Lemma 21.28.3. 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 $D(\mathcal{O}_\mathcal {C})$. Assume

1. $f$ is flat,

2. $K$ is bounded below,

3. $f^*Rf_*H^ q(K) \to H^ q(K)$ is an isomorphism.

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

Proof. Observe that $f^*Rf_*K \to K$ is an isomorphism if and only if it is an isomorphism on cohomology sheaves $H^ j$. Observe that $H^ j(f^*Rf_*K) = f^*H^ j(Rf_*K) = f^*H^ j(Rf_*\tau _{\leq j}K) = H^ j(f^*Rf_*\tau _{\leq j}K)$. Hence we may assume that $K$ is bounded. Then property (3) tells us the cohomology sheaves are in the triangulated subcategory $D' \subset D(\mathcal{O}_\mathcal {C})$ of Lemma 21.28.2. Hence $K$ is in it too. $\square$

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