The Stacks project

Proof. Let $V \to Y$ be an étale morphism where $V$ is an affine scheme. Set $U = V \times _ Y X$ and denote $f' : U \to V$ the induced morphism. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. By Properties of Spaces, Lemma 66.26.2 we have $R^ if'_*(\mathcal{F}|_ U) = (R^ if_*\mathcal{F})|_ V$. Since the property of being a quasi-coherent module is local in the étale topology on $Y$ (see Properties of Spaces, Lemma 66.29.6) we may replace $Y$ by $V$, i.e., we may assume $Y$ is an affine scheme.

Assume $Y$ is affine. 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 $g : U \to X$, see Properties of Spaces, Lemma 66.6.3. Picture

The morphism $g : U \to X$ is representable, separated and quasi-compact because $X$ is quasi-separated. Hence the lemma holds for $g$ (by the discussion above the lemma). It also holds for $f \circ g : U \to Y$ (as this is a morphism of affine schemes).

In the situation described in the previous paragraph we will show by induction on $n$ that $IH_ n$: for any quasi-coherent sheaf $\mathcal{F}$ on $X$ the sheaves $R^ if\mathcal{F}$ are quasi-coherent for $i \leq n$. The case $n = 0$ follows from Morphisms of Spaces, Lemma 67.11.2. Assume $IH_ n$. In the rest of the proof we show that $IH_{n + 1}$ holds.

Let $\mathcal{H}$ be a quasi-coherent $\mathcal{O}_ U$-module. Consider the Leray spectral sequence

Cohomology on Sites, Lemma 21.14.7. As $R^ qg_*\mathcal{H}$ is quasi-coherent by $IH_ n$ all the sheaves $R^ pf_*R^ qg_*\mathcal{H}$ are quasi-coherent for $p \leq n$. The sheaves $R^{p + q}(f \circ g)_*\mathcal{H}$ are all quasi-coherent (in fact zero for $p + q > 0$ but we do not need this). Looking in degrees $\leq n + 1$ the only module which we do not yet know is quasi-coherent is $E_2^{n + 1, 0} = R^{n + 1}f_*g_*\mathcal{H}$. Moreover, the differentials $d_ r^{n + 1, 0} : E_ r^{n + 1, 0} \to E_ r^{n + 1 + r, 1 - r}$ are zero as the target is zero. Using that $\mathit{QCoh}(\mathcal{O}_ X)$ is a weak Serre subcategory of $\textit{Mod}(\mathcal{O}_ X)$ (Properties of Spaces, Lemma 66.29.7) it follows that $R^{n + 1}f_*g_*\mathcal{H}$ is quasi-coherent (details omitted).

Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Set $\mathcal{H} = g^*\mathcal{F}$. The adjunction mapping $\mathcal{F} \to g_*g^*\mathcal{F} = g_*\mathcal{H}$ is injective as $U \to X$ is surjective étale. Consider the exact sequence

where $\mathcal{G}$ is the cokernel of the first map and in particular quasi-coherent. Applying the long exact cohomology sequence we obtain

The cokernel of the first arrow is quasi-coherent and we have seen above that $R^{n + 1}f_*g_*\mathcal{H}$ is quasi-coherent. Thus $R^{n + 1}f_*\mathcal{F}$ has a $2$-step filtration where the first step is quasi-coherent and the second a submodule of a quasi-coherent sheaf. Since $\mathcal{F}$ is an arbitrary quasi-coherent $\mathcal{O}_ X$-module, this result also holds for $\mathcal{G}$. Thus we can choose an exact sequence $0 \to \mathcal{A} \to R^{n + 1}f_*\mathcal{G} \to \mathcal{B}$ with $\mathcal{A}$, $\mathcal{B}$ quasi-coherent $\mathcal{O}_ Y$-modules. Then the kernel $\mathcal{K}$ of $R^{n + 1}f_*g_*\mathcal{H} \to R^{n + 1}f_*\mathcal{G} \to \mathcal{B}$ is quasi-coherent, whereupon we obtain a map $\mathcal{K} \to \mathcal{A}$ whose kernel $\mathcal{K}'$ is quasi-coherent too. Hence $R^{n + 1}f_*\mathcal{F}$ sits in an exact sequence

with all modules quasi-coherent except for possibly $R^{n + 1}f_*\mathcal{F}$. We conclude that $R^{n + 1}f_*\mathcal{F}$ is quasi-coherent, i.e., $IH_{n + 1}$ holds as desired. $\square$