Linkage of finite Gorenstein dimension modules

TL;DR

This paper explores the relationships between invariants like reduced grade, Gorenstein dimension, and depth for horizontally linked modules over certain rings, revealing conditions under which these properties are interconnected.

Contribution

It establishes new links between depth, reduced grade, and Gorenstein dimension for linked modules, and discusses Serre conditions and local cohomology vanishing.

Findings

Depth equals reduced grade under certain conditions

Connections between Serre conditions and local cohomology vanishings

New relationships between Gorenstein dimension and module invariants

Abstract

For a horizontally linked module, over a commutative semiperfect Noetherian ring $R$, the connections of its invariants reduced grade, Gorenstein dimension and depth are studied. It is shown that under certain conditions the depth of a horizontally linked module is equal to the reduced grade of its linked module. The connection of the Serre condition $(S_n)$ on an $R$--module of finite Gorenstein dimension with the vanishing of the local cohomology groups of its linked module is discussed.