The Stacks project

Proof. This lemma is a generalization of Groupoids, Lemma 39.15.7 which deals with the case of a groupoid in schemes. We can't quite use the same argument, so we use the material on “crystals of quasi-coherent sheaves” we developed above.

Choose a scheme $W$ and a surjective étale morphism $W \to U$. Choose a scheme $V$ and a surjective étale morphism $V \to W \times _{U, s} R$. Choose a scheme $V'$ and a surjective étale morphism $V' \to R \times _{t, U} W$. Consider the collection of schemes

and the set of morphisms of schemes

Set $X = (I, \Phi )$. Recall that we have defined a category $\textit{CQC}(X)$ of crystals of quasi-coherent sheaves on $X$. There is a functor

which assigns to $(\mathcal{F}, \alpha )$ the sheaf $\mathcal{F}|_ W$ on $W$, the sheaf $\mathcal{F}|_{W \times _ U W}$ on $W \times _ U W$, the pullback of $\mathcal{F}$ via $V \to W \times _{U, s} R \to W \to U$ on $V$, the pullback of $\mathcal{F}$ via $V' \to R \times _{t, U} W \to W \to U$ on $V'$, and finally the pullback of $\mathcal{F}$ via $V \times _ R V' \to V \to W \times _{U, s} R \to W \to U$ on $V \times _ R V'$. As comparison maps $\{ \alpha _\phi \} _{\phi \in \Phi }$ we use the obvious ones (coming from associativity of pullbacks) except for the map $\phi = \text{pr}_{V'} : V \times _ R V' \to V'$ we use the pullback of $\alpha : t^*\mathcal{F} \to s^*\mathcal{F}$ to $V \times _ R V'$. This makes sense because of the following commutative diagram

The functor displayed above isn't an equivalence of categories. However, since $W \to U$ is surjective étale it is faithful¹. Since all the morphisms in the diagram above are flat we see that it is an exact functor of abelian categories. Moreover, we claim that given $(\mathcal{F}, \alpha )$ with image $(\{ \mathcal{F}_ i\} _{i \in I}, \{ \alpha _\phi \} _{\phi \in \Phi })$ there is a $1$-to-$1$ correspondence between quasi-coherent submodules of $(\mathcal{F}, \alpha )$ and $(\{ \mathcal{F}_ i\} _{i \in I}, \{ \alpha _\phi \} _{\phi \in \Phi })$. Namely, given a submodule of $(\{ \mathcal{F}_ i\} _{i \in I}, \{ \alpha _\phi \} _{\phi \in \Phi })$ compatibility of the submodule over $W$ with the projection maps $W \times _ U W \to W$ will guarantee the submodule comes from a quasi-coherent submodule of $\mathcal{F}$ (by Properties of Spaces, Proposition 66.32.1) and compatibility with $\alpha _{\text{pr}_{V'}}$ will insure this subsheaf is compatible with $\alpha $ (details omitted).

Choose a cardinal $\kappa $ as in Lemma 78.14.1 for the system $X = (I, \Phi )$. It is clear from Properties, Lemma 28.23.2 that there is a set of isomorphism classes of $\kappa $-generated crystals in quasi-coherent sheaves on $X$. Hence the result is clear. $\square$