The Cohen Macaulay property for noncommutative rings

TL;DR

This paper explores the implications of the Cohen-Macaulay property in noncommutative rings, reviewing known results and introducing new findings, with a focus on homological grade symmetry as a key extension tool.

Contribution

It introduces the hypothesis of homological grade symmetry as a crucial extension for noncommutative Cohen-Macaulay rings, expanding classical homological hierarchies.

Findings

Reviewed classical results on Cohen-Macaulay rings

Proved new results under the maximal Cohen-Macaulay hypothesis

Proposed homological grade symmetry as an extension mechanism

Abstract

Let R be a noetherian ring which is a finite module over its centre Z(R). This paper studies the consequences for R of the hypothesis that it is a maximal Cohen Macaulay Z(R)-module. Old results are reviewed and a number of new results are proved. The additional hypothesis of homological grade symmetry is proposed as the appropriate extra lever needed to extend the classical commutative homological hierarchy to this setting, and results are given offering evidence in support of this proposal.