Let $(R, \mathfrak{m})$ be a commutative Noetherian local ring and $M$ a finitely generated $R$-module. Let $x\_1,...,x\_t$ be an $M$-regular sequence and $I = (x\_1,...,x\_t)$. Is it true that
$$\mathrm{Tor}\_1^R(R/I^n, M) = 0$$
for all $n \geq 1$?



Vanishing of Tor
I've posted a proof [here](https://math.stackexchange.com/questions/268542/ideal-generated-by-a-regular-sequence-and-the-vanishing-of-tor) for the special case when $M$ is cyclic. Furthermore, I've mentioned that the result holds for finitely generated modules when the sequence is $R$-regular and $M$-regular.


