Hilbert-Samuel functions of modules over Cohen-Macaulay rings

For a finitely generated, non-free module $M$ over a CM local ring $(R,\mathfrak{m},k)$, it is proved that for $n\gg 0$ the length of $\operatorname{Tor}_1^R(M,R/\mathfrak{m}^{n+1})$ is given by a polynomial of degree $\dim R-1$. The vanishing of $\operatorname{Tor}_ i^R(M,N/\mathfrak{m}^{n+1}N)$ 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 $N/\mathfrak{m}^{n+1}N$ is finite, then is $R$ regular? Upper bounds are provided for $n$ beyond which the question has an affirmative answer.