Lemma 30.4.6. Let $f : X \to S$ be a morphism of schemes. Assume that $f$ is quasi-separated and quasi-compact. Assume $S$ is affine. For any quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$ we have

$H^ q(X, \mathcal{F}) = H^0(S, R^ qf_*\mathcal{F})$

for all $q \in \mathbf{Z}$.

Proof. Consider the Leray spectral sequence $E_2^{p, q} = H^ p(S, R^ qf_*\mathcal{F})$ converging to $H^{p + q}(X, \mathcal{F})$, see Cohomology, Lemma 20.13.4. By Lemma 30.4.5 we see that the sheaves $R^ qf_*\mathcal{F}$ are quasi-coherent. By Lemma 30.2.2 we see that $E_2^{p, q} = 0$ when $p > 0$. Hence the spectral sequence degenerates at $E_2$ and we win. See also Cohomology, Lemma 20.13.6 (2) for the general principle. $\square$

Comment #3031 by Brian Lawrence on

Suggested slogan: The cohomology of a scheme over an affine base agrees with the global sections of its derived pushforward.

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