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

Let \((A,\mathfrak {m} )\) $(A,\mathfrak{m})$ be a Cohen–Macaulay local ring of dimension \(d \ge 1\) $d\ge 1$ . Suppose there exists be a non-zero A module M of finite length and finite projective dimension such that \(\ell \ell (M)\) $\ell \ell (M)$ , the Lowey length of M, is equal to \(\lambda (M)\) $\lambda (M)$ , 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 \(\ell \ell (M) = \lambda (M)-1\) $\ell \ell (M)=\lambda (M)-1$ .