Z Given a graded module M, define the shift operator (M n ) i = M i n 3 2 1 0 1 2 3 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 December 4, 2014 5 / 41
Categories A category, C, consists of a collection of objects, Obj(C ), and for any two objects X, Y a class of morphisms Hom C (X, Y) Example Set, whose objects are sets and morphisms are functions A (covariant) functor, F : C D is a mapping that associates to each: object X Obj(C ) an object F(X) Obj(D), and morphism f Hom C (X, Y) a morphism F(f ) Hom D (X, Y) An equivalence of categories is a functor F : C D and a functor G : D C such that F G = Id D and G F = Id C. Preliminaries December 4, 2014 6 / 41
The graded module category gr-a Objects: finitely generated graded right A-modules Hom sets: 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 Preliminaries December 4, 2014 7 / 41
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 with 1, 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 December 4, 2014 8 / 41
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 December 4, 2014 9 / 41
The Weyl algebra A noncommutative graded k-algebra, the first Weyl algebra A 1 (k) = A 1 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 Exists an outer automorphism ω, reversing the grading ω(x) = y ω(y) = x Preliminaries December 4, 2014 10 / 41
Sierra (2009) Sue Sierra, Rings graded equivalent to the Weyl algebra Classified all rings graded Morita equivalent to A Examined the graded module category gr-a: 3 2 1 0 1 2 3 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 Prior work December 4, 2014 11 / 41
Sierra (2009) Computed Pic(gr-A) Shift functor, S: Autoequivalence ω: Prior work December 4, 2014 12 / 41
Sierra (2009) Theorem (Sierra) 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 }. An autoequivalence is numerically trivial if a = 1, b = 0. Denote the group of numerically trivial autoequivalences Pic 0 (gr-a). Pic(gr-A) = Pic 0 (gr-a) D. Prior work December 4, 2014 13 / 41
Sierra (2009) Theorem (Sierra) There exist ι n, autoequivalences of gr-a, 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 0 (gr-a) = (Z/2Z) (Z) = Z fin. Prior work December 4, 2014 14 / 41
Smith (2011) Paul Smith, A quotient stack related to the Weyl algebra Proves that gr-a 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 gr(c, Z fin ) Qcohχ Prior work December 4, 2014 15 / 41
Smith (2011) Z fin the group of finite subsets of Z, operation XOR Constructs a Z fin graded ring C := hom(a, ι J A) = 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 gr(c, Z fin ). Prior work December 4, 2014 16 / 41
Generalized Weyl algebras 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 There are also higher dimensional generalized Weyl algebras Prior work December 4, 2014 17 / 41
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) Prior work December 4, 2014 18 / 41
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 Prior work December 4, 2014 19 / 41
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 (or otherwise nicer) Γ-graded ring C such that gr(c, Γ) gr-a(f )? Strategy: First quadratic f Distinct, non-congruent roots Congruent roots Double root Results December 4, 2014 20 / 41
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 Results December 4, 2014 21 / 41
Distinct, non-congruent roots Graded simple modules: α 2 α 1 α α+1 2 1 0 1 2 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 Results December 4, 2014 22 / 41
Distinct, non-congruent roots Theorem (W) There exist numerically trivial autoequivalences, ι 0 n permuting X 0 n and Y 0 n and fixing all other simple modules. Similarly, there exist ι α n permuting X α n and Y α n. Theorem (W) Pic(gr-A) = Pic 0 (gr-a) D = (Z/2Z) (Z) D Results December 4, 2014 23 / 41
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. Results December 4, 2014 24 / 41
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 Results December 4, 2014 25 / 41
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 Results December 4, 2014 26 / 41
