The Stacks project

Lemma 78.14.2. Let $B \to S$ as in Section 78.3. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. If $s$, $t$ are flat, then there exists a set $T$ and a family of objects $(\mathcal{F}_ t, \alpha _ t)_{t \in T}$ of $\mathit{QCoh}(U, R, s, t, c)$ such that every object $(\mathcal{F}, \alpha )$ is the directed colimit of its submodules isomorphic to one of the objects $(\mathcal{F}_ t, \alpha _ t)$.

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

\[ I = \{ W, W \times _ U W, V, V', V \times _ R V'\} \]

and the set of morphisms of schemes

\[ \Phi = \{ \text{pr}_ i : W \times _ U W \to W, V \to W, V' \to W, V \times _ R V' \to V, V \times _ R V' \to V'\} \]

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

\[ \mathit{QCoh}(U, R, s, t, c) \longrightarrow \textit{CQC}(X) \]

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

\[ \xymatrix{ & V \times _ R V' \ar[ld] \ar[rd] \\ V \ar[rd] \ar[dd] & & V' \ar[ld] \ar[dd] \\ & R \ar@<-1ex>[dd]_ s \ar@<1ex>[dd]^ t \\ W \ar[rd] & & W \ar[ld] \\ & U } \]

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. $\square$

[1] In fact the functor is fully faithful, but we won't need this.

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