The Gerstenhaber-Schack complex for prestacks and non-commutative deformation theory
22 June 2016
UAntwerp, Campus Middelheim, G0.06 - Middelheimlaan 1 - 2020 Antwerpen
Hoang Dinh Van
PhD defence Hoang Dinh Van - Faculty of Science - Dept. of Mathematics and Computer Science
This thesis is devoted to the study of deformation theory for prestacks. In the first part, we generalize the Gerstenhaber-Schack complex for presheaves into a complex for arbitrary prestacks A, by adding components to the differential. The cohomology of the new complex is shown to compute the Ext-groups in the category of A-bimodules. In particular, we show that the first order deformations of the prestack A are classified by the second cohomology group of the new complex, in analogy with the case of a single algebra. The second part of the thesis contains our study of deformations of a certain class of prestacks in the light of abelian deformation theory. We describe constructions to obtaining the category of quasi-coherent modules over a prestack and we identify a certain class of prestacks which behave well under deformation. In application to smooth quasi-compact separated schemes, we show that our construction is equivalent to Y. Toda's construction in this case.