What is noncommutative algebraic geometry?

What is noncommutative algebraic geometry? Robert Won University of California, San Diego Graduate Algebraic Geometry Seminar, August 2015 August 14, 2015 1 / 20

Overview In the great tradition of algebra, let k = k, char k = 0. 1 From commutative to noncommutative Noncommutative is not commutative Sheaves to the rescue 2 Noncommutative projective schemes 3 Shameless self-plug August 14, 2015 2 / 20

Noncommutative is not commutative Localization is different. Given R commutative and S R multiplicatively closed, r 1 s 1 1 r 2s 1 2 = r 1 r 2 s 1 1 s 1 2 If R noncommutative, can only form RS 1 if S is an Ore set. Definition S is an Ore set if for any r R, s S sr rs. From commutative to noncommutative August 14, 2015 3 / 20

Noncommutative is not commutative Forget localization. Who needs it? Even 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 ring k x, y /(xy qyx) is a noncommutative P 1 but for q n 1 has only three homogeneous ideals (namely (x), (y), and (x, y)). From commutative to noncommutative August 14, 2015 4 / 20

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 s not even a set. Theorem (Reyes, 2012) Suppose F : Ring Set extends the functor Spec : CommRing Set. Then for n 3, F(M n (C)) =. From commutative to noncommutative August 14, 2015 5 / 20

All you need is sheaves 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. Note: From here on, we stick with projective geometry. From commutative to noncommutative August 14, 2015 6 / 20

All you need is sheaves All you need is modules? Theorem Let X = Proj R for a commutative, f.g. k-algebra R generated in degree 1. (1) Every coherent sheaf on X is isomorphic to M for some f.g. graded R-module M. (2) M = Ñ as sheaves if and only if there is an isomorphism M n = N n. Let gr-r be the category of f.g. R-modules. Take the quotient category qgr-r = gr-r/fdim-r The above says qgr-r coh(proj R). From commutative to noncommutative August 14, 2015 7 / 20

All you need is sheaves 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 H 0 (X, L n ). n 1 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). From commutative to noncommutative August 14, 2015 8 / 20

Noncommutative projective schemes A (not necessarily commutative) connected graded k-algebra A is A = k A 1 A 2 such that dim k A i < and A is a f.g. k-algebra. Definition (Artin-Zhang, 1994) The noncommutative projective scheme Proj NC 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 Proj NC A = qgr-a to study A. Noncommutative projective schemes August 14, 2015 9 / 20

Noncommutative projective schemes X a (commutative) projective scheme, L a line bundle, and σ Aut(X). Define L σ = σ L and L n = L L σ L σn 1. The twisted homogeneous coordinate ring is B(X, L, σ) = k n 1 H 0 (X, L n ). 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 projective schemes August 14, 2015 10 / 20

Noncommutative curves Stafford: We take the Hartshorne approach. A a k-algebra, V A a k-subspace generating A spanned by {1, a 1,..., a m }. V 0 = k, V n spanned by monomials of length n in the a i. The Gelfand-Kirillov dimension of A. GKdim k[x 1,..., x m ] = m. GKdim A = lim sup log n (dim k V n ) n So noncommutative projective curves should have GKdim 2. Noncommutative projective schemes August 14, 2015 11 / 20

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. Noncommutative projective schemes August 14, 2015 12 / 20

Noncommutative surfaces The right definition of a noncommutative polynomial ring? Definition A a f.g. connected graded k-algebra is Artin-Schelter regular if (1) gldim A = d < (2) GKdim A < and (3) Ext i A(k, A) = δ i,d k. Behaves homologically like a commutative polynomial ring. k[x 1,..., x m ] is AS-regular of dimension m. Noncommutative P 2 s should be qgr-a for A AS-regular of dimension 3. Noncommutative projective schemes August 14, 2015 13 / 20

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) A = B(X, L, σ) for X = P 2 or X = P 1 P 1 or (b) A B(E, L, σ) for an elliptic curve E. Or noncommutative P 2 s are either commutative or contain a commutative curve. Other noncommutative surfaces (noetherian connected graded domains of GKdim 3)? Noncommutative P 3 (AS-regular of dimension 4)? Noncommutative projective schemes August 14, 2015 14 / 20

All you need is modules Can attempt to do any geometry that only relies on the module category. If X is a commutative projective k-scheme, for each x X, there is the skyscraper sheaf k(x) coh(x). So the simple objects of coh(x) correspond to points of X. You can do cohomology (although I can t). Noncommutative projective schemes August 14, 2015 15 / 20

All you need is modules A point module is a graded right module M such that M is cyclic, generated in degree 0, and has dim k M n = 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 Proj NC A = simple modules. A is called strongly noetherian if A k R is noetherian for all commutative R. If A strongly noetherian then its point modules are parametrized by a commutative projective scheme. There exist noetherian rings that are not strongly noetherian (see Rogalski s thesis). Noncommutative projective schemes August 14, 2015 16 / 20

Z-graded rings Throughout, graded really meant N-graded. Way back in our first example A 1 = k x, y /(xy yx 1) is graded by Z with deg x = 1, deg y = 1. Interesting geometry! Theorem (Smith, 2011) There is a Z fin -graded commutative ring C and a quotient stack χ such that gr-a gr-c qcoh(χ). Shameless self-plug August 14, 2015 17 / 20

Z-graded rings Other Z-graded geometry? Generalized Weyl algebras A, generalizations of the Weyl algebra, GKdim A = 2. Theorem (Bell-Rogalski, 2015) Every simple Z-graded domain of GKdim 2 is graded Morita equivalent to a GWA. Theorem (W) If A is a GWA defined by a quadratic polynomial, there exists a Z fin -graded commutative ring B such that qgr-a gr-b. Shameless self-plug August 14, 2015 18 / 20

Z-graded rings Some of these rings B are interesting. B = Graded by Z fin with deg x n = {n}. k[z][x n n Z] (x 2 n (z + n) 2 n Z) B non-noetherian, non-domain, infinitely many prime ideals. B reduced, Kdim B = 1. (Thanks to Cal and Daniel!) [ ] qgr-a gr-b? Spec B qcoh Spec kz fin What is the geometry of this stack? Shameless self-plug August 14, 2015 19 / 20

References Artin and Stafford, Noncommutative graded domains with quadratic growth, Inv. Math. (1995). Artin, Tate, and Van den Bergh, Some algebras associated to automorphisms of elliptic curves (1990). Artin and Van den Bergh, Twisted Homogeneous Coordinate Rings, J. Alg. (1990). Artin and Zhang, Noncommutative Projective Schemes, Adv. Math. (1994). Brandenburg, Rosenberg s Reconstruction Theorem (after Gabber), (2013). arxiv:1310.5978 Reyes, Obstructing extensions of the functor Spec to noncommutative rings, Isr. J. Math. (2012). arxiv:1101.2239 Rogalski, An Introduction to Noncommutative Projective Geometry (2014). arxiv:1403.3065 Sierra, Rings graded equivalent to the Weyl algebra, J. Alg. (2009). arxiv:0711.1494 Smith, A quotient stack related to the Weyl algebra, J. Alg. (2011). arxiv:0806.0176 Stafford, Noncommutative Projective Geometry, Proc. ICM, (2002). arxiv:math/0304210 August 14, 2015 20 / 20