The Stacks project

Lemma 51.21.4. 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 cohomology sheaves $H^ j(Lp^*\widetilde{M})$ are annihilated by $I^ c$ for $j \leq -t$.

Proof. Say $I = (a_1, \ldots , a_ t)$. The question is affine local on $X$. For $1 \leq i \leq t$ let $B_ i = A[\frac{I}{a_ i}]$ be the affine blowup algebra. Then $X$ has an affine open covering by the spectra of the rings $B_ i$, see Divisors, Lemma 31.32.2. By the description of derived pullback given in Derived Categories of Schemes, Lemma 36.3.8 we conclude it suffices to prove that for each $i$ there exists a $c \geq 0$ such that

\[ \text{Tor}_ j^ A(B_ i, M) \]

is annihilated by $I^ c$ for $j \geq t$. This is Lemma 51.21.2. $\square$

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