Hilbert coefficients of a Cohen–Macaulay module

Let be a d-dimensional Noetherian local ring, M a finite Cohen–Macaulay A-module of dimension r, and let I be an ideal of definition for M. We define the notion of minimal multiplicity of Cohen–Macaulay modules with respect to I and show that if M has minimal multiplicity with respect to I then the associated graded module GI(M) is Cohen–Macaulay. When A is Cohen–Macaulay, M is maximal Cohen–Macaulay, and I is -primary, we find a relation between the first Hilbert coefficient of M, A, and Syz1A(M).