The second Hilbert coefficients and the homological torsions of parameters

TL;DR

This paper establishes bounds for the second Hilbert coefficients of modules over local rings using homological invariants, and provides criteria for when these coefficients equal certain torsions.

Contribution

It introduces bounds and criteria relating second Hilbert coefficients to homological degrees and torsions of modules over Noetherian local rings.

Findings

Bounds for ${ m e}_Q^2(M)$ in terms of homological degrees and torsions.

A criterion for equality between second Hilbert coefficients and homological torsions.

Insights into the structure of modules via Hilbert coefficients and homological invariants.

Abstract

Let $M$ be a finitely generated module over a Noetherian local ring. This paper gives, for a given parameter ideal $Q$ for $M$, bounds for the second Hilbert coefficients ${\mathrm{e}}_Q^2(M)$ in terms of the homological degrees and torsions of modules. We also report a criterion for a certain equality of the second Hilbert coefficients of parameters and the homological torsions of modules.