Lemma 68.8.3. 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 68.8.2 and the Leray spectral sequence. See Cohomology on Sites, Lemma 21.14.6. $\square$

Comment #5517 by Shend Zhjeqi on

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
should we have

Thank you.

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