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The Stacks project

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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