Hochschild cohomology of quasiprojective schemes
This paper studies a new definition (studied independently by \textit{M. Kontsevich}) of Hochschild cohomology for schemes. Namely, let \(X\) be a separated scheme of finite type over a field \(k\), and let \({\mathcal F}\) be a sheaf of \({\mathcal O}_X\)-modules. Let \(\delta: X\to X\times X\) be the diagonal morphism, and identify \({\mathcal O}_X\) with \(\delta_*({\mathcal O}_X)\), \({\mathcal F}\) with \(\delta_*({\mathcal F})\). Define the Hochschild cohomology of \(X\) with coefficients in \({\mathcal F}\) to be \(H^n({\mathcal O}_X, {\mathcal F}): =\text{Ext}^n_{{\mathcal O}_{X\times X}} ({ \mathcal O}_X, {\mathcal F})\). This definition is motivated by the definition \(H^n(A,M) =\text{Ext}^n_{A^e} (A,M)\) of Hochschild cohomology for an \(A\)-module \(M\) \((A\) a commutative \(k\)-algebra, \(A^e=A \otimes_kA)\). Two other definitions of the Hochschild cohomology of \(X\) with coefficients in \({\mathcal F}\) are also discussed, namely \(HH^n(X,{\mathcal F}): =\mathbb{E}\text{xt}^n_{{\mathcal O}_X} ({\mathcal C}., {\mathcal F})\), where \({\mathcal C}.\) is the sheafification of the cyclic bar complex and \(\mathbb{E}\)xt denotes hyperext (an approach suggested by Loday for Hochschild homology), and a definition given by Gerstenhaber and Schack [cf. \textit{M. Gerstenhaber} and \textit{S. D. Schack}, ``Algebraic cohomology and deformation theory'', in: Deformation theory of algebras and structures and applications, Nato ASI Ser., Ser. C, 11-264 (1988; Zbl 0676.16022)]. Cyclic cohomology \(HC^n(X, {\mathcal F})\) is defined in a manner similar to \(HH^n(X, {\mathcal F})\). The main result of this paper is that \(H^n({\mathcal O}_X, {\mathcal F})\) is isomorphic to both of the other two definitions of Hochschild cohomology. Each of the three definitions has an associated Hodge spectral sequence. It is shown that if \(X\) is smooth than these spectral sequences are isomorphic. Furthermore it is immediate from the \(H^n({\mathcal O}_X, {\mathcal F})\) definition that if \(X\) is projective and \({\mathcal F}\) is coherent then the Hochschild cohomology and cyclic cohomology are finite-dimensional \(k\)-vector spaces.
scientific article; zbMATH DE number 896564
scientific article