Partial generalized crossed products, Brauer groups and a comparison of seven-term exact sequences

Abstract

Given a unital partial action $\alpha $ of a group $G$ on a commutative ring $R$ we denote by $ {\bf PicS} _{R^{\alpha}}(R) $ the Picard monoid of the isomorphism classes of partially invertible $R$-bimodules, which are central over the subring $R^{\alpha} \subseteq R$ of $\alpha$-invariant elements, and consider a specific unital partial representation $\Theta : G \to {\bf PicS} _{R^{\alpha}}(R), $ along with the abelian group $\mathcal {C}(\Theta/R)$ of the isomorphism classes of partial generalized crossed products related to $\Theta,$ which already showed their importance in obtaining a partial action analogue of the Chase-Harrison-Rosenberg seven-term exact sequence. We give a description of $\mathcal {C}(\Theta/R)$ in terms partial generalized products of the form $\mathcal D(f \Theta)$ where $f$ is partial $1$-cocycle of $G$ with values in a submonoid of $ {\bf PicS}_{R^{\alpha}}(R).$ Assuming that $G$ is finite and that $R^{\alpha} \subseteq R$ is a partial Galois extension, we prove that any Azumaya $R^\alpha$-algebra, containing $R$ as a maximal commutative subalgebra, is isomorphic to a partial generalized crossed product. Furthermore, we show that the relative Brauer group $\mathcal B(R/R^\alpha)$ can be seen as a quotient of $\mathcal {C}(\Theta/R)$ by a subgroup isomorphic to the Picard group of $R.$ Finally, we prove that the analogue of the Chase-Harrison-Rosenberg sequence, obtained earlier for partial Galois extensions of commutative rings, can be derived from a recent seven-term exact sequence established in a non-commutative setting.