Ramification in modular invariant rings

TL;DR

This paper investigates the ramification properties of invariant rings under $p$-group actions generated by pseudo-reflections, focusing on the splitting of extensions and the structure of the Dedekind different.

Contribution

It establishes a criterion for extension splitting in invariant rings using the Dedekind different and provides an example showing unexpected generators.

Findings

Condition for extension splitting via Dedekind different

Construction of an example with non-standard generators

Analysis of ramification in invariant rings under $p$-group actions

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.