Local Homology, Cohen-Macaulayness and Cohen-Macaulayfications

TL;DR

This paper explores the relationship between local homology and Cohen-Macaulay properties, introduces a new Cohen-Macaulayfication concept, and demonstrates its uniqueness and generalization of previous notions.

Contribution

It establishes a new Cohen-Macaulayfication framework, proves its uniqueness, and generalizes earlier concepts by Goto, advancing understanding of Cohen-Macaulay modules.

Findings

H^x_d(X) is finite, answering Tang's question positively

Provides a necessary condition for modules to be Cohen-Macaulay

Introduces and proves the uniqueness of Cohen-Macaulayfication

Abstract

Let (R,m) be a local, complete ring, X an artinian R-module of Noetherian dimension d; let x_1,...,x_d\in m be such that 0:_X (x_1,...,x_d)R has finite length. Then H^x_d(X) is a finite R-module, providing a positive answer to a question posed by Tang. As a first application of this result corollory 1 contains a necessary condition for a finite module to be CM; secondly we propose a notion of Cohen-Macaulayfication and prove its uniqueness (th. 3); finally we show that this new notion of Cohen-Macaulayfication is a direct generalization of a notion of Cohen-Macaulayfication introduced by Goto (th. 4).