Regular sequences and ZD-modules

TL;DR

This paper investigates the properties of S-sequences in ZD-modules over Noetherian rings, establishing that all maximal S-sequences in an ideal have equal length and linking this length to local cohomology.

Contribution

It introduces the concept of S-depth for ZD-modules and proves its equivalence to the infimum of local cohomology degrees not in a Melkersson subcategory.

Findings

All maximal S-sequences on M in I have equal length.

S-depth equals the infimum of degrees where local cohomology escapes the subcategory.

Provides a new characterization of depth in the context of ZD-modules.

Abstract

Let R be a Noetherian ring, I an ideal of R and M a ZD-module. Let S be a Melkersson subcategory with respect to I such that M/IM doesn't belong to S. We show that all maximal S-sequences on M in I, have equal length. If this common length is denoted by S-depth_I(M), then S-depth_I(M) = inf{i : H^i_I(M) doesn't belong to S}.