The second Hilbert coefficients and the homological torsions of parameters

Let M be a finitely generated module over a Noetherian local ring. This paper gives, for a given parameter ideal Q for M  , bounds for the second Hilbert coefficients eQ2(M) in terms of the homological degrees and torsions of modules. We also report a criterion for a certain equality of the second Hilbert coefficients of parameters and the homological torsions of modules.