algebra seminar - university of california, san...

Z-graded noncommutative projective geometryAlgebra Seminar

Robert WonUniversity of California, San Diego

November 9, 2015 1 / 43

Overview

1 PreliminariesPre-talk catchupNoncommutative things

2 Noncommutative projective schemes

3 Z-graded ringsPrior workGeneralized Weyl algebras

4 Future direction

November 9, 2015 2 / 43

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

Aγ

such that AγAδ ⊆ Aγ+δ

• A graded right A-module M:

M =⊕γ∈Γ

Mγ

such that Mγ · Aδ ⊆Mγ+δ

Preliminaries November 9, 2015 3 / 43

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:

homgr-A(M,N) = {f ∈ Hommod-A(M,N) | f (Mi) ⊆ Ni}

• The shift functor:

S i : gr-A −→ gr-AM 7→M〈i〉

−3 −2 −1 0 1 2 3M M−3 M−2 M−1 M0 M1

M〈1〉 M−3 M−2 M−1 M0 M1M〈2〉 M−3 M−2 M−1 M0 M1


The Picard group

• For a category of graded modules (e.g. gr-A) we define the Picardgroup, Pic(gr-A) to be the group of autoequivalences of gr-Amodulo natural isomorphism.

• (If R is a commutative k-algebra, Pic(mod-R) ∼= Pic(R), theisomorphism classes of invertible unitary R-bimodules)

• For a noncommutative ring, A, Pic(gr-A) can be nonabelian.


Morita equivalence

• Two rings R and S are called Morita equivalent if mod-R isequivalent 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 andonly if R ∼= S.

ExampleR a ring, then Mn(R) (the ring of n× n matrices) is Moritaequivalent to R.

• R and S are graded Morita equivalent if gr-R is equivalent to gr-S.


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 ringThe N-graded ring

k〈x, y〉/(xy− qyx)

is a “noncommutative P1” but for qn 6= 1 has only fourhomogeneous prime ideals (namely (0), (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 R for a commutative, f.g. k-algebra R generated indegree 1.

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

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

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

qgr-R = gr-R/fdim-R

• The above says qgr-R ≡ coh(Proj R).


Noncommutative projective schemes

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

A = k⊕ A1 ⊕ A2 ⊕ · · ·

such that dimk Ai <∞ and A is a f.g. k-algebra.

Definition (Artin-Zhang, 1994)The noncommutative projective scheme ProjNC 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 ProjNC A = qgr-A to study A.

Noncommutative projective schemes November 9, 2015 10 / 43

All you need is modules

• Can attempt to do any geometry that only relies on the modulecategory.

• If X is a commutative projective k-scheme, for each x ∈ X, there isthe 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 dimk Mn = 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 ProjNC A = simple modules.


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⊕⊕n≥1

H0(X,L⊗n).

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).


Twisted homogeneous coordinate rings

• X a (commutative) projective scheme, L a line bundle, andσ ∈ Aut(X). Define

Lσ = σ∗L and Ln = L ⊗ Lσ ⊗ · · · ⊗ Lσn−1.

• The twisted homogeneous coordinate ring is

B(X,L, σ) = k⊕⊕n≥1

H0(X,Ln).

• 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 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 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 X, anautomorphism σ and invertible sheaf L such that up to a f.dvector space

A = B(X,L, σ)

• As a corollary, 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) gldim A = d <∞

(2) GKdim A <∞ and

(3) ExtiA(k,A) = δi,dk.

• Behaves homologically like a commutative polynomial ring.• k[x1, . . . , xm] is AS-regular of dimension m.• Noncommutative P2s should be qgr-A for A AS-regular of

dimension 3.


Noncommutative surfaces

Theorem (Artin-Tate-Van den Bergh, 1990)Let A be an AS-regular ring of dimension 3 generated in degree1. Either(a) A = B(X,L, σ) for X = P2 or X = P1 × P1 or(b) A� B(E,L, σ) 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)?


