The Stacks project

Lemma 68.3.2. Let $S$ be a scheme. Let $f : X \to Y$ be a quasi-separated and quasi-compact morphism of algebraic spaces over $S$. For any quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$ and any affine object $V$ of $Y_{\acute{e}tale}$ we have

\[ H^ q(V \times _ Y X, \mathcal{F}) = H^0(V, R^ qf_*\mathcal{F}) \]

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

Proof. Since formation of $Rf_*$ commutes with ├ętale localization (Properties of Spaces, Lemma 65.26.2) we may replace $Y$ by $V$ and assume $Y = V$ is affine. Consider the Leray spectral sequence $E_2^{p, q} = H^ p(Y, R^ qf_*\mathcal{F})$ converging to $H^{p + q}(X, \mathcal{F})$, see Cohomology on Sites, Lemma 21.14.5. By Lemma 68.3.1 we see that the sheaves $R^ qf_*\mathcal{F}$ are quasi-coherent. By Cohomology of Schemes, 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. $\square$


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