Z-Graded Noncommutative Algebraic Geometry University of Washington Algebra/Algebraic Geometry Seminar

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, 2018 1 / 40 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 M A = Aγ γ∈Γ

such that AγAδ ⊆ Aγ+δ.

• If a ∈ Aγ, 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 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.

• Mn(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 Graded modules and morphisms

• A Γ-graded right A-module M: M M = Mi i∈Γ

such that Mi · Aj ⊆ Mi+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 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

V(a) = {p ∈ Proj A | a ⊆ p}

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

February 13, 2018 Graded rings and things 5 / 40 Noncommutative is not commutative

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

−1 −1 −1 −1 a1s1 a2s2 = a1a2s1 s2 .

• 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 6= ∅.

February 13, 2018 Graded rings and things 6 / 40 Noncommutative is not commutative

• Forget localization. Who needs it? • Even worse, not enough (prime) ideals.

The Weyl algebra

khx, yi/(xy − yx − 1) is a noncommutative analogue of k[x, y] but is simple.

The quantum polynomial ring

khx, yi/(xy − qyx) is a “noncommutative P1” but for qn 6= 1 has only three proper homogeneous ideals (namely (x), (y), and (x, y)).

February 13, 2018 Graded rings and things 7 / 40 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(Mn(C)) = ∅.

February 13, 2018 Graded rings and things 8 / 40 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 Me for some f.g. graded A-module M.

(2) Me =∼ Ne 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 Noncommutative projective schemes

A (not necessarily commutative) connected graded k-algebra A is k A = ⊕ A1 ⊕ A2 ⊕ · · ·

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

Deﬁnition (Artin and 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.

Can do geometry with the module category.

February 13, 2018 Noncommutative projective schemes 10 / 40 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 M B(X, L) = k ⊕ H0(X, L⊗n). n≥1 Theorem (Serre)

Let L = OX(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 From schemes to rings

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

σ ∗ σ σn−1 L = σ L and Ln = L ⊗ L ⊗ · · · ⊗ L .

• The twisted homogeneous coordinate ring is

M 0 B(X, L, σ) = k ⊕ H (X, Ln). n≥1 • 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 Noncommutative curves

• “The Hartshorne approach” • Noncommutative dimension? • 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 sup logn(dimk Vn). n→∞ k • GKdim [x1,..., xm] = m. • So noncommutative projective curves should have GKdim 2.

February 13, 2018 Noncommutative projective schemes 13 / 40 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 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 i k k (3) ExtA( , A) = δi,d . • Behaves homologically like a commutative polynomial ring. k • Commutative AS regular of dimension d ⇔ [x1,..., xd]. • Interesting invariant theory for these rings. • Noncommutative P2s should be qgr-A for A AS regular of dimension 3.

February 13, 2018 Noncommutative projective schemes 15 / 40 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 = P2 or X = P1 × P1 (and hence qgr-A ≡ coh(P2) or qgr-A ≡ coh(P1 × P1)), (b) A B(E, L, σ) for an elliptic curve E (and hence qgr-B ≡ coh(E)).

• Or “noncommutative P2s are either commutative or contain a commutative curve”. • Other noncommutative surfaces (noetherian connected graded domains of GKdim 3)? • Noncommutative P3 (AS regular of dimension 4)?

February 13, 2018 Noncommutative projective schemes 16 / 40 The ﬁrst Weyl algebra

• Throughout, “graded” really meant N-graded. • Recall the ﬁrst Weyl algebra: k A1 = hx, yi/(xy − yx − 1).

• Simple noetherian domain of GK dim 2.

• A1 does not admit a nontrivial N-grading.

February 13, 2018 Z-graded rings 17 / 40 The ﬁrst Weyl algebra

k A1 = hx, yi/(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 ∼ k k • (A1)0 = [xy] 6= • Z-graded module category gr-A?

February 13, 2018 Z-graded rings 18 / 40 The ﬁrst Weyl algebra

• Sue Sierra, Rings graded equivalent to the Weyl algebra (2009)

• 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, Xhni and Yhni

X Y

• For each n, exists a nonsplit extension of Xhni by Yhni and a nonsplit extension of Yhni by Xhni

February 13, 2018 Z-graded rings 19 / 40 The ﬁrst Weyl algebra

• Computed Pic(gr-A1), the Picard group (group of autoequivalences) of gr-A1 • 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 The ﬁrst Weyl algebra

Theorem (Sierra 2009)

There exist ιn, autoequivalences of gr-A1, permuting Xhni and Yhni and ﬁxing all other simple modules.

2 ∼ Also ιiιj = ιjιi and ιn = Idgr-A

Theorem (Sierra 2009) ∼ (Z) ∼ Pic(gr-A1) = (Z/2Z) o D∞ = Zﬁn o D∞.

