Decomposition of the Hochschild complex of a scheme in arbitrary characteristic

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Abstract

Let X be a smooth separated scheme over a noetherian base ring K. The decomposition we are interested in is an isomorphism R ℋomoX×KX(OX, OX) ≅ ⊕qoxq τX/K)[-q] in the derived category D(Mod Ox × K x). Here τX/K is the relative tangent sheaf. Upon passing to cohomology sheaves such an isomorphism recovers the Hochschild-Kostant-Rosenberg Theorem. If K has characteristic 0 there is a decomposition that relies on a particular homomorphism of complexes from poly-tangents to continuous Hochschild cochains. We discuss sheaves of continuous Hochschild cochains on schemes and show why this approach to decomposition fails in positive characteristics. Our main result is a proof of the decomposition valid for any Gorenstein noetherian ring K of finite Krull dimension, regardless of characteristic. The proof is based on properties of minimal injective resolutions.

Original languageEnglish
Pages (from-to)866-896
Number of pages31
JournalCanadian Journal of Mathematics
Volume54
Issue number4
StatePublished - 1 Aug 2002

Keywords

  • Derived categories
  • Gorenstein rings
  • Hochschild cohomology
  • Schemes

ASJC Scopus subject areas

  • Mathematics (all)

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