The Hilbert Function of a Maximal Cohen-Macaulay Module

We study Hilbert functions of maximal Cohen-Macaulay(=CM) modules over CM local rings. We show that if $A$ is a hypersurface ring with dimension $d > 0$ then the Hilbert function of $M$ \wrt $\m$ is non-decreasing. If $A = Q/(f)$ for some regular local ring $Q$, we determine a lower bound for $e_0(M)$ and $e_1(M)$. We analyze the case when equality holds and prove that in this case $G(M)$ is CM. Furthermore in this case we also determine the Hilbert function of $M$. When $A$ is Gorenstein then $M$ is the first syzygy of $S^A(M) = (\Syz^{A}_{1}(M^*))^*$. A relation between the second Hilbert coefficient of $M$, $A$ and $S^A(M)$ is found when $G(M)$ is \CM and $\depth G(A) \geq d-1$. We give bounds for the first Hilbert coefficients of the canonical module of a CM local ring and analyse when equality holds. We also give good bounds on Hilbert coefficients of $M$ when $M$ is maximal CM and $G(M)$ is CM.