On associated graded modules of maximal Cohen-Macaulay modules over hypersurface rings

Let $(A,\mathfrak{m})$ be a hypersurface ring with dimension $d$, and $M$ a MCM $A-$module with red$(M)\le 2$ and $\mu (M)=2$ or $3$ then we have proved that depth $G(M)\ge d-\mu (M)+1$. If $e(A)=3$ and $\mu (M)=4$ then in this case we have proved that depth$G(M)\ge d-3$. Next we consider the case when $e(M)=\mu (M)i(M)+1$ and prove that depth $G(M)\ge d-1$. When $A=Q\mathrm{/}(f)$ where $Q=k[[X_1,\cdots ,X_{d+1}]]$ then we give estimates for $\text{depth}G(M)$ in terms of a minimal presentation of $M$. Our paper is the first systematic study of depth of associated graded modules of MCM modules over hypersurface rings.