The Stacks project

Proof. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. We have to show that $f_*\mathcal{F}$ is a quasi-coherent sheaf on $Y$. For this it suffices to show that for any affine scheme $V$ and étale morphism $V \to Y$ the restriction of $f_*\mathcal{F}$ to $V$ is quasi-coherent, see Properties of Spaces, Lemma 66.29.6. Let $f' : V \times _ Y X \to V$ be the base change of $f$ by $V \to Y$. Note that $f'$ is also quasi-compact and quasi-separated, see Lemmas 67.8.4 and 67.4.4. By (67.11.1.1) we know that the restriction of $f_*\mathcal{F}$ to $V$ is $f'_*$ of the restriction of $\mathcal{F}$ to $V \times _ Y X$. Hence we may replace $f$ by $f'$, and assume that $Y$ is an affine scheme.

Assume $Y$ is an affine scheme. Since $f$ is quasi-compact we see that $X$ is quasi-compact. Thus we may choose an affine scheme $U$ and a surjective étale morphism $U \to X$, see Properties of Spaces, Lemma 66.6.3. By Lemma 67.11.1 we get an exact sequence

where $R = U \times _ X U$. As $X \to Y$ is quasi-separated we see that $R \to U \times _ Y U$ is a quasi-compact monomorphism. This implies that $R$ is a quasi-compact separated scheme (as $U$ and $Y$ are affine at this point). Hence $a : U \to Y$ and $b : R \to Y$ are quasi-compact and quasi-separated morphisms of schemes. Thus by Descent, Proposition 35.9.4 the sheaves $a_*(\mathcal{F}|_ U)$ and $b_*(\mathcal{F}|_ R)$ are quasi-coherent (see also the discussion preceding this lemma). This implies that $f_*\mathcal{F}$ is a kernel of quasi-coherent modules, and hence itself quasi-coherent, see Properties of Spaces, Lemma 66.29.7. $\square$