  For a finitely generated, non-free module $M$ over a CM local ring
$(R,\fm,k)$, it is proved that for $n\gg 0$ the length of $\tor
1RM{R/\fm^{n+1}}$ is given by a polynomial of degree $\dim R-1$. The vanishing
of $\tor iRM{N/\fm^{n+1}N}$ is studied, with a view towards answering the
question: if there exists a finitely generated $R$-module $N$ with $\dim N\ge
1$ such that the projective dimension or the injective dimension of
$N/\fm^{n+1}N$ is finite, then is $R$-regular? Upper bounds are provided for
$n$ beyond which the question has an affirmative answer.
