2003/03/03 by Frank Schuhmacher, Schuhmacher, Frank
Mathematics · #13D03 #13D10 #14F43 #18G10 #18G30 #32A99 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Symplectic Geometry (math.SG) #math.AG #math.SG #msc:13D03 #msc:13D10 #msc:14F43 #msc:18G10 #msc:18G30 #msc:32A99
paper · pdf · doi:10.48550/arxiv.math/0303029
41 pages
arxiv created 2003/03/03 · openalex publication_date 2003/03/03 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The classical HKR-theorem gives an isomorphism of the n-th Hochschild cohomology of a smooth algebra and the n-th exterior power of its module of Kähler differentials. Here we generalize it for simplicial, graded and anticommutative objects in ``good pairs of categories''. We apply this generalization to complex spaces and noetherian schemes and deduce two decomposition theorems for their (relative) Hochschild cohomology (special cases of those were recently shown by Buchweitz-Flenner and Yekutieli). The first one shows that Hochschild cohomology contains tangent cohomology: \HHn(X/Y,\sM)=\coprodi-j=n\Exti(\dachj\LL(X/Y),\sM). The left side is the n-th Hochschild cohomology of X over Y with values in \sM. The right hand-side contains the n-th relative tangent cohomology \Extn(\LL(X/Y),\sM) as direct factor. The second consequence is a decomposition theorem for Hochschild cohomology of complex analytic manifolds and smooth schemes in characteristic zero: \HHn(X)=\coprodi-j=nHi(X,\dachj\sTX). On the right hand-side we have the sheaf cohomology of the exterior powers of the tangent complex.