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Global Theory: Chiral Homology

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Global Theory: Chiral Homology CHAPTER 4 Global Theory: Chiral Homology Qinovnik umiraet, i ordena ego ostats na lice zemli. Koz~ma Prutkov, “Mysli i Aforizmy”, 1860 † 4.1. The cookware This section collects some utensils and implements needed for the construction of chiral homology. A sensible reader should skip it, returning to the material when necessary. Here is the inventory together with a brief comment on the employment mode. (i) In 4.1.1–4.1.2 we consider homotopy direct limits of complexes and discuss their multiplicative properties. The material will be used in 4.2 where the de Rham complex of a D-module on Ran’s space R(X) is defined as the homotopy direct limit of de Rham complexes of the corresponding D-modules on all the Xn’s with respect to the family of all diagonal embeddings. See [BK], [Se]. (ii) In 4.1.3 and 4.1.4 we discuss the notion of the Dolbeault algebra which is an algebraic version of the ∂¯-resolution. Dolbeault resolutions are functorial and have nice multiplicative properties, so they are very convenient for computing the global de Rham cohomology of D-modules on R(X), in particular, the chiral homology of chiral algebras. An important example of Dolbeault algebras comes from the Thom-Sullivan construction; see [HS] §4. (iii) In 4.1.5 we recall the definitions of semi-free DG modules, semi-free com- mutative DG algebras, and the cotangent complex following [H] (see also [Dr1] and [KrM]). The original construction of the cotangent complex, due to Grothendieck [Gr1], [Il], was performed in the setting of simplicial algebras. We follow the DG L setting of [H] (using the notation ΩF instead of the standard LF/k). (iv) Sections 4.1.6 and 4.1.7 deal with Batalin-Vilkovisky algebras and the corresponding homotopy categories. BV algebras are “quantum deformations” of odd Poisson (alias braid, alias Gerstenhaber) algebras. We will see in 4.3.1 that chiral chain complexes of, respectively, commutative DX -algebras, coisson algebras, and general chiral algebras are naturally commutative DG algebras, odd Poisson algebras, and BV algebras. For a more lively account, see, e.g., [Ge] and [Schw]. (v) A typical example of BV algebra is the Chevalley homology complex of a Lie algebra; from the BV viewpoint it is the BV envelope of the Lie algebra (see 4.1.8). In 4.1.9–4.1.10 we explain what BV envelopes of Lie algebroids are. To † “A civil servant dies, and regalia of his stay on the face of the earth.” Koz’ma Proutkoff, “Thoughts and Aphorisms”, 1860. 275 276 4. GLOBAL THEORY: CHIRAL HOMOLOGY define it, an extra structure – that of the BV extension – is needed. The situation is parallel to that of chiral envelopes from 3.9; we will see in 4.8 that the chiral homology functor transforms chiral envelopes to BV envelopes. The homotopy properties of this construction are discussed in 4.1.11 and 4.1.12; in 4.1.13 we prove a technical statement (to be used in 4.1.18) asserting that while considering BV Lie R-algebroids from the homotopy point of view, one can change R at will by any cofibrant representative in its homotopy class. If R is a plain commutative algebra (i.e., a DG algebra sitting in degree 0) and L a plain Lie R-algebroid, then BV extensions of L are the same as right L-module structures on R. Then the BV envelope is the de Rham-Chevalley complex for this L-module (see 2.9.1); as a mere graded algebra it equals Sym(L[1]). They were considered in [Kos] (the case of the tangent algebroid), [Hue], [X], and (under the name of Calabi-Yau or vertex 0-algebroid structures) in §11 of [GMS2]. (vi) In 4.1.14 and 4.1.15 we consider homotopy unital commutative algebras and BV algebras and show that the corresponding homotopy categories are the same as the homotopy categories of the corresponding strictly unital objects. We need this material since the chiral chain complexes of unital chiral algebras are naturally homotopy unital BV algebras (not the strict ones); see 4.3.4. (vii) In 4.1.16–4.1.18 we discuss perfect complexes, perfect commutative DG algebras, and perfect BV algebras. Perfect commutative algebras are immediate counterparts in the homotopy DG world of the usual smooth algebras, and perfect BV algebras correspond