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| Natura: | Preprint |
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2024
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| Accesso online: | https://arxiv.org/abs/2411.00494 |
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| _version_ | 1866910680608669696 |
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| author | Dokuchaev, Mikhailo Pinedo, Hector Rocha, Itailma |
| author_facet | Dokuchaev, Mikhailo Pinedo, Hector Rocha, Itailma |
| contents | Given a unital partial action $α$ of a group $G$ on a commutative ring $R$ we denote by $ {\bf PicS} _{R^α}(R) $ the Picard monoid of the isomorphism classes of partially invertible $R$-bimodules, which are central over the subring $R^α \subseteq R$ of $α$-invariant elements, and consider a specific unital partial representation $Θ: G \to {\bf PicS} _{R^α}(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 PicS}_{R^α}(R).$ Assuming that $G$ is finite and that $R^α \subseteq R$ is a partial Galois extension, we prove that any Azumaya $R^α$-algebra, containing $R$ as a maximal commutative subalgebra, is isomorphic to a partial generalized crossed product. Furthermore, we show that the relative Brauer group $\mathcal B(R/R^α)$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_00494 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Partial generalized crossed products, Brauer groups and a comparison of seven-term exact sequences Dokuchaev, Mikhailo Pinedo, Hector Rocha, Itailma Rings and Algebras Given a unital partial action $α$ of a group $G$ on a commutative ring $R$ we denote by $ {\bf PicS} _{R^α}(R) $ the Picard monoid of the isomorphism classes of partially invertible $R$-bimodules, which are central over the subring $R^α \subseteq R$ of $α$-invariant elements, and consider a specific unital partial representation $Θ: G \to {\bf PicS} _{R^α}(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 PicS}_{R^α}(R).$ Assuming that $G$ is finite and that $R^α \subseteq R$ is a partial Galois extension, we prove that any Azumaya $R^α$-algebra, containing $R$ as a maximal commutative subalgebra, is isomorphic to a partial generalized crossed product. Furthermore, we show that the relative Brauer group $\mathcal B(R/R^α)$ 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. |
| title | Partial generalized crossed products, Brauer groups and a comparison of seven-term exact sequences |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2411.00494 |