Ramification in modular invariant rings

Abstract

Let $p$ be a prime number, $\mathbb{k}$ a field of characteristic $p$ and $G$ a finite $p$-group acting on a standard graded polynomial ring $S=\mathbb{k}[x_1,\dots ,x_n]$ as degree-preserving $\mathbb{k}$-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 $S^G\subseteq S^{G^{\prime }}$ to split in terms of the Dedekind different ${\mathcal{D}}_D(S^{G^{\prime }}/S^G)$. We construct an example illustrating that ${\mathcal{D}}_D(S^{G^{\prime }}/S^G)$ need not have `expected' generators.