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 (Theta) under bar : G -> Pic(R) to the Picard group of R. We study under which conditions (Theta) under bar 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 subset of S with the same set of local units and assuming that (Theta) under bar is induced by a homomorphism of groups G -> 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 subset of S, (Theta) under bar), 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. (C) 2012 Elsevier Inc. All rights reserved.