The Stacks project

Proof. In the statement of the proposition, and in this proof we think of a quasi-coherent sheaf on a scheme as a quasi-coherent sheaf on the small étale site of that scheme. This is permissible by the results of Descent, Sections 35.8, 35.9, and 35.10.

The existence of $\alpha $ comes from the fact that $\varphi \circ t = \varphi \circ s$ and that pullback is functorial in the morphism, see discussion surrounding Equation (66.26.0.1). In exactly the same way, i.e., by functoriality of pullback, we see that the isomorphism $\alpha $ satisfies condition (1) of Groupoids, Definition 39.14.1. To see condition (2) of the definition it suffices to see that $\alpha $ is an isomorphism which is clear. The construction $\mathcal{F} \mapsto (\varphi ^*\mathcal{F}, \alpha )$ is clearly functorial in the quasi-coherent sheaf $\mathcal{F}$. Hence we obtain the functor from left to right in the displayed formula of the lemma.

Conversely, suppose that $(\mathcal{F}, \alpha )$ is a quasi-coherent sheaf on $(U, R, s, t, c)$. Let $V \to X$ be an object of $X_{\acute{e}tale}$. In this case the morphism $V' = U \times _ X V \to V$ is a surjective étale morphism of schemes, and hence $\{ V' \to V\} $ is an étale covering of $V$. Moreover, the quasi-coherent sheaf $\mathcal{F}$ pulls back to a quasi-coherent sheaf $\mathcal{F}'$ on $V'$. Since $R = U \times _ X U$ with $t = \text{pr}_0$ and $s = \text{pr}_0$ we see that $V' \times _ V V' = R \times _ X V$ with projection maps $V' \times _ V V' \to V'$ equal to the pullbacks of $t$ and $s$. Hence $\alpha $ pulls back to an isomorphism $\alpha ' : \text{pr}_0^*\mathcal{F}' \to \text{pr}_1^*\mathcal{F}'$, and the pair $(\mathcal{F}', \alpha ')$ is a descend datum for quasi-coherent sheaves with respect to $\{ V' \to V\} $. By Descent, Proposition 35.5.2 this descent datum is effective, and we obtain a quasi-coherent $\mathcal{O}_ V$-module $\mathcal{F}_ V$ on $V_{\acute{e}tale}$. To see that this gives a quasi-coherent sheaf on $X_{\acute{e}tale}$ we have to show (by Lemma 66.29.3) that for any morphism $f : V_1 \to V_2$ in $X_{\acute{e}tale}$ there is a canonical isomorphism $c_ f : \mathcal{F}_{V_1} \to \mathcal{F}_{V_2}$ compatible with compositions of morphisms. We omit the verification. We also omit the verification that this defines a functor from the category on the right to the category on the left which is inverse to the functor described above. $\square$

Proof. This proof is a repeat of the proof in the case of schemes, see Properties, Proposition 28.23.4. We advise the reader to read that proof first.

Part (1) means $\mathit{QCoh}(\mathcal{O}_ X)$ (a) has all colimits, (b) filtered colimits are exact, and (c) has a generator, see Injectives, Section 19.10. By Lemma 66.29.7 colimits in $\mathit{QCoh}(\mathcal{O}_ X)$ exist and agree with colimits in $\textit{Mod}(\mathcal{O}_ X)$. By Modules on Sites, Lemma 18.14.2 filtered colimits are exact. Hence (a) and (b) hold.

To construct a generator, choose a presentation $X = U/R$ so that $(U, R, s, t, c)$ is an étale groupoid scheme and in particular $s$ and $t$ are flat morphisms of schemes. Pick a cardinal $\kappa $ as in Groupoids, Lemma 39.15.7. Pick a collection $(\mathcal{E}_ t, \alpha _ t)_{t \in T}$ of $\kappa $-generated quasi-coherent modules on $(U, R, s, t, c)$ as in Groupoids, Lemma 39.15.6. Let $\mathcal{F}_ t$ be the quasi-coherent module on $X$ which corresponds to the quasi-coherent module $(\mathcal{E}_ t, \alpha _ t)$ via the equivalence of categories of Proposition 66.32.1. Then we see that every quasi-coherent module $\mathcal{H}$ is the directed colimit of its quasi-coherent submodules which are isomorphic to one of the $\mathcal{F}_ t$. Thus $\bigoplus _ t \mathcal{F}_ t$ is a generator of $\mathit{QCoh}(\mathcal{O}_ X)$ and we conclude that (c) holds. The assertions on limits and injectives hold in any Grothendieck abelian category, see Injectives, Theorem 19.11.7 and Lemma 19.13.2.

Proof of (2). To construct $Q$ we use the following general procedure. Given an object $\mathcal{F}$ of $\textit{Mod}(\mathcal{O}_ X)$ we consider the functor

This functor transforms colimits into limits, hence is representable, see Injectives, Lemma 19.13.1. Thus there exists a quasi-coherent sheaf $Q(\mathcal{F})$ and a functorial isomorphism $\mathop{\mathrm{Hom}}\nolimits _ X(\mathcal{G}, \mathcal{F}) = \mathop{\mathrm{Hom}}\nolimits _ X(\mathcal{G}, Q(\mathcal{F}))$ for $\mathcal{G}$ in $\mathit{QCoh}(\mathcal{O}_ X)$. By the Yoneda lemma (Categories, Lemma 4.3.5) the construction $\mathcal{F} \leadsto Q(\mathcal{F})$ is functorial in $\mathcal{F}$. By construction $Q$ is a right adjoint to the inclusion functor. The fact that $Q(\mathcal{F}) \to \mathcal{F}$ is an isomorphism when $\mathcal{F}$ is quasi-coherent is a formal consequence of the fact that the inclusion functor $\mathit{QCoh}(\mathcal{O}_ X) \to \textit{Mod}(\mathcal{O}_ X)$ is fully faithful. $\square$