The Hilbert function of a maximal Cohen-Macaulay module

We study Hilbert functions of maximal CM modules over CM local rings. When A is a hypersurface ring with dimension d>0, we show that the Hilbert function of M with respect to is non-decreasing. If A=Q/(f) for some regular local ring Q, we determine a lower bound for e₀(M) and e₁(M) and analyze the case when equality holds. When A is Gorenstein a relation between the second Hilbert coefficient of M, A and S^A(M)= (Syz^A₁(M*))* is found when G(M) is CM and depthG(A)≥d−1. We give bounds for the first Hilbert coefficients of the canonical module of a CM local ring and analyze when equality holds. We also give good bounds on Hilbert coefficients of M when M is maximal CM and G(M) is CM.