Cohen-Macaulay modules and holonomic modules over filtered rings

TL;DR

This paper investigates the relationships between Gorenstein dimension and grade of modules over filtered rings, establishing key inequalities and equalities, and applies these results to Cohen-Macaulay and holonomic modules with associated graded rings.

Contribution

It introduces new inequalities and equalities connecting Gorenstein dimension and grade for filtered modules, enhancing the understanding of their structure and properties.

Findings

Proves G-dim M ≤ G-dim gr M for filtered modules.

Establishes grade M = grade gr M when Gorenstein dimension of gr M is finite.

Applies results to Cohen-Macaulay and holonomic modules over filtered rings.

Abstract

We study Gorenstein dimension and grade of a module $M$ over a filtered ring whose assosiated graded ring is a commutative Noetherian ring. An equality or an inequality between these invariants of a filtered module and its associated graded module is the most valuable property for an investigation of filtered rings. We prove an inequality G-dim$M\leq{G-dim gr}M$ and an equality ${\rm grade}M={\rm grade gr}M$, whenever Gorenstein dimension of ${\rm gr}M$ is finite (Theorems 2.3 and 2.8). We would say that the use of G-dimension adds a new viewpoint for studying filtered rings and modules. We apply these results to a filtered ring with a Cohen-Macaulay or Gorenstein associated graded ring and study a Cohen-Macaulay, perfect or holonomic module.