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

TL;DR

This paper develops a framework for partial generalized crossed products over non-commutative rings, establishing a seven-term exact sequence that generalizes previous results by Kanzaki and Miyashita.

Contribution

It introduces a new abelian group of partial crossed products and identifies a second partial cohomology group, extending the Chase-Harrison-Rosenberg sequence to non-commutative ring extensions.

Findings

Constructed an abelian group of partial generalized crossed products.

Identified a second partial cohomology group with a subgroup of the crossed products.

Established a seven-term exact sequence generalizing prior work.

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.