Lemma 96.14.2. Let $S$ be a scheme. Let $\mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ be a category fibred in groupoids which is representable by an algebraic space $F$. The functor (96.10.2.1) defines an equivalence

\[ \mathit{QCoh}(\mathcal{O}_\mathcal {X}) \to \mathit{QCoh}(\mathcal{O}_ F),\quad \mathcal{F} \longmapsto \mathcal{F}|_{F_{\acute{e}tale}} \]

with quasi-inverse given by $\mathcal{G} \mapsto \pi _ F^*\mathcal{G}$. This equivalence is compatible with pullback for morphisms between categories fibred in groupoids representable by algebraic spaces.

**Proof.**
By Lemma 96.11.4 we may work with the étale topology. We will use the notation and results of Lemma 96.10.1 without further mention. Recall that the restriction functor $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X}) \to \textit{Mod}(F_{\acute{e}tale}, \mathcal{O}_ F)$, $\mathcal{F} \mapsto \mathcal{F}|_{F_{\acute{e}tale}}$ is given by $i_ F^*$. By Lemma 96.14.1 or by Modules on Sites, Lemma 18.23.4 we see that $\mathcal{F}|_{F_{\acute{e}tale}}$ is quasi-coherent if $\mathcal{F}$ is quasi-coherent. Hence we get a functor as indicated in the statement of the lemma and we get a functor $\pi _ F^*$ in the opposite direction. Since $\pi _ F \circ i_ F = \text{id}$ we see that $i_ F^*\pi _ F^*\mathcal{G} = \mathcal{G}$.

For $\mathcal{F}$ in $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$ there is a canonical map $\pi _ F^*(\mathcal{F}|_{F_{\acute{e}tale}}) \to \mathcal{F}$, namely the map adjoint to the identification $\mathcal{F}|_{F_{\acute{e}tale}} = \pi _{F, *}\mathcal{F}$. We will show that this map is an isomorphism if $\mathcal{F}$ is a quasi-coherent module on $\mathcal{X}$. Choose a scheme $U$ and a surjective étale morphism $U \to F$. Denote $x : U \to \mathcal{X}$ the corresponding object of $\mathcal{X}$ over $U$. It suffices to show that $\pi _ F^*(\mathcal{F}|_{F_{\acute{e}tale}}) \to \mathcal{F}$ is an isomorphism after restricting to $\mathcal{X}_{\acute{e}tale}/x = (\mathit{Sch}/U)_{\acute{e}tale}$. Since $U \to F$ is étale, it follows from Remark 96.10.2 that

\[ \pi _ F^*(\mathcal{F}|_{F_{\acute{e}tale}})|_{\mathcal{X}_{\acute{e}tale}/x} = \pi _ U^*(\mathcal{F}|_{U_{\acute{e}tale}}) \]

and that the restriction of the map $\pi _ F^*(\mathcal{F}|_{F_{\acute{e}tale}}) \to \mathcal{F}$ to $\mathcal{X}_{\acute{e}tale}/x = (\mathit{Sch}/U)_{\acute{e}tale}$ is equal to the corresponding map $\pi _ U^*(\mathcal{F}|_{U_{\acute{e}tale}}) \to \mathcal{F}|_{(\mathit{Sch}/U)_{\acute{e}tale}}$. Since we have seen the result is true for schemes in Descent, Section 35.8^{1} we conclude.

Compatibility with pullbacks follows from the fact that the quasi-inverse is given by $\pi _ F^*$ and the commutative diagram of ringed topoi in Lemma 96.10.3.
$\square$

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