The Stacks project

Proposition 65.32.1. With $S$, $\varphi : U \to X$, and $(U, R, s, t, c)$ as above. For any quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$ the sheaf $\varphi ^*\mathcal{F}$ comes equipped with a canonical isomorphism

\[ \alpha : t^*\varphi ^*\mathcal{F} \longrightarrow s^*\varphi ^*\mathcal{F} \]

which satisfies the conditions of Groupoids, Definition 39.14.1 and therefore defines a quasi-coherent sheaf on $(U, R, s, t, c)$. The functor $\mathcal{F} \mapsto (\varphi ^*\mathcal{F}, \alpha )$ defines an equivalence of categories

\[ \begin{matrix} \text{Quasi-coherent} \\ \mathcal{O}_ X\text{-modules} \end{matrix} \longleftrightarrow \begin{matrix} \text{Quasi-coherent modules} \\ \text{on }(U, R, s, t, c) \end{matrix} \]

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 (65.26.0.1). 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 65.29.3) 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. $\square$


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