Lemma 20.55.10. In Situation 20.55.2 let $M$ be an object of $D(\mathcal{O}_ X)$. Let $x \in X$ with $\mathcal{O}_{X, x}$ nonzero. If $H^ i(M)_ x$ is finite free over $\mathcal{O}_{X, x}$, then $H^ i(L\eta _\mathcal {I}M)_ x$ is finite free over $\mathcal{O}_{X, x}$ of the same rank.

**Proof.**
Namely, say $f \in \mathcal{O}_{X, x}$ generates the stalk $\mathcal{I}_ x$. Then $f$ is a nonzerodivisor in $\mathcal{O}_{X, x}$ and hence $H^ i(M)_ x[f] = 0$. Thus by Lemma 20.55.5 we see that $H^ i(L\eta _\mathcal {I}M)_ x$ is isomorphic to $\mathcal{I}^ i_ x \otimes _{\mathcal{O}_{X, x}} H^ i(M)_ x$ which is free of the same rank as desired.
$\square$

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