The first Euler characteristics versus the homological degrees

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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 χ₁(Q;M)=hdegQ(M)-eQ⁰(M), where χ₁(Q;M), hdegQ(M), and eQ⁰(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.