On the depth of quotients of modular invariant rings by transfer ideals

TL;DR

This paper investigates the depth of quotients of modular invariant rings by transfer ideals, providing explicit regular sequences and computing depths for specific groups like cyclic and Klein 4, advancing understanding in modular invariant theory.

Contribution

It introduces an explicit regular sequence in quotients of modular invariant rings by transfer ideals for p-groups and applies this to compute depths in specific cases, including Klein 4 groups.

Findings

Established a regular sequence of length equal to the dimension of fixed points in certain quotients.

Identified conditions where the regular sequence suffices to determine the depth.

Computed the depth of quotients for Klein 4 group representations.

Abstract

Let $G$ be a finite group, and $V$ a finite dimensional vector space over a field $k$ of characteristic dividing the order of $G$. Let $H \leq G$. The transfer map $k[V]^H \rightarrow k[V]^G$ is an important feature of modular invariant theory. Its image is called a transfer ideal $I^G_H$ of $k[V]^G$, and this ideal, along with the quotients $k[V]^G/I^G_H$ are widely studied. In this article we study $k[V]^G/I$, where $I$ is any sum of transfer ideals. Our main result gives an explicit regular sequence of length $\dim(V^G)$ in $k[V]^G/I$ when $G$ is a $p$-group. We identify situations where this is sufficient to compute the depth of $k[V]^G/I$, in particular recovering a result of Totaro. We also study the cases where $G$ is cyclic or isomorphic to the Klein 4 group in greater detail. In particular we use our results to compute the depth of $k[V]^G/I^G_{\{1\}}$ for an arbitrary indecomposable representation of the Klein 4-group.