The Continuous Hochschild Cochain Complex of a Scheme

Abstract

Let X be a separated finite type scheme over a noetherian base ring K. There is a complex C · (X) of topological O X -modules, called the complete Hochschild chain complex of X. To any O X -module M -not necessarily quasi-coherent -we assign the complex Hom cont O X C · (X), M of continuous Hochschild cochains with values in M. Our first main result is that when X is smooth over K there is a functorial isomorphismThe second main result is that if X is smooth of relative dimension n and n! is invertible in K, then the standard maps π :When M = O X this is the quasi-isomorphism underlying the Kontsevich Formality Theorem.Combining the two results above we deduce a decomposition of the global Hochschild cohomologywhere T X/K is the relative tangent sheaf. Introduction and Statement of ResultsLet K be a noetherian commutative ring and X a separated K-scheme of finite type. The diagonal morphism ∆ : X → X 2 = X × K X is then a closed embedding. This allows us to identify the category Mod O X of O X -modules with its image inside Mod O X 2 under the functor ∆ * .We shall use derived categories freely in this paper, following the reference [RD].Definition 0.1 (Hochschild Cohomology, First Definition). Given an. This definition of Hochschild cohomology was considered by Kontsevich [Ko] and Swan [Sw] among others. We observe that if K is a field, A is a commutative Kalgebra, A e := A⊗ K A, X := Spec A, M is an A-module and M is the quasi-coherent Date: 8.11.01.

Derived categories of sheaves: a skimming

“…Based on these calculations, it should come as no surprise that we have Theorem 6.3 (Hochschild-Kostant-Rosenberg [16], Swan [26], Kontsevich [20], Yekutieli [29]). If X is a smooth, quasi-projective variety over C, then…”

Derived equivalences of K3 surfaces and orientation

“…The algebraic case was studied in [36], for the absolute case see also [28,7]. The very general situation of an arbitrary analytic morphism has been discussed in [4,5].…”

The Atiyah class, Hochschild cohomology and the Riemann-Roch theorem

“…We introduce a global analog of the Hochschild-Kostant-Rosenberg isomorphism ( [9]). This construction has appeared in the literature, see [7], [14] and [15]. Our definition of the global HKR isomorphism appeared in the preprint of the present paper and was used in [4] and [13].…”