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

Let $R$ be a ring with a set of local units, and a homomorphism of groups $\underset{‾}{\mathrm{\Theta }}:\text{G}\to \text{Picar}R$ to the Picard group of $R$. We study under which conditions $\underset{‾}{\mathrm{\Theta }}$ 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 $\underset{‾}{\mathrm{\Theta }}$ is induced by a homomorphism of groups $\text{G}\to \text{Inv}RS$ 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 $\text{G}$ 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.