Lemma 69.8.3 (0D2U)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 69: Cohomology of Algebraic Spaces
Section 69.8: Vanishing for higher direct images
Lemma 69.8.3 (cite)

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Lemma 69.8.3. Let $S$ be a scheme. Let $f : X \to Y$ be an affine morphism of algebraic spaces over $S$ . Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$ -module. Then $H^ i(X, \mathcal{F}) = H^ i(Y, f_*\mathcal{F})$ for all $i \geq 0$ .

Proof. Follows from Lemma 69.8.2 and the Leray spectral sequence. See Cohomology on Sites, Lemma 21.14.6. $\square$

Comments (2)

Comment #5517 by Shend Zhjeqi on September 24, 2020 at 05:10

I might be mistaken but shouldn't the second cohomology( in lemma 67.8.3) be over $Y$ and not over $S?$ . I.e. instead of

$H^i(X,\mathcal{F})= H^i(S, f_* \mathcal{F}),$ should we have $H^i(X,\mathcal{F})= H^i(Y, f_*\mathcal{F})?$

Thank you.

Feel free to delete the comment afterwards.

Comment #5710 by Johan on November 15, 2020 at 22:46

THanks and fixed here.

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