Z-graded rings

• Throughout, “graded” really meant N-graded• Way back to our first example:

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• 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, 2015 18 / 43

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)

• 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 ≡ Qcohχ• 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) ≡ Qcohχ


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 = x2

m + m | m,n ∈ Z)

∼= k[z |√

z− n]

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) ≡ Qcoh(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.


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〉(

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

) ∼= k〈x, y〉(xy− yx− 1)

= A1.


The main object

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 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


Distinct, non-congruent roots

Let f (z) = z(z + α) for some α ∈ k \ Z.

A = A(f ) ∼=k〈x, y, z〉(

xy = z(z + α) 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



Graded simple modules:

−2

α− 2

−1

α− 1

0

α

1

α+1

2

• One simple graded module Mλ for each λ ∈ k \ (Z ∪ Z + α)

• For each n ∈ Z two simple modules X0〈n〉 and Y0〈n〉• For each n ∈ Z two simple modules Xα〈n〉 and Yα〈n〉• A nonsplit extension of X0〈n〉 by Y0〈n〉 and vice versa• A nonsplit extension of Xα〈n〉 by Yα〈n〉 and vice versa



Theorem (W)There exist numerically trivial autoequivalences, ι(n,∅)permuting X0〈n〉 and Y0〈n〉 and fixing all other simple modules.Similarly, there exist ι(∅,n) permuting Xα〈n〉 and Yα〈n〉.

Theorem (W)

Pic(gr-A) ∼= (Z/2Z)(Z) o D∞



• Define a Zfin × Zfin graded ring C:

C = k[an, bn | n ∈ Z]

modulo the relations

a2n + n = a2

m + m and a2n = b2

n + α for all 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.


Multiple root

Let f (z) = z2.

A = A(f ) ∼=k〈x, y, z〉(

xy = z2 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) =∞


Multiple root

The graded simple modules

−3 −2 −1 0 1 2 3

• For each λ ∈ k \ Z, Mλ

• For each n ∈ Z, X〈n〉 and Y〈n〉

X Y

• Nonsplit extensions of X〈n〉 by Y〈n〉 and vice versa. Alsoself-extensions of X〈n〉 by X〈n〉 and Y〈n〉 and Y〈n〉.


Multiple root

Theorem (W)There exist numerically trivial autoequivalences, ιn permutingX〈n〉 and Y〈n〉 and fixing all other simple modules.

Theorem (W)

Pic(gr-A) ∼= (Z/2Z)(Z) o D∞


Multiple root

Define a Zfin graded ring B:

B =⊕

J∈Zfin

homA(A, ιJA) ∼=k[z][an | 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.


Two congruent rootsLet 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


Two congruent roots

The graded simple modules• For each λ ∈ k \ Z, Mλ

• For each n ∈ Z, X〈n〉, Y〈n〉, and Z〈n〉

X YZ

• Z〈n〉 is finite dimensional• proj.dim Z〈n〉 = 2

• Leads to “more” finite length modules than in previous cases


Two congruent roots

Theorem (W)If F is an autoequivalence of gr-A then there exists a = ±1 and b ∈ Zsuch that

{F(X〈n〉),F(Y〈n〉)} ∼= {X〈an + b〉,Y〈an + b〉}

andF(Z〈n〉) ∼= Z〈an + b〉.

Theorem (W)There exist numerically trivial autoequivalences, ιn permuting X〈n〉and Y〈n〉 and fixing all other simple modules.


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,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 )) ∼= (Z2)Z 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

]• 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, 2015 42 / 43

Questions

Theorem (Smith, 2011)A1 a Z-graded domain of GK dim 2. Stack χ such that

Gr-A1 ≡ Gr(C,Zfin) ≡ 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 =⊕F∈Γ

Homqgr-A(A,FA)

• Take Γ = 〈S〉 = Z then B = A.• Opposite view: O a quasicoherent sheaf on χ:⊕

n∈ZHomQcoh(χ)(O,SnO)

Future direction November 9, 2015 43 / 43

algebra seminar - university of california, san...

Documents