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.

在前面的文章中，我们为对称有限张量类别构造了la Villamayor-Zelinsky的无限精确序列。它由以三种类型的系数评估的同调性组组成，这些系数定期重复。在本文中，我们解释了序列第二级中的中间同调群。我们引入了取芯类别的概念，并得到了上述中间同调群与Azumaya准取芯类别的相对群是同构的。该结果是经典交叉积定理的分类概括，该定理将相对的Brauer组和第二个Galois同调群与Galois场扩展相关。vec。我们证明后两组是同构的。