The Stacks project

Proof. Let $U$ be a scheme étale over $F$. Then $\mathcal{F}|_{U_{\acute{e}tale}} = (\mathcal{F}|_{F_{\acute{e}tale}})|_{U_{\acute{e}tale}}$. This is clear but see also Remark 96.10.2. Thus the assertion follows from the definitions. $\square$

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

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 35.8¹ 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. $\square$

Proof. We will construct quasi-inverse functors

where $\mathit{QCoh}(U, R, s, t, c)$ denotes the category of quasi-coherent modules on the groupoid $(U, R, s, t, c)$.

Let $\mathcal{F}$ be an object of $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$. Denote $\mathcal{U}$, $\mathcal{R}$ the categories fibred in groupoids corresponding to $U$ and $R$. Denote $x$ the (defining) object of $\mathcal{X}$ over $U$. Recall that we have a $2$-commutative diagram

See Groupoids in Spaces, Lemma 78.20.3. By Lemma 96.3.3 the $2$-arrow inherent in the diagram induces an isomorphism $\alpha : t^*x^*\mathcal{F} \to s^*x^*\mathcal{F}$ which satisfies the cocycle condition over $\mathcal{R} \times _{s, \mathcal{U}, t} \mathcal{R}$; this is a consequence of Groupoids in Spaces, Lemma 78.23.1. Thus if we set $\mathcal{G} = x^*\mathcal{F}|_{U_{\acute{e}tale}}$ then the equivalence of categories in Lemma 96.14.2 (used several times compatibly with pullbacks) gives an isomorphism $\alpha : t_{small}^*\mathcal{G} \to s_{small}^*\mathcal{G}$ satisfying the cocycle condition on $R \times _{s, U, t} R$, i.e., $(\mathcal{G}, \alpha )$ is an object of $\mathit{QCoh}(U, R, s, t, c)$. The rule $\mathcal{F} \mapsto (\mathcal{G}, \alpha )$ is our functor from left to right.

Construction of the functor in the other direction. Let $(\mathcal{G}, \alpha )$ be an object of $\mathit{QCoh}(U, R, s, t, c)$. According to Lemma 96.13.2 the stackification map $[U/_{\! p}R] \to [U/R]$ (see Groupoids in Spaces, Definition 78.20.1) induces an equivalence of categories of quasi-coherent sheaves. Thus it suffices to construct a quasi-coherent module $\mathcal{F}$ on $[U/_{\! p}R]$.

Recall that an object $x = (T, u)$ of $[U/_{\! p}R]$ is given by a scheme $T$ and a morphism $u : T \to U$. A morphism $(T, u) \to (T', u')$ is given by a pair $(f, r)$ where $f : T \to T'$ and $r : T \to R$ with $s \circ r = u$ and $t \circ r = u' \circ f$. Let us call a special morphism any morphism of the form $(f, e \circ u' \circ f) : (T, u' \circ f) \to (T', u')$. The category of $(T, u)$ with special morphisms is just the category of schemes over $U$.

With this notation in place, given an object $(T, u)$ of $[U/_{\! p}R]$, we set

Given a morphism $(f, r) : (T, u) \to (T', u')$ we get a map

where the first arrow is pullback along $f$ and the second arrow is $\alpha $. Note that if $(T, r)$ is a special morphism, then this map is just pullback along $f$ as $e_{small}^*\alpha = \text{id}$ by the axioms of a sheaf of quasi-coherent modules on a groupoid. The cocycle condition implies that $\mathcal{F}$ is a presheaf of modules (details omitted). We see that the restriction of $\mathcal{F}$ to $(\mathit{Sch}/T)_{fppf}$ is quasi-coherent by the simple description of the restriction maps of $\mathcal{F}$ in case of a special morphism. Hence $\mathcal{F}$ is a sheaf on $[U/_{\! p}R]$ and quasi-coherent (Lemma 96.11.3).

We omit the verification that the functors constructed above are quasi-inverse to each other. $\square$

Proof. According to Lemma 96.13.2 the stackification map $[U/_{\! p}R] \to [U/R]$ (see Groupoids in Spaces, Definition 78.20.1) induces an equivalence of categories of quasi-coherent sheaves and of fppf $\mathcal{O}$-modules. Thus it suffices to prove the lemma with $\mathcal{X} = [U/_{\! p}R]$. By Proposition 96.14.3 and its proof there exists a quasi-coherent module $(\mathcal{G}, \alpha )$ on $(U, R, s, t, c)$ such that $\mathcal{F}$ is given by the rule $\mathcal{F}(T, u) = \Gamma (T, u^*\mathcal{G})$. In particular $x^*\mathcal{F}|_{U_{\acute{e}tale}} = \mathcal{G}$ and it is clear that the map of the statement of the lemma is injective. Moreover, given a map $\varphi : \mathcal{G} \to x^*\mathcal{H}|_{U_{\acute{e}tale}}$ and given any object $y = (T, u)$ of $[U/_{\! p}R]$ we can consider the map

where the second arrow is the comparison map (96.9.4.1) for the sheaf $\mathcal{H}$. This assignment is compatible with the restriction mappings of the sheaves $\mathcal{F}$ and $\mathcal{G}$ for morphisms of $[U/_{\! p}R]$ if the cocycle condition of the lemma is satisfied. Proof omitted. Hint: the restriction maps of $\mathcal{F}$ are made explicit in terms of $(\mathcal{G}, \alpha )$ in the proof of Proposition 96.14.3. $\square$