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

Let $A=Q/(f)$ where $(Q,\mathfrak{n})$ be a complete regular local ring of dimension $d+1$, $f\in \mathfrak{n}^i\setminus\mathfrak{n}^{i+1}$ for some $i\geq 2$ and $M$ an MCM $A-$module with $e(M)=\mu(M)i(M)+1$ then we prove that depth $G(M)\geq d-1$. If $(A,\mathfrak{m})$ is a complete hypersurface ring of dimension $d$ with infinite residue field and $e(A)=3$, let $M$ be an MCM $A$-module with $\mu(M)=2$ or $3$ then we prove that depth $G(M)\geq d-\mu(M)+1$. Our paper is the first systematic study of depth of associated graded modules of MCM modules over hypersurface rings.