TITLE: regularity of local ring
QUESTION [10 upvotes]: 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 $pd(R/\mathfrak{m}^2)< \infty$?

REPLY [7 votes]: If $R$ has positive dimension, then for any $t\ge 1$, the ideal $I=\mathfrak m^t$ is a so called Burch ideal (as defined by Dao,  Kobayashi, Takahashi; Burch ideals and Burch rings )  i.e. $\mathfrak m(I:\mathfrak m)\ne \mathfrak mI$. 
It is a Theorem of Burch that if $I$ is a Burch ideal and $R/I$ has finite projective dimension, then $R$ is regular.