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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2508.12517 |
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| _version_ | 1866912541293150208 |
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| author | Tushev, Anatolii V. |
| author_facet | Tushev, Anatolii V. |
| contents | Let $N$ be a minimax nilpotent torsion-free normal subgroup of a soluble group $G$ of finite rank, $R$ be a finitely generated commutative domain and $R*N$ be a crossed product of $R$ and $N$. In the paper we construct a correspondence between an $R*N$-module $W$ and a finite set $M$ of equivalent classes of prime ideals minimal over $Ann_{kA}(W/WI)$, where $kA$ is a group algebra of an abelian minimax group $A$ and $I$ is an appropriative $G$-invariant ideal of $RG$. It is shown that if $Wg \cong W$ for all $ g \in g $ then the action of the group $G$ by conjugations on $N$ can be extended to an action of the group $G$ on the set $M$. The results allow us to apply methods of commutative algebra to the study of $W$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_12517 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On commutative invariants for modules over crossed products of minimax nilpotent linear groups Tushev, Anatolii V. Group Theory Representation Theory 16S34, 20C07, 11R27 Let $N$ be a minimax nilpotent torsion-free normal subgroup of a soluble group $G$ of finite rank, $R$ be a finitely generated commutative domain and $R*N$ be a crossed product of $R$ and $N$. In the paper we construct a correspondence between an $R*N$-module $W$ and a finite set $M$ of equivalent classes of prime ideals minimal over $Ann_{kA}(W/WI)$, where $kA$ is a group algebra of an abelian minimax group $A$ and $I$ is an appropriative $G$-invariant ideal of $RG$. It is shown that if $Wg \cong W$ for all $ g \in g $ then the action of the group $G$ by conjugations on $N$ can be extended to an action of the group $G$ on the set $M$. The results allow us to apply methods of commutative algebra to the study of $W$. |
| title | On commutative invariants for modules over crossed products of minimax nilpotent linear groups |
| topic | Group Theory Representation Theory 16S34, 20C07, 11R27 |
| url | https://arxiv.org/abs/2508.12517 |