Cohen-Macaulay modules of covariants for cyclic $p$-groups
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866913874957041664 |
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| author | Elmer, Jonathan |
| author_facet | Elmer, Jonathan |
| contents | Let $G$ be a a finite group, $k$ a field of characteristic dividing $|G|$ and and $V,W$ $kG$-modules. Broer and Chuai showed that if $\mathrm{codim}(V^G) \leq 2$ then the module of covariants $k[V,W]^G = (k[V]\otimes W)^G$ is a Cohen-Macaulay module, hence free over a homogeneous system of parameters for the invariant ring $k[V]^G$. In the present article we prove a general result which allows us to determine whether a set of elements of a free $A$-module is a generating set, for any $k$-algebra $A$. We use this result to find generating sets for all modules of covariants $k[V,W]^G$ over a homogeneous system of parameters, where $\mathrm{codim}(V^G) \leq 2$ and $G$ is a cyclic $p$-group. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_03677 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cohen-Macaulay modules of covariants for cyclic $p$-groups Elmer, Jonathan Commutative Algebra Rings and Algebras 13A50, 13A02 Let $G$ be a a finite group, $k$ a field of characteristic dividing $|G|$ and and $V,W$ $kG$-modules. Broer and Chuai showed that if $\mathrm{codim}(V^G) \leq 2$ then the module of covariants $k[V,W]^G = (k[V]\otimes W)^G$ is a Cohen-Macaulay module, hence free over a homogeneous system of parameters for the invariant ring $k[V]^G$. In the present article we prove a general result which allows us to determine whether a set of elements of a free $A$-module is a generating set, for any $k$-algebra $A$. We use this result to find generating sets for all modules of covariants $k[V,W]^G$ over a homogeneous system of parameters, where $\mathrm{codim}(V^G) \leq 2$ and $G$ is a cyclic $p$-group. |
| title | Cohen-Macaulay modules of covariants for cyclic $p$-groups |
| topic | Commutative Algebra Rings and Algebras 13A50, 13A02 |
| url | https://arxiv.org/abs/2506.03677 |