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

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.