February 13, 2018 Z-graded rings 21 / 40 The ﬁrst Weyl algebra

• Paul Smith, A quotient stack related to the Weyl algebra (2011)

• Proves that gr-A1 ≡ 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 BZ2 supported at each integer point”

• gr-A1 ≡ gr(C, Zﬁn) ≡ coh(χ)

February 13, 2018 Z-graded rings 22 / 40 The ﬁrst Weyl algebra

• Zﬁn the group of ﬁnite subsets of Z, operation XOR

• Constructs a Zﬁn-graded ring k M [xn | n ∈ Z] C := hom(A , ι A ) =∼ 1 J 1 (x2 + n = x2 + m | m, n ∈ ) J∈ n m Z Zﬁn√ ∼ = k[z][ z − n | n ∈ Z]

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

Theorem (Smith 2011) There is an equivalence of categories

gr-A1 ≡ gr-(C, Zﬁn).

February 13, 2018 Z-graded rings 23 / 40 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)

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

gr-A1 ≡ gr(C, Zﬁn) ≡ coh(χ).

Question (Smith) What happens in positive characteristic?

February 13, 2018 Z-graded rings 24 / 40 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

Rhx, yi R(σ, a) = 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 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× Rhx, yi R(σ, a) =∼ xy = a yx = σ−1(a) xz = σ(z)x yz = σ−1(z)y

Two roots α, β of a are congruent if σi(α) = β for some i ∈ Z.

Example (The ﬁrst Weyl algebra) Take R = k[z], σ(z) = z + 1, and a = z k k [z]hx, yi ∼ hx, yi R(σ, z) = = = A1. xy = z yx = z − 1 (xy − yx − 1) xz = (z + 1)x yz = (z − 1)y

February 13, 2018 Z-graded rings 26 / 40 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 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:

−2 −1 2 σ (α2) σ (α2) α2 σ(α2) σ (α2)

−2 −1 α1 2 σ (α1) σ (α1) σ(α1) σ (α1)

• Parametrized by MaxSpec R (mostly).

• Along the σ-orbit of each root αi of f , two modules analogous to Xhni and Yhni.

• If the multiplicity of αi is ni, there are two length ni-irreducible modules whose factors are all “X” or all “Y”.

February 13, 2018 Z-graded rings 28 / 40 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 “Xhni” and “Yhni” and ﬁxing all other simple modules.

Corollary (W, Spicer) Qn ∼ i=1 Zﬁn = Zﬁn ≤ Pic(gr-A).

February 13, 2018 Z-graded rings 29 / 40 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 Fγ. As long as Fγ, Fδ, and Fγ+δ satisfy some technical condition for each γ, δ ∈ Γ, then M R = Homgr-(S,G)(S, FγS) γ∈Γ L is an associative Γ-graded ring. Further, if γ∈Γ FγS 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 L γ∈Γ FγS generates but small enough that R is nice.

February 13, 2018 Z-graded rings 30 / 40 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 Zﬁn-graded ring B such that

gr-A ≡ gr-(B, Zﬁn).

Corollary (W, Spicer) Spec B gr-A ≡ coh k . Spec Zﬁn

Question What are the properties of the commutative rings B and the quotient stacks?

February 13, 2018 Z-graded rings 31 / 40 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, z2).

Conjecture For a non-simple GWA A, there is a simple GWA A0 such that

qgr-A ≡ gr-A0.

February 13, 2018 Z-graded rings 32 / 40 Non-simple rings?

Fact (Sierra) There is a Morita context between k[z](σ, z(z − m)) and k[z](σ, z2).

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, z2)?

Question What about other GWAs? Different base rings? Higher degree GWAs?

February 13, 2018 Z-graded rings 33 / 40 (A: Should be hard.)

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 ringQ gr(A). −1 • Fact: Qgr(A) = D[t, t ; σ] for a division ring D and automorphism σ : D → D. −1 • If Qgr(A) = K[t, t ; σ] for a ﬁeld K, A is birationally commutative. • Restrict to birationally commutative A.

February 13, 2018 Z-graded rings 34 / 40 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 ringQ gr(A). −1 • Fact: Qgr(A) = D[t, t ; σ] for a division ring D and automorphism σ : D → D. −1 • If Qgr(A) = K[t, t ; σ] for a ﬁeld K, A is birationally commutative. • Restrict to birationally commutative A.

February 13, 2018 Z-graded rings 34 / 40 • Let R[t, t−1; σ] denote the skew-Laurent ring with tr = σ(r)t and t−1r = σ−1(r)t−1 for all r ∈ R. i • Z a σ-lonely closed subset of X: σ∗(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)t2 ⊕ · · · | {z } | {z } |{z} |{z} | {z } 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