The Stacks project

Lemma 51.21.6. Let $I$ be an ideal of a Noetherian ring $A$. Let $t \geq 0$ be an upper bound on the number of generators of $I$. There exist $N, c \geq 0$ such that the maps

\[ \text{Tor}_{t + 1}^ A(M, A/I^ n) \to \text{Tor}_{t + 1}^ A(M, A/I^{n - c}) \]

are zero for any $A$-module $M$ and all $n \geq N$.

Proof. Let $c_1$ be the constant found in Lemma 51.21.5. Please keep in mind that this constant $c_1$ works for $\text{Tor}_ i$ for all $i > t$ simultaneously.

Say $I = (a_1, \ldots , a_ t)$. For an $A$-module $M$ we set

\[ \ell (M) = \# \{ i \mid 1 \leq i \leq t,\ a_ i^{c_1}\text{ is zero on }M\} \]

This is an element of $\{ 0, 1, \ldots , t\} $. We will prove by descending induction on $0 \leq s \leq t$ the following statement $H_ s$: there exist $N, c \geq 0$ such that for every module $M$ with $\ell (M) \geq s$ the maps

\[ \text{Tor}_{t + 1 + i}^ A(M, A/I^ n) \to \text{Tor}_{t + 1 + i}^ A(M, A/I^{n - c}) \]

are zero for $i = 0, \ldots , s$ for all $n \geq N$.

Base case: $s = t$. If $\ell (M) = t$, then $M$ is annihilated by $(a_1^{c_1}, \ldots , a_ t^{c_1}\} $ and hence by $I^{t(c_1 - 1) + 1}$. We conclude from Lemma 51.21.1 that $H_ t$ holds by taking $c = N$ to be the maximum of the integers $c(A, I, t(c_1 - 1) + 1, t + 1), \ldots , c(A, I, t(c_1 - 1) + 1, 2t + 1)$ found in the lemma.

Induction step. Say $0 \leq s < t$ we have $N, c$ as in $H_{s + 1}$. Consider a module $M$ with $\ell (M) = s$. Then we can choose an $i$ such that $a_ i^{c_1}$ is nonzero on $M$. It follows that $\ell (M[a_ i^ c]) \geq s + 1$ and $\ell (M/a_ i^{c_1}M) \geq s + 1$ and the induction hypothesis applies to them. Consider the exact sequence

\[ 0 \to M[a_ i^{c_1}] \to M \xrightarrow {a_ i^{c_1}} M \to M/a_ i^{c_1}M \to 0 \]

Denote $E \subset M$ the image of the middle arrow. Consider the corresponding diagram of Tor modules

\[ \xymatrix{ & & \text{Tor}_{i + 1}(M/a_ i^{c_1}M, A/I^ q) \ar[d] \\ \text{Tor}_ i(M[a_ i^{c_1}], A/I^ q) \ar[r] & \text{Tor}_ i(M, A/I^ q) \ar[r] \ar[rd]^0 & \text{Tor}_ i(E, A/I^ q) \ar[d] \\ & & \text{Tor}_ i(M, A/I^ q) } \]

with exact rows and columns (for every $q$). The south-east arrow is zero by our choice of $c_1$. We conclude that the module $\text{Tor}_ i(M, A/I^ q)$ is sandwiched between a quotient module of $\text{Tor}_ i(M[a_ i^{c_1}], A/I^ q)$ and a submodule of $\text{Tor}_{i + 1}(M/a_ i^{c_1}M, A/I^ q)$. Hence we conclude $H_ s$ holds with $N$ replaced by $N + c$ and $c$ replaced by $2c$. Some details omitted. $\square$

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