Deformations of Affine Varieties and the Deligne Crossed Groupoid
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.
Keywords
Cite
@article{arxiv.1209.6243,
title = {Deformations of Affine Varieties and the Deligne Crossed Groupoid},
author = {Amnon Yekutieli},
journal= {arXiv preprint arXiv:1209.6243},
year = {2012}
}
Comments
29 pages. This material was part of arXiv:0905.0488v3, now made into a separate paper with improved presentation