Lemma 30.4.6 (01XK)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 30: Cohomology of Schemes
Section 30.4: Quasi-coherence of higher direct images
Lemma 30.4.6 (cite)

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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. $\square$

Comments (2)

Comment #3031 by Brian Lawrence on December 13, 2017 at 03:07

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

Comment #3144 by Johan on February 01, 2018 at 20:51

Need a pithier slogan!

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