Lecture 25: Final Class: Noncommutative Geometry

Lecture 25: Final Class: Noncommutative Geometry 25 May 16 - Final class: noncommutative geometry Recall that commutative algebra is closely related to algebraic geometry. A commutative ring 𝑅 corresponds to the affine scheme Spec 𝑅, and modules over 𝑅 correspond to sheaves on Spec 𝑅. In algebraic geometry these concepts are extended to more general non-affine schemes, while also creating powerful geometric intuition and techniques that have had a strong impact on commutative algebra. Noncommutative geometry is an area that grew out of attempts to tie noncommutative algebra to geometry in a similar way. This has not led to as comprehensive a theory as exists in the commutative case. However, it did lead to emergence of a number of different directions, some leading to impressive results. In this lecture we will briefly survey some of these directions. Our list is by no means complete; for example we don’t discuss the direction involving tools from functional analysis (𝐶 ∗ -algebras) developed by A. Connes et al. 25.1 Representation varieties ¯ Then 𝑘-points of Spec(𝑅) correspond to the homomorLet 𝑅 be a finitely generated commutative ring over 𝑘 = 𝑘. phisms of 𝑘-algebras Hom(𝑅, 𝑘) i.e. to one-dimensional representations of the algebra 𝑅. Note that every simple module over a finitely generated commutative algebra 𝑅 is one-dimensional (Hilbert’s Nullstellensatz). If 𝑅 is instead a finitely generated noncommutative algebra over 𝑘, it is natural to consider the space of all finite-dimensional representations of 𝑅. Let us describe this space (to be denoted Rep(𝑅)). à First of all note that Rep(𝑅) = 𝑛∈Z⋇0 Rep𝑛 (𝑅), where Rep𝑛 (𝑅) is the space of 𝑛-dimensional representations of 𝑅. Every element of Rep𝑛 (𝑅) corresponds to a homomorphism 𝜑 : 𝑅 → Mat𝑛 (𝑘) i.e. 𝜑 ∈ Hom(𝑅, Mat𝑛 (𝑘)). Two homomorphisms 𝜑 1 , 𝜑 2 define isomorphic representations iff they lie in the same orbit of GL𝑛 , acting naturally on the space Hom(𝑅, Mat𝑛 (𝑘)) (via its action on Mat𝑛 (𝑘)). Since 𝑅 is finitely generated, 𝑅 = 𝑘 ⟨𝑥 1, . . . , 𝑥𝑚 ⟩/𝐼 , and 2 2 Hom(𝑅, Mat𝑛 (𝑘)) ⊂ (Mat𝑛 )𝑚 = 𝑘 𝑛 𝑚 = A𝑛 𝑚 is a subset of the affine variety (Mat𝑛 )𝑚 cut out by polynomial equations. We can consider this subset as an alge2 braic subvariety of A𝑛 𝑚 . We see that Rep𝑛 (𝑅) = Hom(𝑅, Mat𝑛 (𝑘))/GL𝑛 (𝑘) is the quotient of the algebraic variety Hom(𝑅, Mat𝑛 (𝑘)) by the action of the algebraic group GL𝑛 (𝑘). Space Rep𝑛 (𝑅) is an example