Lemma 96.14.2 (0GQD)—The Stacks project

Lemma 96.14.2. Let $S$ be a scheme. Let $\mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ be a category fibred in groupoids wich 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. $\square$

[1] Namely, if $U$ is a scheme and $\mathcal{F}$ is quasi-coherent on $(\mathit{Sch}/U)_{\acute{e}tale}$, then $\mathcal{F} = \mathcal{H}^ a$ for some quasi-coherent module $\mathcal{H}$ on the scheme $U$ by Descent, Proposition 35.8.9. In other words, $\mathcal{F} = (\text{id}_{{\acute{e}tale},Zar})^*\mathcal{H}$ by Descent, Remark 35.8.6 with notation as in Descent, Lemma 35.8.5. Then we have $\text{id}_{{\acute{e}tale},Zar} = \pi _ U \circ \text{id}_{small,{\acute{e}tale},Zar}$ and hence we see that $\mathcal{F} = \pi _ U^*\mathcal{G}$ where $\mathcal{G} = (\text{id}_{small,{\acute{e}tale},Zar})^*\mathcal{H}$ is quasi-coherent. Then $\pi _ U^*i_ U^*\mathcal{F} = \pi _ U^*i_ U^*\pi _ U^*\mathcal{G} = \pi _ U^*\mathcal{G} = \mathcal{F}$ as desired.

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