  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.