of an algebraic stack. This is a replacement for Spec(𝑅). Preprojective algebras are examples of explicit algebras with interesting representation varieties Rep 𝑅. Let 𝑄 be an oriented quiver and let 𝑄 be the corresponding double quiver. For an edge 𝑒 of 𝑄 we will denote by 𝑒 + , 𝑒 − the corresponding edges of 𝑄. Let 𝑃 (𝑄) be the quiver algebra of 𝑄 modulo the relation ∑︁ ∑︁ 𝑒 −𝑒+ − 𝑒 +𝑒 − = 0. (3) 𝑒 𝑒 For example, let 𝑄 be a cyclic quiver consisting of 𝑛 vertices labeled by the elements of Z/𝑛Z (vertices [𝑖], [𝑖 + 1] are connected by the edge). Quiver 𝑄 has vertices labeled by Z/𝑛Z, edges of this quiver are [𝑖] ← [𝑖 +1] and [𝑖 +1] ← [𝑖], [𝑖] ∈ Z/𝑛Z. Pick 𝜁 ∈ 𝑘 of order 𝑛 and consider the action Z/𝑛Z ↷ 𝑘 [𝑥, 𝑦] given by [1] · 𝑥 = 𝜁 𝑥, [1] · 𝑦 = 𝜁 −1𝑦. We have ∼ (Z/𝑛Z)#𝑘 [𝑥, 𝑦] −→ 𝑃 (𝑄), the isomorphism is given by: ∑︁ ∑︁ ∑︁ 1 ⊗ 𝑥 ↦→ 𝑒 [𝑖+1]←[𝑖 ] , 1 ⊗ 𝑦 ↦→ 𝑒 [𝑖 ]←[𝑖+1] , [1] ⊗ 1 ↦→ 𝜁 𝑖 𝑒 [𝑖 ] . [𝑖 ] ∈Z/𝑛Z [𝑖 ] ∈Z/𝑛Z [𝑖 ] ∈Z/𝑛Z The isomorphism above induces the equivalence between the categories of 𝑃 (𝑄) and (Z/𝑛Z)#𝑘 [𝑥, 𝑦]-modules. Let É us describe this equivalence explicitly. Module (𝑀 [𝑖 ] ) [𝑖 ] ∈Z/𝑛Z over 𝑃 (𝑄) goes to 𝑀 := [𝑖 ] ∈Z/𝑛Z 𝑀 [𝑖 ] , where the 𝑖 action of [1] ∈ Z/𝑛Z on 𝑀 [𝑖 ] is given by 𝜁 and the action of 𝑥 : 𝑀 [𝑖 ] → 𝑀 [𝑖+1] is given by 𝑒 [𝑖+1]←[𝑖 ] , the action of Í Í 𝑦 : 𝑀 [𝑖+1] → 𝑀 [𝑖 ] is given by 𝑒 [𝑖 ]←[𝑖+1] . The condition [𝑖 ] ∈Z/𝑛Z 𝑒 [𝑖 ]←[𝑖+1] 𝑒 [𝑖+1]←[𝑖 ] = [𝑖 ] ∈Z/𝑛Z 𝑒 [𝑖+1]←[𝑖 ] 𝑒 [𝑖 ]←[𝑖+1] precisely corresponds to the fact that 𝑥 and 𝑦 commute. This example can be generalized as follows. Recall that finite subgroups Γ in SL(2, 𝑘) correspond to simply laced b be the affine Dynkin graph; the vertices Dynkin graphs 𝐷 (this is known as McKay correspondence, see [15]). Let 𝐷 b of 𝐷 are in bijection with irreps of Γ (see [15]). Then (see [7]) b ∼ Γ#𝑘 [𝑥, 𝑦] 𝑃 ( 𝐷) É where the ∼ is Morita equivalence. It sends a Γ#𝑘 [𝑥, 𝑦]-module 𝑀 to 𝑣 𝑀𝑣 , where 𝑀𝑣 = [𝑀 : 𝜌 𝑣 ] = HomΓ (𝜌 𝑣 , 𝑀) and 𝜌 𝑣 is the irreducible representation of Γ corresponding to the vertex 𝑣. b Γ#𝑘 [𝑥, 𝑦] are not isomorphic in general (they are isomorphic for Remark 25.1: Note that the algebras 𝑃 ( 𝐷), Γ = Z/𝑛Z). Let us describe the representation variety of the algebra 𝑅 = 𝑃 (𝑄). Let us first of all recall that 𝑃 (𝑄) is a certain quotient of the path algebra of the quiver 𝑄 so every representation of 𝑃 (𝑄) can be considered as a representation à of the quiver 𝑄 such that (3) holds. So, Rep(𝑃 (𝑄)) = 𝑑 𝑣 ∈Z⋇0 Rep𝑑 𝑣 (𝑃 (𝑄)), where Rep𝑑 𝑣 (𝑃 (𝑄)) is the space of representations (𝑀𝑣 ) of 𝑄 such that (3) holds and dim 𝑀𝑣 = 𝑑 𝑣 (considered up to an isomorphism). Explicitly, let Ö g 𝑑 (𝑃 (𝑄)) ⊂ (4) Rep Mat𝑑 𝑣 ,𝑑 𝑣′ × Mat𝑑 𝑣′ ,𝑑 𝑣 𝑣 𝑒 : 𝑣→𝑣 ′ be the subvariety consisting of collections of maps such that (3) holds. Then Ö g 𝑑 (𝑃 (𝑄))/ Rep𝑑 𝑣 (𝑃 (𝑄)) = Rep GL(𝑑 𝑣 ). 