A criterion of Cohen Macaulayness of the form module

TL;DR

This paper introduces a new criterion for determining when the form module associated with an ideal in a Noetherian local ring is Cohen-Macaulay, based on local cohomology vanishing conditions.

Contribution

It provides a novel criterion linking Cohen-Macaulayness of the form module to local cohomology, extending existing characterizations.

Findings

Criterion characterizes Cohen-Macaulayness via local cohomology vanishing.

Applicable to Noetherian local rings and finitely generated modules.

Enhances understanding of the structure of form modules in commutative algebra.

Abstract

Let $\mathfrak{q}$ be an ideal of a Noetherian local ring $(A,\mathfrak{m})$ and $M$ a non-zero finitely generated $A$-module. We present a criterion of Cohen-Macaulayness of the form module $G_M(\mathfrak{q})$ in terms of (non-)vanishing of a variation of local cohomology introduced in \cite{KSch}.