Top Local Cohomology and the Catenaricity of the Unmixed Support of a Finitely Generated Module

Abstract

Let (R, 𝔪) be a Noetherian local ring and M a finitely generated R-module with dim M = d. This paper is concerned with the following property for the top local cohomology module :

In this article we will show that this property is equivalent to the catenaricity of the unmixed support USupp M of M which is defined by Usupp M = Supp M/U_M(0), where U_M(0) is the largest submodule of M of dimension less than d. Some characterizations of this property in terms of system of parameters as well as the relation between the unmixed supports of M and of the 𝔪-adic completion are given.

Keywords:

• Artinian module
• Catenaricity
• Top local cohomology
• Unmixed support

AMS Classification:

• 13D45
• 13E10

ACKNOWLEDGMENT

We wish to express our gratitude to the refree for his/her useful suggestions and, especially, the shorter proof of Lemma 4.2.

Notes

Communicated by I. Swanson.

^#Senior Associate Member of ICTP, Trieste, Italy.

*Junior Associate Member of ICTP, Trieste, Italy.