Partial generalized crossed products and a seven term exact sequence (expanded version)

Abstract

Given a non-necessarily commutative unital ring $R$ and a unital partial representation $\Theta$ of a group $G$ into the Picard semigroup $\mathbf{PicS}(R)$ of the isomorphism classes of partially invertible $R$-bimodules, we construct an abelian group $\mathcal{C}(\Theta /R)$ formed by the isomorphism classes of partial generalized crossed products related to $\Theta$ and identify an appropriate second partial cohomology group of $G$ with a naturally defined subgroup ${\mathcal{C}}_0(\Theta /R)$ of $\mathcal{C}(\Theta /R).$ Then we use the obtained results to give an analogue of the Chase-Harrison-Rosenberg exact sequence associated with an extension of non-necessarily commutative rings $R\subseteq S$ with the same unity and a unital partial representation $G\to {\mathcal{S}}_R(S)$ of an arbitrary group $G$ into the monoid ${\mathcal{S}}_R(S)$ of the $R$-subbimodules of $S.$ This generalizes the works by Kanzaki and Miyashita.