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 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. Subject Classification (2010): 18D10, 16W30, 19D23.
Mathematics
A deeper understanding of recent computations of the Brauer group of Hopf algebras is attained by explaining why a direct product decomposition for this group holds and describing the non-interpreted factor occurring in it. For a Hopf algebra B in a braided monoidal category C, and under certain assumptions on the braiding (fulfilled if C is symmetric), we construct a sequence for the Brauer group BM(C; B) of B-module algebras, generalizing Beattie's one. It allows one to prove that BM(C; B) ∼ = Br(C) × Gal(C; B), where Br(C) is the Brauer group of C and Gal(C; B) the group of B-Galois objects. We also show that BM(C; B) contains a subgroup isomorphic to Br(C) × H 2 (C; B, I), where H 2 (C; B, I) is the second Sweedler cohomology group of B with values in the unit object I of C. These results are applied to the Brauer group of a quasi-triangular Hopf algebra that is a Radford biproduct B × H, where H is a usual Hopf algebra over a field K, the Hopf subalgebra generated by the quasi-triangular structure R is contained in H and B is a Hopf algebra in the category H M of left H-modules. The Hopf algebras whose Brauer group was recently computed fit this framework. We finally show that BM(K, H, R)×H 2 ( H M; B, K) is a subgroup of the Brauer group BM(K, B ×H, R), confirming the suspicion that a certain cohomology group of B × H (second lazy cohomology group was conjectured) embeds into BM(K, B × H, R). New examples of Brauer groups of quasi-triangular Hopf algebras are computed using this sequence.
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