Sectional genera of parameter ideals

TL;DR

This paper establishes a criterion relating the sectional genus of a parameter ideal to various algebraic invariants of a finitely generated module over a Noetherian local ring, deepening understanding of their interrelations.

Contribution

It provides a new criterion for the equality involving sectional genus and homological invariants of modules over Noetherian local rings.

Findings

Criterion for equality involving sectional genus and homological invariants.

Deeper insight into the relationships between algebraic invariants.

Potential applications to the study of parameter ideals and module structure.

Abstract

Let $M$ be a finitely generated module over a Noetherian local ring. This paper reports, for a given parameter ideal $Q$ for $M$, a criterion for the equality ${\mathrm{g}}_s(Q;M)=\operatorname{hdeg}_Q(M)-{\mathrm{e}}_Q^0(M)-{\mathrm{T}}_Q^1(M)$, where ${\mathrm{g}}_s(Q;M)$, ${\mathrm{e}}_Q^0(M)$, ${\mathrm{e}}_Q^1(M)$, and ${\mathrm{T}}_Q^1(M)$ respectively denote the sectional genus, the multiplicity, the first Hilbert coefficient, and the Homological torsion of $M$ with respect to $Q$.