Invertible unital bimodules over rings with local units, and related exact sequences of groups II

TL;DR

This paper investigates conditions under which a group homomorphism to the Picard group of a ring with local units determines a generalized crossed product, and constructs an exact sequence relating cohomology groups in this context.

Contribution

It generalizes existing theories by establishing conditions for factor maps and constructing an analogue of the Chase-Harrison-Rosenberg sequence for rings with local units.

Findings

Conditions for homomorphisms to be determined by factor maps

Construction of an exact sequence involving cohomology groups

Extension of previous unital ring results to rings with local units

Abstract

Let $R$ be a ring with a set of local units, and a homomorphism of groups $\underline{\Theta} : \G \to \Picar{R}$ to the Picard group of $R$. We study under which conditions $\underline{\Theta}$ is determined by a factor map, and, henceforth, it defines a generalized crossed product with a same set of local units. Given a ring extension $R \subseteq S$ with the same set of local units and assuming that $\underline{\Theta}$ is induced by a homomorphism of groups $\G \to \Inv{R}{S}$ to the group of all invertible $R$-sub-bimodules of $S$, then we construct an analogue of the Chase-Harrison-Rosenberg seven terms exact sequence of groups attached to the triple $(R \subseteq S, \underline{\Theta})$, which involves the first, the second and the third cohomology groups of $\G$ with coefficients in the group of all $R$-bilinear automorphisms of $R$. Our approach generalizes the works by Kanzaki and Miyashita in the unital case.