Deformations of affine varieties and the Deligne crossed groupoid

JOURNAL OF ALGEBRA

卷:382

Article

Yekutieli, Amnon¹ 

[1] Ben Gurion Univ Negev, Dept Math, IL-84105 Beer Sheva, Israel

关键词: Deformation quantization;    Algebraic varieties;    Stacks;    Gerbes;    DG Lie algebras;   

DOI  :  10.1016/j.jalgebra.2013.01.038

来源: Elsevier

【 摘 要 】

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. (C) 2013 Elsevier Inc. All rights reserved.

【 授权许可】

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