𝑣 𝑣 ∗ Note now that the RHS Îof (4) is a symplectic vector space (we identify Mat𝑑 𝑣 ,𝑑 𝑣′ × Mat𝑑 𝑣′ ,𝑑 𝑣 with 𝑇 Mat𝑑 𝑣 ,𝑑 𝑣′ via the trace form), and 𝐺 = GL(𝑑 𝑣 ) acts on it preserving the symplectic structure; the equations (3) are zeroes of the Î Î Î moment map 𝜇 : 𝑒 : 𝑣→𝑣 ′ 𝑇 ∗ Mat𝑑 𝑣 ,𝑑 𝑣′ → 𝑣 Mat𝑑 𝑣 for the 𝑣 GL𝑑 𝑣 action. So, Rep𝑑 𝑣 (𝑃 (𝑄)) is obtained from the RHS by Hamiltonian reduction i.e. Ö Rep𝑑 𝑣 (𝑃 (𝑄)) = 𝜇 −1 (0)/ GL𝑑 𝑣 𝑣 is the Hamiltonian reduction of the symplectic vector space Î 𝑒 : 𝑣→𝑣 ′ 𝑇 ∗ Mat 𝑑 𝑣 ,𝑑 𝑣 ′ by Î 𝑣 GL𝑑 𝑣 . Remark 25.2: Ginzburg introduced a notion of a quiver with potential (see [8, Section 4.2]). Potential is an element of the vector subspace 𝐹 cyc of the quiver algebra 𝐹 = 𝑘𝑄 of a quiver 𝑄 generated by cyclic paths. 𝜕 : 𝐹 cyc → 𝐹 defined as follows: given a cyclic path Φ = 𝑒𝑖 1 𝑒𝑖 2 . . . 𝑒𝑖𝑟 , we For any edge 𝑒 ∈ 𝐷 there exists a map 𝜕𝑒 put ∑︁ 𝜕Φ := 𝑒𝑖𝑠+1 𝑒𝑖𝑠+2 . . . 𝑒𝑖𝑟 𝑒𝑖 1 𝑒𝑖 2 . . . 𝑒𝑖𝑠 −1 . 𝜕𝑒 {𝑠 | 𝑒𝑖𝑠 =𝑒 } 𝜕Φ One can then consider the quotient algebra 𝔄(𝑄, Φ) := 𝐹 /( 𝜕Φ 𝜕𝑒 )𝑒 ∈𝑄 , where ( 𝜕𝑒 )𝑒 ∈𝐷 is the two-sided ideal generated by the elements 𝜕Φ 𝜕𝑒 ∈ 𝐹 (see [8, Equation (4.2.1)]). g 𝑑 𝔄(𝑄, Φ) can be described as follows. Consider the space Rep g 𝑑 (𝐹 ) (that is just a vector space), The variety Rep 𝑣 𝑣 Î g b element Φ defines a map Φ : Rep𝑑 𝑣 (𝐹 ) → 𝑣 Mat𝑑 𝑣 sending a representation 𝜌 to 𝜌 (Φ) (recall that Φ ∈ 𝐹 cyc ). g 𝑑 (𝐹 ) → C. One can show that the variety Rep g 𝑑 𝔄(𝑄, Φ) is the critical We obtain a functional 𝜙 := tr b Φ : Rep 𝑣 𝑣 b locus of tr Φ (cf. [8, Section 2.3]). Let us describe how to obtain preprojective algebras via algebras with potentials (see [8, Example 4.3.3]). Con- sider the quiver 𝑄 loop obtained from 𝑄 by attaching an additional edge loop, 𝑡 𝑣 , for every vertex 𝑣. We can loop identify the quiver algebra 𝑘𝑄 with Í 𝑘𝑄 ∗ 𝑘 [𝑡], where ∗ corresponds to the free product of associative 𝑘algebras (this isomorphism sends 𝑡 to 𝑣 𝑡 𝑣 ). Consider the potential ∑︁ ∑︁ ∑︁ Φ := 𝑡𝑣 · (𝑒 +𝑒 − − 𝑒 − 𝑒 + ) = 𝑡 (𝑒 +𝑒 − − 𝑒 − 𝑒 + ). 