The Stacks project

Proof. This proof is a repeat of the proof in the case of schemes, see Properties, Proposition 28.23.4 and the case of algebraic spaces, see Properties of Spaces, Proposition 66.32.2. We advise the reader to read either of those proofs first.

Part (1) means $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$ (a) has all colimits, (b) filtered colimits are exact, and (c) has a generator, see Injectives, Section 19.10. By Lemma 96.15.1 colimits in $\mathit{QCoh}(\mathcal{O}_ X)$ exist and agree with colimits in $\textit{Mod}(\mathcal{O}_ X)$. By Modules on Sites, Lemma 18.14.2 filtered colimits are exact. Hence (a) and (b) hold.

Choose a presentation $\mathcal{X} = [U/R]$ so that $(U, R, s, t, c)$ is a smooth groupoid in algebraic spaces and in particular $s$ and $t$ are flat morphisms of algebraic spaces. By Lemma 96.15.1 above we have $\mathit{QCoh}(\mathcal{O}_\mathcal {X}) = \mathit{QCoh}(U, R, s, t, c)$. By Groupoids in Spaces, Lemma 78.14.2 there exists a set $T$ and a family $(\mathcal{F}_ t)_{t \in T}$ of quasi-coherent sheaves on $\mathcal{X}$ such that every quasi-coherent sheaf on $\mathcal{X}$ is the directed colimit of its subsheaves which are isomorphic to one of the $\mathcal{F}_ t$. Thus $\bigoplus _ t \mathcal{F}_ t$ is a generator of $\mathit{QCoh}(\mathcal{O}_ X)$ and we conclude that (c) holds. The assertions on limits and injectives hold in any Grothendieck abelian category, see Injectives, Theorem 19.11.7 and Lemma 19.13.2.

Proof of (2). To construct $Q$ we use the following general procedure. Given an object $\mathcal{F}$ of $\textit{Mod}(\mathcal{O}_\mathcal {X})$ we consider the functor

This functor transforms colimits into limits, hence is representable, see Injectives, Lemma 19.13.1. Thus there exists a quasi-coherent sheaf $Q(\mathcal{F})$ and a functorial isomorphism $\mathop{\mathrm{Hom}}\nolimits _\mathcal {X}(\mathcal{G}, \mathcal{F}) = \mathop{\mathrm{Hom}}\nolimits _\mathcal {X}(\mathcal{G}, Q(\mathcal{F}))$ for $\mathcal{G}$ in $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$. By the Yoneda lemma (Categories, Lemma 4.3.5) the construction $\mathcal{F} \leadsto Q(\mathcal{F})$ is functorial in $\mathcal{F}$. By construction $Q$ is a right adjoint to the inclusion functor. The fact that $Q(\mathcal{F}) \to \mathcal{F}$ is an isomorphism when $\mathcal{F}$ is quasi-coherent is a formal consequence of the fact that the inclusion functor $\mathit{QCoh}(\mathcal{O}_\mathcal {X}) \to \textit{Mod}(\mathcal{O}_\mathcal {X})$ is fully faithful. $\square$