The second Hilbert coefficient of modules with almost maximal depth

TL;DR

This paper establishes bounds and conditions for the second Hilbert coefficient of modules with almost maximal depth, extending previous results from Cohen-Macaulay modules to more general modules.

Contribution

It generalizes bounds on the second Hilbert coefficient to modules with almost maximal depth and relates these bounds to the depth of the associated graded module.

Findings

Provides an upper bound for $e_2( ext{M})$ when $depth(M) \,\geq\, d-1$

Establishes a condition for equality involving the depth of the associated graded module

Proves a lower bound on $e_2( ext{M})$ generalizing Rees and Narita's results

Abstract

Let $\mathbb{M} = \{ M_n \}$ be a good $\mathfrak{q}$-filtration of a finitely generated $R$-module $M$ of dimension $d$, where $(R,\mathfrak{m})$ is a local ring and $\mathfrak{q}$ is an $\mathfrak{m}$-primary ideal of $R$. In case $depth(M) \geq d-1$, we give an upper bound for the second Hilbert coefficient $e_2(\mathbb{M})$ generalizing results by Huckaba-Marley and Rossi-Valla proved assuming that $M$ is Cohen-Macaulay. We also give a condition for the equality, which relates to the depth of the associated graded module $gr_{\mathbb{M}}(M)$. A lower bound on $e_2(\mathbb{M})$ is proved generalizing a result by Rees and Narita.