2024/11/01 by Mikhailo Dokuchaev, Dokuchaev, Mikhailo, Héctor Pinedo +3
Mathematics · #Advanced Algebra and Geometry #Analytic and geometric function theory #FOS: Mathematics #Finite Group Theory Research #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2411.00494
openalex publication_date 2024/11/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a unital partial action α of a group G on a commutative ring R we denote by \bf PicS Rα(R) the Picard monoid of the isomorphism classes of partially invertible R-bimodules, which are central over the subring Rα ⊆ R of α-invariant elements, and consider a specific unital partial representation Θ: G → \bf PicS Rα(R), along with the abelian group \mathcal C(Θ/R) of the isomorphism classes of partial generalized crossed products related to Θ, which already showed their importance in obtaining a partial action analogue of the Chase-Harrison-Rosenberg seven-term exact sequence. We give a description of \mathcal C(Θ/R) in terms partial generalized products of the form \mathcal D(f Θ) where f is partial 1-cocycle of G with values in a submonoid of \bf PicSRα(R). Assuming that G is finite and that Rα ⊆ R is a partial Galois extension, we prove that any Azumaya Rα-algebra, containing R as a maximal commutative subalgebra, is isomorphic to a partial generalized crossed product. Furthermore, we show that the relative Brauer group \mathcal B(R/Rα) can be seen as a quotient of \mathcal C(Θ/R) by a subgroup isomorphic to the Picard group of R. Finally, we prove that the analogue of the Chase-Harrison-Rosenberg sequence, obtained earlier for partial Galois extensions of commutative rings, can be derived from a recent seven-term exact sequence established in a non-commutative setting.