On the Lowey length of modules of finite projective dimension-II

Let $(A,\mathfrak{m})$ be a Gorenstein local ring of dimension $d\ge 1$. Suppose there exists be a non-zero $A$ module $M$ of finite length and finite projective dimension such that $\mathrm{\ell }\mathrm{\ell }(M)$, the Lowey length of $M$, is equal to $\lambda (M)$, the length of $M$. Then we show that necessarily $A$ is at worst a hypersurface singularity. We also characterize Gorenstein local rings having a non-zero module $M$ of finite length and finite projective dimension with $\mathrm{\ell }\mathrm{\ell }(M)=\lambda (M)-1$.