The Stacks project

Lemma 51.21.5. In the situation above, let $t$ be an upper bound on the number of generators for $I$. There exists an integer $c = c(A, I) \geq 0$ such that for any $A$-module $M$ the tor modules $\text{Tor}_ i^ A(M, A/I^ q)$ are annihilated by $I^ c$ for $i > t$ and all $q \geq 0$.

Proof. Let $q(A, I)$ be as above. For $q \geq q(A, I)$ we have

\[ R\Gamma (X, Lp^*\widetilde{M}(q)) = M \otimes _ A^\mathbf {L} I^ q \]

by Lemma 51.21.3. We have a bounded and convergent spectral sequence

\[ H^ a(X, H^ b(Lp^*\widetilde{M}(q))) \Rightarrow \text{Tor}_{-a - b}^ A(M, I^ q) \]

by Derived Categories of Schemes, Lemma 36.4.4. Let $d$ be an integer as in Cohomology of Schemes, Lemma 30.4.4 (actually we can take $d = t$, see Cohomology of Schemes, Lemma 30.4.2). Then we see that $H^{-i}(X, Lp^*\widetilde{M}(q)) = \text{Tor}_ i^ A(M, I^ q)$ has a finite filtration with at most $d$ steps whose graded are subquotients of the modules

\[ H^ a(X, H^{- i - a}(Lp^*\widetilde{M})(q)),\quad a = 0, 1, \ldots , d - 1 \]

If $i \geq t$ then all of these modules are annihilated by $I^ c$ where $c = c(A, I)$ is as in Lemma 51.21.4 because the cohomology sheaves $H^{- i - a}(Lp^*\widetilde{M})$ are all annihilated by $I^ c$ by the lemma. Hence we see that $\text{Tor}_ i^ A(M, I^ q)$ is annihilated by $I^{dc}$ for $q \geq q(A, I)$ and $i \geq t$. Using the short exact sequence $0 \to I^ q \to A \to A/I^ q \to 0$ we find that $\text{Tor}_ i(M, A/I^ q)$ is annihilated by $I^{dc}$ for $q \geq q(A, I)$ and $i > t$. We conclude that $I^ m$ with $m = \max (dc, q(A, I) - 1)$ annihilates $\text{Tor}_ i^ A(M, A/I^ q)$ for all $q \geq 0$ and $i > t$ as desired. $\square$

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