Ramification in modular invariant rings
Manoj Kummini, Mandira Mondal
TL;DR
The paper investigates height-one ramification for modular invariant rings under a finite $p$-group action generated by pseudo-reflections, focusing on the extension $S^{G}\subseteq S^{G'}$ along a penultimate member $G'$ of a composition series. It develops a framework using inertia/decomposition groups and the Dedekind and Noether differents to formulate a splitting criterion for $S^{G}\subseteq S^{G'}$ in terms of $\mathscr D_D(S^{G'}/S^G)$, and it derives explicit factorizations for $\Delta_{A/R}$ in transvection-driven setups. The results include two equivalent expressions for $\Delta_{A/R}$ in a special class of groups and constructive examples showing that $\mathscr D_D(S^{G'}/S^G)$ may lack the expected generators, highlighting inseparability as the source of height-one ramification in the modular setting. The work clarifies ramification phenomena in modular invariant theory and demonstrates limitations of direct-summand intuition for invariant rings.
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.