Maximal depth property of finitely generated modules

TL;DR

This paper investigates finitely generated modules over Noetherian local rings that have the maximal depth property, exploring their preservation under module operations, classifying generalized Cohen–Macaulay modules with this property, and analyzing local cohomology attached primes.

Contribution

It introduces the concept of maximal depth modules, studies their stability under operations, classifies related Cohen–Macaulay modules, and examines local cohomology attached primes.

Findings

Maximal depth property is preserved under certain module operations.

Generalized Cohen–Macaulay modules with maximal depth are classified.

Attached primes of local cohomology modules are analyzed for modules with maximal depth.

Abstract

Let $(R,\mathfrak{m})$ be a Noetherian local ring and $M$ a finitely generated $R$-module. We say $M$ has maximal depth if there is an associated prime $\mathfrak{p}$ of $M$ such that depth $M=\dim R/\mathfrak{p}$. In this paper, we study finitely generated modules with maximal depth. It is shown that the maximal depth property is preserved under some important module operations. Generalized Cohen--Macaulay modules with maximal depth are classified. Finally, the attached primes of $H^i_{\mathfrak{m}}(M)$ are considered for $i<\mathrm{dim} M$.