Ramification in modular invariant rings
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916627801440256 |
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| author | Kummini, Manoj Mondal, Mandira |
| author_facet | Kummini, Manoj Mondal, Mandira |
| contents | Let $p$ be a prime number, $\Bbbk$ a field of characteristic $p$ and $G$ a finite $p$-group acting on a standard graded polynomial ring $S = \Bbbk[x_1, \ldots, x_n]$ as degree-preserving $\Bbbk$-algebra automorphisms. Assume that $G$ is generated by pseudo-reflections. In our earlier work (\emph{J. Pure Appl. Algebra}, vol. 228, no. 12, 2024) we introduced a composition series of $G$. In this note, we study the height-one ramification for the invariant rings at the consecutive stages of this composition series. We prove a condition for the extension $S^{G}\subseteq S^{G'}$ to split in terms of the Dedekind different $\mathscr{D}_D(S^{G'}/S^G)$. We construct an example illustrating that $\mathscr{D}_D(S^{G'}/S^G)$ need not have `expected' generators. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_17228 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Ramification in modular invariant rings Kummini, Manoj Mondal, Mandira Commutative Algebra 13A50 Let $p$ be a prime number, $\Bbbk$ a field of characteristic $p$ and $G$ a finite $p$-group acting on a standard graded polynomial ring $S = \Bbbk[x_1, \ldots, x_n]$ as degree-preserving $\Bbbk$-algebra automorphisms. Assume that $G$ is generated by pseudo-reflections. In our earlier work (\emph{J. Pure Appl. Algebra}, vol. 228, no. 12, 2024) we introduced a composition series of $G$. In this note, we study the height-one ramification for the invariant rings at the consecutive stages of this composition series. We prove a condition for the extension $S^{G}\subseteq S^{G'}$ to split in terms of the Dedekind different $\mathscr{D}_D(S^{G'}/S^G)$. We construct an example illustrating that $\mathscr{D}_D(S^{G'}/S^G)$ need not have `expected' generators. |
| title | Ramification in modular invariant rings |
| topic | Commutative Algebra 13A50 |
| url | https://arxiv.org/abs/2502.17228 |