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 to the Picard group of R. We study under which conditions 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⊆S with the same set of local units and assuming that is induced by a homomorphism of groups G→InvR(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 , 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.