Let $(R, \mathfrak{m})$ be a Noetherian local ring. It is well know that

$R$ is regular iff $pd(R/\mathfrak{m}) < \infty$ (i.e. $R/\mathfrak{m}$ has finite projective dimension).

Assume that $\dim R > 0$. Is $R$ regular, if $pd(R/\mathfrak{m}^2)< \infty$?

every$R$-module has finite projective dimension. $\endgroup$ – user91132 Sep 10 '12 at 9:14