Quasi-pure resolutions and some lower bounds of Hilbert coefficients of Cohen-Macaulay modules

Let $(A,\mathfrak{m})$ be a Gorenstein local ring and let $M$ be a finitely generated Cohen Macaulay $A$ module. Let $G(A)={⨁}_{n\ge 0}{\mathfrak{m}}^n\mathrm{/}{\mathfrak{m}}^{n+1}$ be the associated graded ring of $A$ and $G(M)={⨁}_{n\ge 0}{\mathfrak{m}}^{n}M\mathrm{/}{\mathfrak{m}}^{n+1}M$ 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 $A=Q\mathrm{/}(f_1,\dots ,f_c)$ 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 $e_1(M)$.