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

$f$ is flat,

$K$ is bounded below,

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

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

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