The graded module category of a generalized Weyl algebra › ~rwon › files › talks › RWon_Defense.pdfLet X = ProjA for a commutative, f.g. k-algebra A generated in degree 1

The graded module category of a generalized Weyl algebraFinal Defense

Robert WonAdvised by: Daniel Rogalski

May 2, 2016 1 / 39

Overview

1 Graded rings and things

2 Noncommutative is not commutative

3 Noncommutative projective schemes

4 Z-graded ringsThe first Weyl algebraGeneralized Weyl algebras

5 Future direction

May 2, 2016 2 / 39

Graded k-algebras

• Throughout, k an algebraically closed field, chark = 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, Mn(k)• The complex numbers, C = R · 1 + R · i

Graded rings and things May 2, 2016 3 / 39

Graded k-algebras

• Γ an abelian (semi)group• A Γ-graded k-algebra A has k-space decomposition

A =⊕γ∈Γ

Aγ

such that AγAδ ⊆ Aγ+δ.• If a ∈ Aγ , deg a = γ and a is homogeneous of degree γ.


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

RemarkUsually, “graded ring” means N-graded.


Graded modules and morphisms

• A graded right A-module M:

M =⊕i∈Z

Mi

such that Mi · Aj ⊆Mi+j.• Finitely generated graded right A-modules with graded module

homomorphisms form a category gr-A.

ExampleN-graded ring k[x, y]

• Homogeneous ideals (x), (x, x + y), (x2 − xy) are gradedsubmodules

• Nonhomogeneous ideals (x + 3), (x2 + y) are not


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• Mn(A) is Morita equivalent to A• Morita invariant properties of A: simple, semisimple,

noetherian, artinian, hereditary, etc.


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

V(a) = {p ∈ Proj A | a ⊆ p}

for homogeneous ideals a of A• Structure sheaf: localize at homogeneous prime ideals

Noncommutative is not commutative May 2, 2016 8 / 39

Proj of a graded k-algebra

Interplay beteween algebra and geometry

algebra geometrygraded k-algebra A space Proj A

homogeneous prime ideals pointsI ⊆ J V(I) ⊇ V(J)

factor rings subschemeshomogeneous elements of A functions on Proj A


Noncommutative is not commutative

Noncommutative rings are ubiquitous.

Examples• Ring A, nonabelian group G, the group algebra A[G]• Mn(A), n× n matrices• Differential operators on k[t], generated by t· and d/dt is

isomorphic to the first Weyl algebra

A1 = k〈x, y〉/(xy− yx− 1).

If you like physics, position and momentum operators don’t commutein quantum mechanics.


Noncommutative is not commutative

• Localization is different.• Given A commutative and S ⊂ A multiplicatively closed,

a1s−11 a2s−12 = a1a2s

−11 s−12

• If A noncommutative, can only form AS−1 if S is an Ore set.

DefinitionS is an Ore set if for any a ∈ A, s ∈ S

sA ∩ aS 6= ∅.


Noncommutative is not commutativeEven 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 ringThe N-graded ring

k〈x, y〉/(xy− qyx)

is a “noncommutative P1” but for qn 6= 1 has only threehomogeneous ideals (namely (x), (y), and (x, y)).


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.


Sheaves to the rescue

• All you need is modules

TheoremLet X = Proj A for a commutative, f.g. k-algebra A generated indegree 1.

(1) Every coherent sheaf on X is isomorphic to M̃ for some f.g.graded A-module M.

(2) M̃ ∼= Ñ as sheaves if and only if there is an isomorphismM≥n ∼= N≥n.

• Let gr-A be the category of f.g. A-modules. Take the quotientcategory

qgr-A = gr-A/fdim-A

• The above says qgr-A ≡ coh(Proj A).


Noncommutative projective schemes

A (not necessarily commutative) connected graded k-algebra A is

A = k⊕ A1 ⊕ A2 ⊕ · · ·

such that dimk Ai

Noncommutative curves

• “The Hartshorne approach”• A a k-algebra, V ⊆ A a k-subspace generating A spanned by{1, a1, . . . , am}.

• V0 = k, Vn spanned by monomials of length ≤ n in the ai.• The Gelfand-Kirillov dimension of A

GKdim A = lim supn→∞

logn(dimk Vn)

• GKdimk[x1, . . . , xm] = m.• So noncommutative projective curves should have GKdim 2.

Noncommutative projective schemes May 2, 2016 16 / 39

Noncommutative curves

Theorem (Artin-Stafford, 1995)Let A be a f.g. connected graded domain generated in degree 1with GKdim(A) = 2. Then there exists a projective curve Xsuch that

qgr-A ≡ coh(X)

Or “noncommutative projective curves are commutative”.


Noncommutative surfaces

• The “right” definition of a noncommutative polynomial ring?

DefinitionA a f.g. connected graded k-algebra is Artin-Schelter regular if

(1) gl.dimA = d

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 = P2 or X = P1 × P1, or(b) A � B and qgr-B ≡ coh(E) for an elliptic curve E.

• Or “noncommutative P2s are either commutative or contain acommutative curve”.

• Other noncommutative surfaces (noetherian connected gradeddomains of GKdim 3)?

• Noncommutative P3 (AS-regular of dimension 4)?


