*Title:*
**Deformations of Affine
Varieties and the Deligne Crossed Groupoid***Author:
*Amnon
Yekutieli*Publication
status: Journal of Algebra*
**382**
(2013),
115–143.

*Abstract:
*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[[h]]). We consider associative (resp. Poisson) R-deformations
of the structure sheaf O_X. 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 O_X. 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.

published paper (full text requires permission)

updated 15 March 2013