Relative Cohen-Macaulay filtered modules with a view toward relative Cohen-Macaulay modules

TL;DR

This paper introduces and studies relative Cohen-Macaulay filtered modules over Noetherian rings, providing characterizations and properties that extend the understanding of relative Cohen-Macaulay modules and their filtrations.

Contribution

It defines relative Cohen-Macaulay filtered modules, offers non-zerodivisor characterizations, and establishes new properties and filtrations related to these modules.

Findings

Characterization of non-zerodivisors on relative Cohen-Macaulay modules

Introduction of relative Cohen-Macaulay filtered modules and their properties

Cohomological dimension filtration related to associated prime ideals

Abstract

Let R be a commutative Noetherian ring, a a proper ideal of R and M a finite R-module. It is shown that, if (R;m) is a complete local ring, then under certain conditions a contains a regular element on DR(Hc a(M)), where c = cd(a;M). A non-zerodivisor characterization of relative Cohen-Macaulay modules w.r.t a is given. We introduce the concept of relative Cohen-Macaulay filtered modules w.r.t a and study some basic properties of such modules. In paticular, we provide a non-zerodivisor characterization of relative Cohen-Macaulay filtered modules w.r.t a. Furthermore, a characterization of cohomological dimension filtration of M by the associated prime ideals of its factors is established. As a consequence, we present a cohomological dimension filtration for those modules whose zero submodule has a primary decomposition. Finally, we bring some new results about relative Cohen- Macaulay modules w.r.t a.