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arXiv · 2502.17228

Ramification in modular invariant rings

Manoj Kummini·Mandira Mondal
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Abstract

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.

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Manoj Kummini, Mandira Mondal. 2025-02-24. Ramification in modular invariant rings. https://arxiv.org/abs/2502.17228

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