Lemma 15.27.3. Let $A$ be a Noetherian ring. Let $I$ be an ideal of $A$. Let $M$ be a finite $A$-module. For every $p > 0$ there exists a $c > 0$ such that $\text{Tor}_ p^ A(M, A/I^ n) \to \text{Tor}_ p^ A(M, A/I^{n - c})$ is zero for all $n \geq c$.
This is [Lemma 9.9, quillenhomology]; note that the author forgot the word “strict” in the statement although it was clearly intended.
Proof.
Proof for $p = 1$. Choose a short exact sequence $0 \to K \to A^{\oplus t} \to M \to 0$. Then $\text{Tor}_1^ A(M, A/I^ n) = K \cap (I^ n)^{\oplus t}/I^ nK$. By Artin-Rees (Algebra, Lemma 10.51.2) there is a constant $c \geq 0$ such that $K \cap (I^ n)^{\oplus t} \subset I^{n - c}K$ for $n \geq c$. Thus the result for $p = 1$. For $p > 1$ we have $\text{Tor}_ p^ A(M, A/I^ n) = \text{Tor}^ A_{p - 1}(K, A/I^ n)$. Thus the lemma follows by induction.
$\square$
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