Deformations of affine varieties and the Deligne crossed groupoid

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3 Scopus citations


Let X be a smooth affine algebraic variety over a field K of characteristic 0, and let R be a complete parameter K-algebra (e.g. R=K). We consider associative (resp. Poisson) R-deformations of the structure sheaf OX. The set of R-deformations has a crossed groupoid (i.e. strict 2-groupoid) structure. Our main result is that there is a canonical equivalence of crossed groupoids from the Deligne crossed groupoid of normalized polydifferential operators (resp. polyderivations) of X to the crossed groupoid of associative (resp. Poisson) R-deformations of OX. The proof relies on a careful study of adically complete sheaves. In the associative case we also have to use ring theory (Ore localizations) and the properties of the Hochschild cochain complex. The results of this paper extend previous work by various authors. They are needed for our work on twisted deformation quantization of algebraic varieties.

Original languageEnglish
Pages (from-to)115-143
Number of pages29
JournalJournal of Algebra
StatePublished - 5 May 2013


  • Algebraic varieties
  • DG Lie algebras
  • Deformation quantization
  • Gerbes
  • Primary
  • Secondary
  • Stacks

ASJC Scopus subject areas

  • Algebra and Number Theory


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