Lemma 21.28.4 (0D7T)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 21: Cohomology on Sites
Section 21.28: Formalities on cohomological descent
Lemma 21.28.4 (cite)

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Lemma 21.28.4. Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ be a morphism of ringed topoi. Let $K$ be an object of $D(\mathcal{O}_\mathcal {D})$ . Assume

$f$ is flat,
$K$ is bounded below,
$H^ q(K) \to Rf_*f^*H^ q(K)$ is an isomorphism.

Then $K \to Rf_*f^*K$ is an isomorphism.

Proof. Observe that $K \to Rf_*f^*K$ is an isomorphism if and only if it is an isomorphism on cohomology sheaves $H^ j$ . Observe that $H^ j(Rf_*f^*K) = H^ j(Rf_*\tau _{\leq j}f^*K) = H^ j(Rf_*f^*\tau _{\leq j}K)$ . Hence we may assume that $K$ is bounded. Then property (3) tells us the cohomology sheaves are in the triangulated subcategory $D' \subset D(\mathcal{O}_\mathcal {D})$ of Lemma 21.28.1. Hence $K$ is in it too. $\square$

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