The Stacks project

Lemma 30.2.4. Let $f : X \to S$ be an affine morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Then $H^ i(X, \mathcal{F}) = H^ i(S, f_*\mathcal{F})$ for all $i \geq 0$.

Proof. Follows from Lemma 30.2.3 and the Leray spectral sequence. See Cohomology, Lemma 20.13.6. $\square$

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