Lemma 69.8.1. Let S be a scheme. Let f : X \to Y be a morphism of algebraic spaces over S. Assume that
f is quasi-compact and quasi-separated, and
Y is quasi-compact.
Then there exists an integer n(X \to Y) such that for any algebraic space Y', any morphism Y' \to Y and any quasi-coherent sheaf \mathcal{F}' on X' = Y' \times _ Y X the higher direct images R^ if'_*\mathcal{F}' are zero for i \geq n(X \to Y).
Proof.
Let V \to Y be a surjective étale morphism where V is an affine scheme, see Properties of Spaces, Lemma 66.6.3. Suppose we prove the result for the base change f_ V : V \times _ Y X \to V. Then the result holds for f with n(X \to Y) = n(X_ V \to V). Namely, if Y' \to Y and \mathcal{F}' are as in the lemma, then R^ if'_*\mathcal{F}'|_{V \times _ Y Y'} is equal to R^ if'_{V, *}\mathcal{F}'|_{X'_ V} where f'_ V : X'_ V = V \times _ Y Y' \times _ Y X \to V \times _ Y Y' = Y'_ V, see Properties of Spaces, Lemma 66.26.2. Thus we may assume that Y is an affine scheme.
Moreover, to prove the vanishing for all Y' \to Y and \mathcal{F}' it suffices to do so when Y' is an affine scheme. In this case, R^ if'_*\mathcal{F}' is quasi-coherent by Lemma 69.3.1. Hence it suffices to prove that H^ i(X', \mathcal{F}') = 0, because H^ i(X', \mathcal{F}') = H^0(Y', R^ if'_*\mathcal{F}') by Cohomology on Sites, Lemma 21.14.6 and the vanishing of higher cohomology of quasi-coherent sheaves on affine algebraic spaces (Proposition 69.7.2).
Choose U \to X, d, V_ p \to U_ p and d_ p as in Lemma 69.7.3. For any affine scheme Y' and morphism Y' \to Y denote X' = Y' \times _ Y X, U' = Y' \times _ Y U, V'_ p = Y' \times _ Y V_ p. Then U' \to X', d' = d, V'_ p \to U'_ p and d'_ p = d is a collection of choices as in Lemma 69.7.3 for the algebraic space X' (details omitted). Hence we see that H^ i(X', \mathcal{F}') = 0 for i \geq \max (p + d_ p) and we win.
\square
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