Let $I$ be an ideal of a Noetherian ring $R$. $M$ is a finitely generated $R$module. I question is:
Does there exist $n_0$ such that for all $n \geq n_0$, the short exact sequence $$I^n/I^{n+1} \otimes_R M \cong I^nM/I^{n+1}M$$
Let $I$ be an ideal of a Noetherian ring $R$. $M$ is a finitely generated $R$module. I question is: Does there exist $n_0$ such that for all $n \geq n_0$, the short exact sequence $$I^n/I^{n+1} \otimes_R M \cong I^nM/I^{n+1}M$$ 


I suppose you intend to ask whether there is isomorphism as in the formula. The answer is negative, in general. For example, take $R = k[x]$, $I = (x)$, $M = R/I^2$. [Edit] What you may have in mind is that the result holds when $M$ is flat over $R$. 

