On the notion of Cohen-Macaulayness for non Noetherian rings

TL;DR

This paper explores different characterizations of Cohen-Macaulay rings beyond the Noetherian setting, compares their equivalences, and generalizes a key result on invariant rings to non-Noetherian rings.

Contribution

It provides a comparison of Cohen-Macaulay characterizations in non-Noetherian rings and generalizes Hochster-Eagon's result to this broader context.

Findings

Comparison of Cohen-Macaulay characterizations in non-Noetherian rings

Generalization of Hochster-Eagon theorem to non-Noetherian invariant rings

Partial resolution of Glaz's conjecture

Abstract

There exist many characterizations of Noetherian Cohen-Macaulay rings in the literature. These characterizations do not remain equivalent if we drop the Noetherian assumption. The aim of this paper is to provide some comparisons between some of these characterizations in non Noetherian case. Toward solving a conjecture posed by Glaz, we give a generalization of the Hochster-Eagon result on Cohen-Macaulayness of invariant rings, in the context of non Noetherian rings.