  Let $(A,\mathfrak{m})$ be a Gorenstein local ring and let $M$ be a finitely
generated Cohen Macaulay $A$ module. Let $G(A)=\bigoplus_{n\geq
0}\mathfrak{m}^n/\mathfrak{m}^{n+1}$ be the associated graded ring of $A$ and
$G(M)=\bigoplus_{n\geq 0}\mathfrak{m}^nM/\mathfrak{m}^{n+1}M$ be the associated
graded module of $M$. If $A$ is regular and if $G(M)$ has a quasi-pure
resolution then we show that $G(M)$ is Cohen-Macaulay.
  If $G(A)$ is Cohen-Macaulay and if $M$ has finite projective dimension then
we give lower bounds on $e_0(M)$ and $e_1(M)$. Finally let $A = Q/(f_1, \ldots,
f_c)$ be a strict complete intersection with $\text{ord}(f_i) = s$ for all $i$.
Let $M$ be an Cohen-Macaulay module with $\text{cx}_A(M) = r < c$. We give
lower bounds on $e_0(M)$ and $e_1(M)$.