𝑣 𝑒 𝑒 loop g 𝑑 (𝑃 (𝑄)) × g 𝑑 𝔄(𝑘𝑄 loop, Φ) = 𝑅𝑒𝑝 We then have 𝔄(𝑘𝑄 , Φ) = 𝑃 (𝑄) [𝑡] (see [8, Equation (4.3.4)]), so Rep 𝑣 𝑣 Î 𝑣 Mat𝑑 𝑣 . The potential 𝜙 that we construct above is given by the formula: Ö Ö 𝑇 ∗ Mat𝑑 𝑣 ,𝑑 𝑣′ × Mat𝑑 𝑣 ∋ (𝑣, 𝜉) ↦→ tr(𝜇 (𝑣)𝜉) ∈ C 𝑒 : 𝑣→𝑣 ′ 𝑣 in this case. 25.2 Weyl algebra and deformations Many interesting algebras have no nonzero finite-dimensional representations (in particular, Rep 𝑅 = {0} is not interesting in this case). For example, the Weyl algebra 𝑊 = C⟨𝑥, 𝑦⟩/⟨𝑥𝑦 −𝑦𝑥 − 1⟩ doesn’t. You can see this by noting that tr(𝑥𝑦 − 𝑦𝑥) = 0 (on every finite-dimensional representation), while tr(1𝑛 ) = 𝑛. We can study noncommutative geometry by deforming from the commutative case, this procedure is also called deformation quantization. Consider 𝑊ℏ = C[ℏ] ⟨𝑥, 𝑦⟩/⟨𝑥𝑦 − 𝑦𝑥 − ℏ⟩. When we take ℏ = 1, we recover 𝑊 , and when we take ℏ = 0, we get C[𝑥, 𝑦], which is commutative. Other examples of deformations: If 𝑋 is a smooth affine algebraic variety over a field 𝑘, we can consider Diff ℏ (𝑋 ), the asymptotic differential operators. This is 𝑊 when 𝑋 = A1 . If ℏ = 0, we get O (𝑇 ∗𝑋 ) = Sym O (𝑋 ) (Der(O (𝑋 ))). If 𝔤 is a Lie algebra, let 𝑈 ℏ (𝔤) = 𝑘 ⟨𝔤⟩/𝑥𝑦 − 𝑦𝑥 = ℏ𝑥. If 𝔤 = 𝔤𝔩𝑛 , then define 𝑈¯ℏ (𝔤) = 𝑈 ℏ ⊗𝑍 (𝑈ℏ (𝔤) ) 𝑘 [ℏ] so 𝑈¯0 = O (N ) where N is the nilpotent matrices. There’s also the spherical rational double affine Hecke algebra (DAHA) 𝐴ℏ , also called rational Cherednik algebra, where 𝐴0 = O ((A2 )𝑛 /𝑆𝑛 ). How can we define deformations for non-affine varieties? Previously we deformed the algebra of functions on 𝑋 , now we need to deform the structure sheaf of 𝑋 . There is no obvious way to make a deformation into a sheaf. Here are some ways: a) We can work with a formal parameter. If 𝐴 is flat over 𝑘 [[ℏ]] and complete in the ℏ-topology, let 𝐴0 = 𝐴/ℏ, which is commutative. Exercise: a) if 𝑎¯ = 𝑎 (mod ℏ) is invertible, then so is 𝑎. b) If 𝑈 ⊂ Spec(𝐴) is open, then ¯ 𝑈 is invertible} is a localizing class. c) Localizations form a Zariski sheaf on Spec(𝐴). {𝑎 ∈ 𝐴|𝑎| b) Diff (𝑋 ) can be made into a sheaf on 𝑇 ∗𝑋 with conical topology (𝑈 ⊂ Spec(𝑇 ∗𝑋 ) is open in a conical topology if it is open in Zariski topology and invariant under dilation). c) In characteristic 𝑝, for 𝑊 = 𝑘 ⟨𝑥, 𝑦⟩/𝑥𝑦 − 𝑦𝑥 = 1 has 𝑥 𝑝 , 𝑦 𝑝 ∈ 𝑍 (𝑊 ). Hence, 𝑊 is a sheaf on Spec 𝑘 [𝑥 𝑝 , 𝑦 𝑝 ]. 