The first Euler characteristics versus the homological degrees

TL;DR

This paper establishes criteria relating Euler characteristics, homological degrees, and multiplicities of modules over Noetherian local rings, advancing understanding of their algebraic invariants and torsions.

Contribution

It provides new criteria for equalities involving Euler characteristics, homological degrees, and Hilbert coefficients in modules over local rings.

Findings

Criterion for the equality $oldsymbol{ ext{chi}_1(Q;M) = ext{hdeg}_Q(M) - e_Q^0(M)}$

Analysis of homological torsions and their relation to Hilbert coefficients

Conditions under which specific algebraic invariants coincide or relate

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 $\chi_1(Q;M)=\operatorname{hdeg}_Q(M)-\mathrm{e}_Q^0(M)$, where $\chi_1(Q;M)$, $\operatorname{hdeg}_Q(M)$, and $\mathrm{e}_Q^0(M)$ respectively denote the first Euler characteristic, the homological degree, and the multiplicity of $M$ with respect to $Q$. We also study homological torsions of $M$ and give a criterion for a certain equality of the first Hilbert coefficients of parameters and the homological torsions of $M$.