On polynomial invariant rings in modular invariant theory
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866913402070237184 |
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| author | Kummini, Manoj Mondal, Mandira |
| author_facet | Kummini, Manoj Mondal, Mandira |
| contents | Let $\Bbbk$ be a field of characteristic $p>0$, $V$ a finite-dimensional $\Bbbk$-vector-space, and $G$ a finite $p$-group acting $\Bbbk$-linearly on $V$. Let $S = \Sym V^*$. We show that $S^G$ is a polynomial ring if and only if the dimension of its singular locus is less than $\rank_\Bbbk V^G$. Confirming a conjecture of Shank-Wehlau-Broer, we show that if $S^G$ is a direct summand of $S$, then $S^G$ is a polynomial ring, in the following cases: \begin{enumerate}
\item $\Bbbk = \bbF_p$ and $\rank_\Bbbk V^G = 4$; or
\item $|G| = p^3$. \end{enumerate} In order to prove the above result, we also show that if $\rank_\Bbbk V^G \geq \rank_\Bbbk V - 2$, then the Hilbert ideal $\hilbertIdeal_{G,S}$ is a complete intersection. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_05945 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On polynomial invariant rings in modular invariant theory Kummini, Manoj Mondal, Mandira Commutative Algebra 13A50 Let $\Bbbk$ be a field of characteristic $p>0$, $V$ a finite-dimensional $\Bbbk$-vector-space, and $G$ a finite $p$-group acting $\Bbbk$-linearly on $V$. Let $S = \Sym V^*$. We show that $S^G$ is a polynomial ring if and only if the dimension of its singular locus is less than $\rank_\Bbbk V^G$. Confirming a conjecture of Shank-Wehlau-Broer, we show that if $S^G$ is a direct summand of $S$, then $S^G$ is a polynomial ring, in the following cases: \begin{enumerate} \item $\Bbbk = \bbF_p$ and $\rank_\Bbbk V^G = 4$; or \item $|G| = p^3$. \end{enumerate} In order to prove the above result, we also show that if $\rank_\Bbbk V^G \geq \rank_\Bbbk V - 2$, then the Hilbert ideal $\hilbertIdeal_{G,S}$ is a complete intersection. |
| title | On polynomial invariant rings in modular invariant theory |
| topic | Commutative Algebra 13A50 |
| url | https://arxiv.org/abs/2210.05945 |