25.3 Coh(𝑋 ) and 𝐷 ♭ (Coh(𝑋 )) The previous two subsections described approaches closely tied to the usual commutative algebraic geometry. The next relies on it as a source of motivation for conjectures rather than trying to relate a noncommutative structure to a specific commutative ring or variety. An algebraic variety 𝑋 can be studied via the category Coh(𝑋 ) on 𝐷 ♭ (Coh(𝑋 )). É If 𝑋 = Spec(𝑅) is affine, then Coh(𝑋 ) = 𝑅-mod. If 𝑌 is projective over 𝑘 and 𝑌 = Proj(𝐴), 𝐴 = 𝑛≥0 𝐴𝑛 , 𝐴0 = 𝑘, then Coh(𝑌 ) is the Serre quotient of graded finitely generated 𝐴-modules by graded finite-dimensional 𝐴-modules. Theorem 25.3 (Serre): Let 𝑋 be an algebraic variety over a field 𝑘. Then 𝑋 is smooth iff Coh(𝑋 ) has finite homological dimension, i.e. Ext𝑛 (𝐹, 𝐺) = 0 for all 𝑛 > 𝑑 and all 𝐹, 𝐺 ∈ Coh(𝑋 ). It is also known that 𝑋 is projective iff Ext𝑛 (𝐹, 𝐺) is finite-dimensional for all 𝑛, 𝐹, 𝐺 ∈ Coh(𝑋 ). Let 𝑋 be a smooth affine variety. Then Ω𝑋𝑖 , the 𝑖-forms, is HH𝑖 (O (𝑋 )). Recall that HH𝑖 and HH𝑖 are Morita invariant. They can also be defined starting from a category: HH𝑖 = Ext𝑖 (Id, Id) where Id is the identity functor, while HH𝑖 relates to the tensor of bimodules. For 𝑋 smooth and projective, HH𝑖 (Coh(𝑋 )) = Ê 𝑖 𝐻 𝑝 (𝑋, Ω𝑝 ) ≃ 𝐻 dR (𝑋 ) 𝑞−𝑝=𝑖 where the last equivalence is from the Hodge theorem, and the first one is known as the Hochschild-KostantRosenberg isomorphism. ∗ for nonprojective 𝑋 , we can use cyclic homology. The bar complex for HH has cyclic symmeTo recover 𝐻 dR ∗ try: 𝐶 : 𝑎 0 ⊗ · · · 𝑎𝑛 → (−1)𝑛 𝑎𝑛 ⊗ 𝑎 0 · · · ⊗ 𝑎𝑛−1 . Then Bar/(𝐶 − Id)Bar inherits the differential from the bar complex. Its cohomology is Ê 𝑛−2𝑖 HC𝑛 (𝐴) = Ω𝑛 /𝑑 Ω𝑛−1 ⊕ 𝐻 dR (𝑋 ) 𝑖 ≥1 and per HC𝑛 = lim HC𝑛+2𝑖 = ∞ Ê → 𝑛+2𝑖 𝐻 dR (𝑋 ). 𝑖=−∞ For a smooth projective dimension 𝑛 variety 𝑋 we have Serre duality: Ext𝑖 (𝐹, 𝐺) ≃ Ext𝑛−𝑖 (𝐺, 𝐹 ⊗ 𝐾 × ) ∗ . Definition 25.4 (Bondal-Kapranov): Let C be a finite type 𝑘-linear triangualated category. (Finite type means dim𝑘 Ext∗ (𝐴, 𝐵) < ∞ ∀𝐴, 𝐵.) For example, C = 𝐷 ♭ (𝐴-mod) where 𝐴 is finite-dimensional and of finite homological dimension. A Serre functor is a functor 𝑆 : C → C and an isomorphism Hom(𝐴, 𝐵) ≃ Hom(𝐵, 𝑆 (𝐴)) ∗ . The Yoneda lemma implies that if 𝑆 exists, it is unique. Example 25.5: For C = 𝐷 ♭ (Coh(𝑋 )), 𝑆 : 𝐹 ↦→ 𝐹 ⊗ 𝐾𝑥 [𝑛] is a Serre functor. per The Hodge theorem can be restated as a claim that a spectral sequence HH∗ (Coh(𝑋 )) =⇒ HC∗ (Coh(𝑋 )) degenerates for smooth projective 𝑋 . The following striking generalization was proposed (in a slightly different form) by Kontsevich and Soibelman (see [11]). Conjecture 25.6: The above spectral sequence degenerates for any dg-category of finite type over 𝑘. This was partly proved by Dmitry Kaledin and Akhil Mathew, see [10], [14]. 