Partial generalized crossed products and a seven term exact sequence

Abstract

Given a non-necessarily commutative unital ring R and a unital partial representation Θ of a group G into the Picard semigroup PicS(R) of the isomorphism classes of partially invertible R-bimodules, we construct an abelian group C(Θ/R) formed by the isomorphism classes of partial generalized crossed products related to Θ and identify an appropriate second partial cohomology group of G with a naturally defined subgroup C0(Θ/R) of C(Θ/R). Then we use the obtained results to give an analogue of the Chase-Harrison-Rosenberg exact sequence associated with an extension of non-necessarily commutative rings R⊆S with the same unity and a unital partial representation G→SR(S) of an arbitrary group G into the monoid SR(S) of the R-subbimodules of S. This generalizes the works by Kanzaki and Miyashita.