The first Euler characteristics versus the homological degrees

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 ${\chi }_1(Q;M)={\mathrm{hdeg}}_Q(M)-{\mathrm{e}}_Q^0(M)$, where ${\chi }_1(Q;M)$, ${\mathrm{hdeg}}_Q(M)$, 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 $M$.