2012/09/27 by Amnon Yekutieli, Yekutieli, Amnon
Mathematics · #16S80 #17B40 #18B05 (Secondary) #53D55 (Primary) 14B10 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA) #Rings and Algebras (math.RA) #math.AG #math.QA #math.RA #msc:14B10 #msc:16S80 #msc:17B40 #msc:18B05 #msc:53D55
paper · pdf · doi:10.48550/arxiv.1209.6243
29 pages. This material was part of arXiv:0905.0488v3, now made into a separate paper with improved presentation
arxiv created 2012/09/27 · openalex publication_date 2012/09/27 · arxiv updated 2012/09/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 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.