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 $\underline{\Theta} : \G \to \Picar{R}$ 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.