On the formal grade of finitely generated modules over local rings

TL;DR

This paper investigates the formal grade of finitely generated modules over local rings, establishing inequalities, characterizations of Cohen-Macaulay modules, and generalizations related to invariant rings.

Contribution

It introduces a new inequality for the formal grade, provides a novel characterization of Cohen-Macaulay modules, and extends results on Cohen-Macaulayness of invariant rings.

Findings

grade(a,M)  \u2265  M - ext{cd}_{a}(M)

Characterization of Cohen-Macaulay modules via formal grade

Generalization of Hochster-Eagon result on invariant rings

Abstract

Let \fa be an ideal of a local ring (R,\fm) and M a finitely generated R-module. This paper concerns the notion \fgrade(\fa,M), the formal grade of M with respect to \fa (i.e. the least integer i such that {\vpl}_nH^i_{\fm}(M/\fa^n M)\neq 0). We show that \fgrade(\fa,M)\geq \depth M-\cd_{\fa}(M), and as a result, we establish a new characterization of Cohen-Macaulay modules. As an application of this characterization, we show that if M is Cohen-Macaulay and L a pure submodule of M with the same support as M, then \fgrade(\fa,L)=\fgrade(\fa,M). Also, we give a generalization of the Hochster-Eagon result on Cohen-Macaulayness of invariant rings.