On Cohen-Macaulayness and depth of ideals in invariant rings

TL;DR

This paper explores the conditions under which ideals in invariant rings are Cohen-Macaulay, revealing that such ideals are generally not Cohen-Macaulay unless the entire invariant ring is, with specific results for modular cyclic groups.

Contribution

It establishes new links between the Cohen-Macaulay property of ideals and the structure of invariant rings, especially in modular representations and cyclic groups.

Findings

Ideals in invariant rings are not Cohen-Macaulay unless the ring itself is.

Non-Cohen-Macaulay factorial rings cannot contain Cohen-Macaulay ideals.

The quotient of invariant rings by transfer ideals is Cohen-Macaulay for modular cyclic groups of prime order.

Abstract

We investigate the presence of Cohen-Macaulay ideals in invariant rings and show that an ideal of an invariant ring corresponding to a modular representation of a $p$-group is not Cohen-Macaulay unless the invariant ring itself is. As an intermediate result, we obtain that non-Cohen-Macaulay factorial rings cannot contain Cohen-Macaulay ideals. For modular cyclic groups of prime order, we show that the quotient of the invariant ring modulo the transfer ideal is always Cohen-Macaulay, extending a result of Fleischmann.