The Stacks project

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. $\square$