The Stacks project

Proof. This follows from Cohomology on Sites, Lemma 21.16.1. Namely, let $\mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (X_{spaces, {\acute{e}tale}})$ be the set of quasi-compact and quasi-separated spaces étale over $X$. Note that if $U \in \mathcal{B}$ then, because $U$ is quasi-compact, the collection of finite coverings $\{ U_ i \to U\}$ with $U_ i \in \mathcal{B}$ is cofinal in the set of coverings of $U$ in $X_{spaces, {\acute{e}tale}}$. By Morphisms of Spaces, Lemma 67.8.10 the set $\mathcal{B}$ satisfies all the assumptions of Cohomology on Sites, Lemma 21.16.1. Since $X \in \mathcal{B}$ we win. $\square$

Proof. We will use that the morphism of topoi $f_{small} : X_{small} \to Y_{small}$ comes from the morphism of sites $f_{spaces, {\acute{e}tale}} : X_{spaces, {\acute{e}tale}} \to Y_{spaces, {\acute{e}tale}}$ corresponding to the continuous functor $V \longmapsto X \times _ Y V$, see Properties of Spaces, Lemma 66.18.8. We will apply Cohomology on Sites, Lemma 21.16.4 to this morphism of sites. Since every object of $Y_{spaces, {\acute{e}tale}}$ has a covering by affine objects, it suffices to show that for $V$ affine and étale over $Y$ we have $H^ p(X \times _ Y V, \mathcal{F}) = \mathop{\mathrm{colim}}\nolimits H^ p(X \times _ Y V, \mathcal{F}_ i)$. Since $V$ is affine, the algebraic space $X \times _ Y V$ is quasi-compact and quasi-separated. Hence we can apply Lemma 69.5.1 to conclude. $\square$

Proof. The displayed equality is a special case of Modules on Sites, Lemma 18.27.12. In order to apply it, we need to check the hypotheses of Sites, Lemma 7.17.8 part (4) for the site $X_{\acute{e}tale}$. In order to do this, we will check hypotheses (2)(a), (2)(b), (2)(c) of Sites, Remark 7.17.9. Namely, let $\mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (X_{\acute{e}tale})$ be the set of affine objects. Then

1. Since $X$ is quasi-compact, there exists a $U \in \mathcal{B}$ such that $U \to X$ is surjective (Properties of Spaces, Lemma 66.6.3), hence $h_ U^\# \to *$ is surjective.

2. For $U \in \mathcal{B}$ every étale covering $\{ U_ i \to U\} _{i \in I}$ of $U$ can be refined by a finite étale covering $\{ U_ j \to U\} _{j = 1, \ldots , m}$ with $U_ j \in \mathcal{B}$ (Topologies, Lemma 34.4.4).

3. For $U, U' \in \mathop{\mathrm{Ob}}\nolimits (X_{\acute{e}tale})$ we have $h_ U^\# \times h_{U'}^\# = h_{U \times _ X U'}^\#$. If $U, U' \in \mathcal{B}$, then $U \times _ X U'$ is quasi-compact because $X$ is quasi-separated, see Morphisms of Spaces, Lemma 67.8.10 for example. Hence we can find a surjective étale morphism $U'' \to U \times _ X U'$ with $U'' \in \mathcal{B}$ (Properties of Spaces, Lemma 66.6.3). In other words, we have morphisms $U'' \to U$ and $U'' \to U'$ such that the map $h_{U''}^\# \to h_ U^\# \times h_{u'}^\#$ is surjective.

For the final statement, observe that the inclusion functor $\mathit{QCoh}(\mathcal{O}_ X) \to \textit{Mod}(\mathcal{O}_ X)$ commutes with colimits and that finitely presented modules are quasi-coherent. See Properties of Spaces, Lemma 66.29.7. $\square$