25.4 Artin-Schelter regular algebras and noncommutative projective geometry A projective variety 𝑋 ⊂ P𝑛𝑘 is determined by its homogeneous coordinate ring 𝐴, a graded commutative ring such that 𝑋 = 𝑃𝑟𝑜 𝑗 (𝐴). An important invariant of 𝑋 is the abelian category 𝐶𝑜ℎ(𝑋 ) of coherent sheaves on 𝑋 . It can 𝑔𝑟 𝑔𝑟 𝑔𝑟 be realized as a Serre quotient 𝐴 − 𝑚𝑜𝑑 𝑓 𝑔 /𝐴 − 𝑚𝑜𝑑 𝑓 𝑑 where 𝐴 − 𝑚𝑜𝑑 𝑓 𝑔 is the category of finitely generated graded 𝑔𝑟 𝐴-modules and 𝐴 − 𝑚𝑜𝑑 𝑓 𝑑 is the Serre subcategory of finite dimensional graded modules. One can study noncommutative graded ring that share basic features with commutative ones, thinking of them as homogeneous coordinate rings of (yet to be defined) noncommutative projective varieties. Some beautiful results in that direction were obtained by Artin, Schelter and others in 1990’s. So consider a nonnegatively graded algebras over a field 𝐴 = ⊕𝐴𝑛 , 𝐴0 = 𝑘. Assuming 𝐴 is Noetherian, the category 𝑔𝑟 𝑔𝑟 𝑔𝑟 𝐴 − 𝑚𝑜𝑑 𝑓 𝑔 is abelian, so one can consider 𝐴 − 𝑚𝑜𝑑 𝑓 𝑔 /𝐴 − 𝑚𝑜𝑑 𝑓 𝑑 , the category of coherent sheaves on the purported noncommutative 𝑃𝑟𝑜 𝑗 of 𝐴. One defines a point module over 𝐴 as a cyclic graded module with Poincare series 1/(1 − 𝑡). In the case when 𝐴 is commutative and generated by 𝐴1 point modules are in bijection with points of 𝑋 = 𝑃𝑟𝑜 𝑗 (𝐴). Several important classification results are achieved by considering point modules and a usual (commutative) algebraic variety arising as the moduli space of point modules. We briefly mention a sample classification problem. Recall that a commutative 𝐴 as above has finite homological dimension (equivalently, is regular) iff it is a polynomial algebra. Assuming it is generated by 𝐴1 , we get the homogeneous coordinate ring of 𝑋 = P𝑛𝑘 . Generalization of this simplest projective variety leads to the following definition. An algebra 𝐴 as above is called Artin-Schelter regular if it has finite homological dimension 𝑑, a finite GK dimension and 𝐸𝑥𝑡𝐴𝑖 (𝑘, 𝐴) = 0 for 𝑖 ≠ 𝑑, while 𝐸𝑥𝑡𝐴𝑑 (𝑘, 𝐴) is one dimensional. The work of Artin-Schelter and Artin-Tate-van den Bergh achieved classification of AS regular algebras of dimensions two and three (noncommutative projective lines and planes). We will not present the answer, but mention that it involves beautiful and rather elementary algebro-geometric data, such as an elliptic curve with an automorphism (see [17] and references therein), arising in the process of classification of point modules over 𝐴.