The first Weyl algebra

• Throughout, “graded” really meant N-graded• Recall the first Weyl algebra:

A1 = k〈x, y〉/(xy− yx− 1)

• Simple noetherian domain of GK dim 2• A1 is not N-graded

Z-graded rings May 2, 2016 20 / 39

The first Weyl algebra

A1 = 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• (A1)0 ∼= k[xy] 6= k• Exists an outer automorphism ω, reversing the grading

ω(x) = y ω(y) = −x


Sierra (2009)

• Sue Sierra, Rings graded equivalent to the Weyl algebra• Classified all rings graded Morita equivalent to A1• Examined the graded module category gr-A1:

−3 −2 −1 0 1 2 3

• For each λ ∈ k \ Z, one simple module Mλ• 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 anonsplit extension of Y〈n〉 by X〈n〉


Sierra (2009)

• Computed Pic(gr-A1), the Picard group (group ofautoequivalences) of gr-A1

• Shift functor, S :

• Autoequivalence ω:


Sierra (2009)

Theorem (Sierra)There exist ιn, autoequivalences of gr-A1, permuting X〈n〉 and Y〈n〉and fixing all other simple modules.

Also ιiιj = ιjιi and ι2n ∼= Idgr-A

Theorem (Sierra)

Pic(gr-A1) ∼= (Z/2Z)(Z) o D∞ ∼= Zfin o D∞.


Smith (2011)

• Paul Smith, A quotient stack related to the Weyl algebra

• Proves that Gr-A1 ≡ 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 pointsBZ2 supported at each integer point”

• Gr-A1 ≡ Gr(C,Zfin) ≡ Qcoh(χ)


Smith (2011)

• Zfin the group of finite subsets of Z, operation XOR• Constructs a Zfin-graded ring

C :=⊕

J∈Zfin

hom(A1, ιJA1) ∼=k[xn | n ∈ Z]

(x2n + n = x2m + m | m,n ∈ Z)

∼= k[z][√

z− n | n ∈ Z]

where deg xn = {n}• C is commutative, integrally closed, non-noetherian PID

Theorem (Smith)There is an equivalence of categories

Gr-A1 ≡ Gr(C,Zfin).


Z-graded rings

Theorem (Artin-Stafford, 1995)Let A be a f.g. connected N-graded domain generated in degree1 with GKdim(A) = 2. Then there exists a projective curve Xsuch that

qgr-A ≡ coh(X).

Theorem (Smith, 2011)A1 is a f.g. Z-graded domain with GKdim(A1) = 2. There existsa commutative ring C and quotient stack χ such that

gr-A1 ≡ gr(C,Zfin) ≡ coh(χ).


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 Moritaequivalent to a GWA.


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〉(

xy = z yx = z− 1zx = x(z− 1) zy = y(z + 1)

) ∼= k〈x, y〉(xy− yx− 1)

= A1.


Generalized Weyl algebras

Properties of A(f ):

• Noetherian domain• Krull dimension 1• Simple if and only if no congruent roots

• gl.dim A(f ) =

1, f has neither multiple nor congruent roots2, 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


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 )?


Results

If α ∈ k \ Z (distinct roots):

−2

α− 2

−1

α− 1

0

α

1

α+1

2

If α = 0 (double root):

−3 −2 −1 0 1 2 3

If α ∈ N+ (congruent roots):


Results

Theorem(s) (W)In all cases, there exist numerically trivial autoequivalences ιnpermuting “X〈n〉” and “Y〈n〉” and fixing all other simple modules.

Corollary (W)For quadratic f , Pic(gr-A(f )) ∼= Zfin o D∞.


Results

Let α ∈ k \ Z. Define a Zfin × Zfin graded ring C:

C =⊕

(J,J′)∈Zfin×Zfin

homA(A, ι(J,J′)A

)∼=

k[an, bn | n ∈ Z](a2n + n = a2m + m, a2n = b2n + α | m,n ∈ Z)

with deg an = ({n}, ∅) and deg bn = (∅, {n})

Theorem (W)There is an equivalence of categories gr(C,Zfin × Zfin) ≡ gr-A.


Results

Let α = 0. Define a Zfin-graded ring B:

B =⊕

J∈Zfin

homA(A, ιJA) ∼=k[z][bn | n ∈ Z]

(b2n = (z + n)2 | n ∈ Z).

Theorem (W)B is a reduced, non-noetherian, non-domain of Kdim 1 withuncountably many prime ideals.

Theorem (W)There is an equivalence of categories gr(B,Zfin) ≡ gr-A.


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,Zfin) ≡ qgr-A.


The upshot

In all cases,• There exist numerically trivial autoequivalences permuting X and

Y and fixing all other simples.• Pic(gr-A(f )) ∼= Zfin o D∞.• There exists a Zfin-graded commutative ring R such that

qgr-A(f ) ≡ gr(R,Zfin).


Questions• Zfin-grading on B gives an action of SpeckZfin on Spec B

χ =

[Spec B

SpeckZfin

]• Other Z-graded domains of GK dimension 2? Other GWAs?

• U(sl(2)), Down-up algebras, quantum Weyl algebra, SimpleZ-graded domains

• Construction of B: {Fγ | γ ∈ Γ} ⊆ Pic(gr-A)

B =⊕γ∈Γ

Homqgr-A(A,FγA)

• Opposite view: O a quasicoherent sheaf on χ:⊕n∈Z

HomQcoh(χ)(O,SnO)

Future direction May 2, 2016 38 / 39

Thank you!

May 2, 2016 39 / 39

Graded rings and thingsNoncommutative is not commutativeNoncommutative projective schemesZ-graded ringsThe first Weyl algebraGeneralized Weyl algebras

Future direction

Documents

The graded module category of a generalized Weyl algebra › ~rwon › files › talks › RWon_Defense.pdfLet X = ProjA for a commutative, f.g. k-algebra A generated in degree 1