Partial generalized crossed products and a seven term exact sequence

Given a non-necessarily commutative unital ring R and a unital partial representation Theta 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(Theta/R) formed by the isomorphism classes of partial generalized crossed products related to Theta and identify an appropriate second partial cohomology group of G with a naturally defined subgroup C0(Theta/R) of C(Theta/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 subset of 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. (c) 2023 Elsevier B.V. All rights reserved.

Automated summary

In this study, we construct an abelian group C(Theta/R) formed by the isomorphism classes of partial generalized crossed products related to a unital partial representation Theta of a group G into the Picard semigroup PicS(R) of a non-necessarily commutative unital ring R. We identify an appropriate second partial cohomology group of G with a naturally defined subgroup C0(Theta/R) of C(Theta/R). Using these results, we generalize the works by Kanzaki and Miyashita by giving an analogue of the Chase-Harrison-Rosenberg exact sequence associated with an extension of rings and a unital partial representation of an arbitrary group into the monoid of R-subbimodules.