The Stacks project

Lemma 96.15.1. Let $\mathcal{X}$ be an algebraic stack over $S$.

1. If $[U/R] \to \mathcal{X}$ is a presentation of $\mathcal{X}$ then there is a canonical equivalence $\mathit{QCoh}(\mathcal{O}_\mathcal {X}) \cong \mathit{QCoh}(U, R, s, t, c)$.

2. The category $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$ is abelian.

3. The inclusion functor $\mathit{QCoh}(\mathcal{O}_\mathcal {X}) \to \textit{Mod}(\mathcal{O}_\mathcal {X})$ is right exact but not exact in general.

4. The category $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$ has colimits and they agree with colimits in the category $\textit{Mod}(\mathcal{O}_\mathcal {X})$.

5. 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 $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$.

6. 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 $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$.

7. 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.