  Let $(R,\m)$ be a Noetherian local ring. Consider the notion of homological
dimension of a module, denoted H-dim, for H= Reg, CI, CI$_*$, G, G$^*$ or CM.
We prove that, if for a finite $R$-module $M$ of positive depth,
$\Hd_R({\m}^iM)$ is finite for some $i \geq \reg(M)$, then the ring $R$ has
property H.
