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Let $M$ be a finitely generated module over a Noetherian local ring. Thispaper reports, for a given parameter ideal $Q$ for $M$, a criterion for theequality $\chi_1(Q;M)=\operatorname{hdeg}_Q(M)-\mathrm{e}_Q^0(M)$, where$\chi_1(Q;M)$, $\operatorname{hdeg}_Q(M)$, and $\mathrm{e}_Q^0(M)$ respectivelydenote the first Euler characteristic, the homological degree, and themultiplicity of $M$ with respect to $Q$. We also study homological torsions of$M$ and give a criterion for a certain equality of the first Hilbertcoefficients of parameters and the homological torsions of $M$.

The first Euler characteristics versus the homological degrees