to de Rham complexes for a flat connection on the line bundle ω−1. The cohomology of a perfect BV algebra is finite-dimensional. We show in 4.6.9 that the chiral homology of a (very) smooth commutative DX -algebra is a perfect commutative algebra, and that of a cdo is a perfect BV algebra (see 4.8.5). We refer to [CFK] for a geometric discussion of commutative DG algebras supported in non-positive degrees. The perfectness property for commutative DG algebras should be compared with a much stronger (and deeper) condition on a DG algebra F – perfectness of F as an F ⊗ F -bimodule – introduced by M. Kontsevich (the latter property makes sense for arbitrary associative DG algebras). 4.1.1. Homotopy direct limits. Below A is a category which we tacitly assume to be closed under direct sums of sufficiently high cardinality. (i) Let B be a simplicial set. So for every n ≥ 0 we have a set Bn and for every monotonous map of intervals α : [0, . , m] → [0, . , n] we have the corresponding map ∂α : Bn → Bm compatible with the composition of the α’s. Denote by C(B, A) the category of homology type coefficient systems on B with coefficients in A. Thus F ∈ C(B, A) is a rule that assigns to every simplex b ∈ Bn F an object Fb ∈ A and to every α as above a morphism ∂α = ∂α : Fb → F∂α(b) compatible with the composition of the α’s. Denote by Cs(A) the category of simplicial objects in A. We have a functor (4.1.1.1) Cs : C(B, A) → Cs(A) where Cs(F )n := ⊕ Fb and the structure maps ∂α : Cs(F )n → Cs(F )m are direct b∈Bn F sums of the corresponding morphisms ∂α . (ii) Let P be a small category. It yields a simplicial set BP (the classifying simplicial set; see [Se] or [Q3]); an n-simplexp ˜ ∈ BPn is a diagram (p0 → · · · → pn) 4.1. THE COOKWARE 277 in P. Consider the category Fun(P, A) of functors F : P → A, p 7→ Fp. There is a fully faithful embedding (4.1.1.2) Fun(P, A) ,→ C(BP, A) which identifies a functor F with the system F(p0→···→pn) = Fp0 . (iii) Suppose that A is a k-category. Let C(A) be the category of complexes in A. By Dold-Puppe one has the fully faithful embedding (4.1.1.3) Norm : Cs(A) ,→ C(A) which identifies Cs(A) with the full subcategory of complexes having degrees ≤ 0. (iv) Suppose that A is a DG (super) k-category.1 We have the functor tot : C(A) → A which sends a complex C = (C·, d) to an object tot C ∈ A such that tot C = ⊕Cn[−n] as a plain object without differential; the structure differential is the sum of d and the structure differentials of Cn[−n]. Set (4.1.1.4) Norm := tot Norm : Cs(A) → A For F ∈ C(B, A) as in (i) set C(B, F ) := Norm Cs(B, F ). For a functor F : P → A as in (ii) set C(P,F ) := C(BP,F ); this is the homotopy direct P-limit of F . Notice that the DG structure on A yields DG structures on all the above categories, and (4.1.1.1)–(4.1.1.4) are DG functors. Remark. Suppose that A = C(B) for an abelian category B. Let−→ lim F ∈ C(B) be the plain direct P-limit of F . There is an obvious canonical morphism of complexes C(P,F ) → −→lim F . If F takes values in B ⊂ A, then the complex C(P,F ) has degrees ≤ 0 and the above morphism yields an isomorphism H0C(P,F ) −→∼ −→lim F . Exercise. Let S ∈ P be an object such that the group G := Aut P acts freely on every set Hom(S, T ), T ∈ P. Let A ∈ A be an object equipped with an Aut S-action, and let F =Ind A be the corresponding induced functor, F (T ) := A[Hom(S, T )]G. Then the complex C(P,F ) computes the homology of G with coefficients in A. In particular, if G is finite and we are dealing with k-categories where k is a field of characteristic 0, then the map C(P,F ) → −→lim F = AG is a quasi-isomorphism. 4.1.2. Operations. From now on A is a pseudo-tensor category; we assume that direct sums in A are compatible with operations.2 (i) Consider the category of k-modules. This is a tensor category, so both categories of simplicial k-modules and k-complexes are tensor categories. Norm is not a tensor functor. However for any finite set of simplicial k-modules Ni there is a canonical functorial morphism of complexes (4.1.2.1) c = c{Ni} : ⊗Norm(Ni) → Norm(⊗Ni) 1We always assume A to be pretriangulated (the cones of morphisms are well defined, see [BoKa] or [Dr1]) and closed under appropriate direct limits. 2 Q I.e., PI ({ ⊕ Fαi},G) = PI ({Fαi},G). α∈Ai 278 4.
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