Two congruent roots Some finite length indecomposables E Z,Y,X X Z Y E X,Z,X X X Z E Y,Z,Y Z Y Y Results December 4, 2014 27 / 41
Two congruent roots We investigate rank 1 projective modules. As a left k[z]-module, A n is generated x n for n 0 and y n for n < 0. Since y n x n = σ 1 (f ) σ n (f ) A = i Z (a i )x i with a i = { 1, i 0 σ 1 (f ) σ i (f ), i < 0 Results December 4, 2014 28 / 41
Two congruent roots Rank 1 projective modules P a rank 1 projective, P A P = (b i )x i i Z such that b i+1 b i and b i b i+1 σ i (f ). Define c i = b i b 1 i+1. We then have c i σ i (f ), so c i {1, σ i (z), σ i (z + m), σ i (f )}. We call this sequence {c i } the structure constants of P. An ideal of A is determined up to isomorphism by its structure constants. Results December 4, 2014 29 / 41
Two congruent roots The structure constants {c i } determine the properties of a right ideal, I: Lemma (W) I X n if and only if c n m {1, σ n m (z)}, I Y n if and only if c n {σ n (z), σ n (f )}, I Z n if and only if c n m {σ n m (z + m), σ n m (f )} and c n {1, σ n (z + m)}. Lemma (W) I is projective if and only if for n Z, it is not the case that both c n m {1, σ n m (z)} and c n {σ n (z), σ n (f )} (if and only if I surjects onto exactly one of X n, Y n, or Z n ). Results December 4, 2014 30 / 41
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. Results December 4, 2014 31 / 41
Two congruent roots Shift functor, S: Autoequivalence ω: Pic(gr-A) = Pic 0 (gr-a) D. Results December 4, 2014 32 / 41
Two congruent roots Theorem (almost) There exist numerically trivial autoequivalences, ι n permuting X n and Y n and fixing all other simple modules. Construction For a rank 1 projective P, we showed P X (or Y or Z). Lift this surjection to a surjection P E X,Z,X (or E Y,Z,Y or E Z,Y,X ) and let ι 0 (P) be the kernel. For a general projective, more technical. Results December 4, 2014 33 / 41
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 As in all previous cases: K.dim(A) = 1 Exists an outer automorphism ω reversing the grading ω(x) = y ω(y) = x ω(z) = 1 z Different from the previous cases: Now gl.dim(a) = Results December 4, 2014 34 / 41
Multiple root The graded simple modules 3 2 1 0 1 2 3 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. Results December 4, 2014 35 / 41
Multiple root The structure constants {c i } determine the properties of a right ideal, I: Lemma (W) I X n if and only if c n {1, σ n (z)}, I Y n if and only if c n {σ n (z), σ n (f )}, Lemma (W) I is projective if and only if for n Z, it is not the case that both c n = σ n (z) (if and only if I surjects onto exactly one of X n, and Y n ). Results December 4, 2014 36 / 41
Multiple root Shift functor, S: Autoequivalence ω: Theorem (W) F an autoequivalence of gr-a then there exists a = ±1 and b Z: {F(X n ), F(Y n )} = {X an + b, Y an + b }. So Pic(gr-A) = Pic 0 (gr-a) D. Results December 4, 2014 37 / 41
Multiple root Theorem (almost) There exist numerically trivial autoequivalences, ι n permuting X n and Y n and fixing all other simple modules. Theorem (W) Pic(gr-A) = Pic 0 (gr-a) D = (Z/2Z) (Z) D Results December 4, 2014 38 / 41
Multiple root Define a Z fin graded ring B: B = hom A (A, ι J A). J Z fin A commutative k-algebra Not a domain Theorem (almost) There is an equivalence of categories gr(b, Z fin ) gr-a. Results December 4, 2014 39 / 41
The upshot For all f, there exist numerically trivial autoequivalences permuting X and Y and fixing all other simples. For all f, Pic(gr-A(f )) = (Z 2 ) Z D. If f has distinct roots, there exists a commutative ring C such that gr-a(f ) gr(c, Z fin Z fin ) If f has a multiple root, there exists a commutative ring B such that gr-a(f ) gr(b, Z fin ) Results December 4, 2014 40 / 41
Questions What are the properties of the commutative ring B? What can we say about the congruent root case? Conjecture (W) qgr-a(z(z + m)) gr(b, Z fin ). where qgr-a is the category of graded A-modules modulo the full subcategory of finite dimensional modules. Can we piece together the picture for general (non-quadratic) f? Future direction December 4, 2014 41 / 41