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 γ2Γ such that AγAδ ⊆ Aγ+δ. • If a 2 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 i2Γ 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) = fp 2 Proj A j a ⊆ pg 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 2 A, s 2 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 < 1 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 2 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 f1; a1;:::; amg. • 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!1 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 < 1 (2) GKdim A < 1 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 2 k n Z, one simple module Mλ • For each n 2 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 !, reversing the grading !(x) = y !(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 D1 = Zfin o D1. 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 j n 2 Z] C := hom(A ; ι A ) =∼ 1 J 1 (x2 + n = x2 + m j m; n 2 ) J2 n m Z Zfinp ∼ = k[z][ z − n j n 2 Z] where deg xn = fng • 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.