Authors: Femić, Bojana 
Affiliations: Mathematics 
Mathematical Institute of the Serbian Academy of Sciences and Arts 
Title: Coring Categories and Villamayor–Zelinsky Sequence for Symmetric Finite Tensor Categories
Journal: Applied Categorical Structures
Volume: 29
First page: 485
Last page: 527
Issue Date: 28-Mar-2021
Rank: ~M23
ISSN: 0927-2852
DOI: 10.1007/s10485-020-09624-8
In the preceeding paper we constructed an infinite exact sequence a la Villamayor–Zelinsky for a symmetric finite tensor category. It consists of cohomology groups evaluated at three types of coefficients which repeat periodically. In the present paper we interpret the middle cohomology group in the second level of the sequence. We introduce the notion of coring categories and we obtain that the mentioned middle cohomology group is isomorphic to the relative group of Azumaya quasi coring categories. This result is a categorical generalization of the classical Crossed Product Theorem, which relates the relative Brauer group and the second Galois cohomology group with respect to a Galois field extension. We construct the colimit over symmetric finite tensor categories of the relative groups of Azumaya quasi coring categories and the full group of Azumaya quasi coring categories over vec. We prove that the latter two groups are isomorphic.
Keywords: Brauer–Picard group | Cohomology groups | Finite tensor category | Symmetric monoidal category
Publisher: Springer Link

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