96.15 Quasi-coherent sheaves on algebraic stacks
Let $\mathcal{X}$ be an algebraic stack over $S$. By Algebraic Stacks, Lemma 94.16.2 we can find an equivalence $[U/R] \to \mathcal{X}$ where $(U, R, s, t, c)$ is a smooth groupoid in algebraic spaces. Then
\[ \mathit{QCoh}(\mathcal{O}_\mathcal {X}) \cong \mathit{QCoh}(\mathcal{O}_{[U/R]}) \cong \mathit{QCoh}(U, R, s, t, c) \]
where the second equivalence is Proposition 96.14.3. Hence the category of quasi-coherent sheaves on an algebraic stack is equivalent to the category of quasi-coherent modules on a smooth groupoid in algebraic spaces. In particular, by Groupoids in Spaces, Lemma 78.12.6 we see that $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$ is abelian!
There is something slightly disconcerting about our current setup. It is that the fully faithful embedding
\[ \mathit{QCoh}(\mathcal{O}_\mathcal {X}) \longrightarrow \textit{Mod}(\mathcal{O}_\mathcal {X}) \]
is in general not exact. However, exactly the same thing happens for schemes: for most schemes $X$ the embedding
\[ \mathit{QCoh}(\mathcal{O}_ X) \cong \mathit{QCoh}((\mathit{Sch}/X)_{fppf}, \mathcal{O}_ X) \longrightarrow \textit{Mod}((\mathit{Sch}/X)_{fppf}, \mathcal{O}_ X) \]
isn't exact, see Descent, Lemma 35.10.2. Parenthetically, the example in the proof of Descent, Lemma 35.10.2 shows that in general the strictly full embedding $\mathit{QCoh}(\mathcal{O}_\mathcal {X}) \to \textit{LQCoh}(\mathcal{O}_\mathcal {X})$ isn't exact either.
We collect all the results obtained so far in a single statement.
Lemma 96.15.1. Let $\mathcal{X}$ be an algebraic stack over $S$.
If $[U/R] \to \mathcal{X}$ is a presentation of $\mathcal{X}$ then there is a canonical equivalence $\mathit{QCoh}(\mathcal{O}_\mathcal {X}) \cong \mathit{QCoh}(U, R, s, t, c)$.
The category $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$ is abelian.
The inclusion functor $\mathit{QCoh}(\mathcal{O}_\mathcal {X}) \to \textit{Mod}(\mathcal{O}_\mathcal {X})$ is right exact but not exact in general.
The category $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$ has colimits and they agree with colimits in the category $\textit{Mod}(\mathcal{O}_\mathcal {X})$.
Given $\mathcal{F}, \mathcal{G}$ in $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$ the tensor product $\mathcal{F} \otimes _{\mathcal{O}_\mathcal {X}} \mathcal{G}$ in $\textit{Mod}(\mathcal{O}_\mathcal {X})$ is an object of $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$.
Given $\mathcal{F}, \mathcal{G}$ in $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$ with $\mathcal{F}$ finite locally free the sheaf $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {X}}(\mathcal{F}, \mathcal{G})$ in $\textit{Mod}(\mathcal{O}_\mathcal {X})$ is an object of $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$.
Given a short exact sequence $0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$ in $\textit{Mod}(\mathcal{O}_\mathcal {X})$ with $\mathcal{F}_1$ and $\mathcal{F}_3$ quasi-coherent, then $\mathcal{F}_2$ is quasi-coherent.
Proof.
Properties (4), (5), and (6) were proven in Lemma 96.12.5. Part (1) is Proposition 96.14.3. Part (2) follows from part (1) and Groupoids in Spaces, Lemma 78.12.6 as discussed above. Right exactness of the inclusion functor in (3) follows from (4); please compare with Homology, Lemma 12.7.2. For the nonexactness of the inclusion functor in part (3) see Descent, Lemma 35.10.2. To see (7) observe that it suffices to check the restriction of $\mathcal{F}_2$ to the big site of a scheme is quasi-coherent (Lemma 96.11.3), hence this follows from the corresponding part of Descent, Lemma 35.10.2.
$\square$
Next we construct the coherator for modules on an algebraic stack.
Proposition 96.15.2. Let $\mathcal{X}$ be an algebraic stack over $S$.
The category $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$ is a Grothendieck abelian category. Consequently, $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$ has enough injectives and all limits.
The inclusion functor $\mathit{QCoh}(\mathcal{O}_\mathcal {X}) \to \textit{Mod}(\mathcal{O}_\mathcal {X})$ has a right adjoint1
\[ Q : \textit{Mod}(\mathcal{O}_\mathcal {X}) \longrightarrow \mathit{QCoh}(\mathcal{O}_\mathcal {X}) \]
such that for every quasi-coherent sheaf $\mathcal{F}$ the adjunction mapping $Q(\mathcal{F}) \to \mathcal{F}$ is an isomorphism.
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
\[ \mathit{QCoh}(\mathcal{O}_\mathcal {X})^{opp} \longrightarrow \textit{Sets}, \quad \mathcal{G} \longmapsto \mathop{\mathrm{Hom}}\nolimits _\mathcal {X}(\mathcal{G}, \mathcal{F}) \]
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.
$\square$
Comments (0)