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The Stacks project

Lemma 36.5.1. Let f : X \to S be an affine morphism of schemes. Let \mathcal{F}^\bullet be a complex of quasi-coherent \mathcal{O}_ X-modules. Then f_*\mathcal{F}^\bullet = Rf_*\mathcal{F}^\bullet .

Proof. Combine Lemma 36.4.2 with Cohomology of Schemes, Lemma 30.2.3. An alternative proof is to work affine locally on S and use Lemma 36.3.7. \square


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