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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