The Stacks project

Lemma 94.14.1. 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 category $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$ has colimits and they agree with colimits in the category $\textit{Mod}(\mathcal{O}_\mathcal {X})$.

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

  5. Given $\mathcal{F}, \mathcal{G}$ in $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$ with $\mathcal{F}$ locally of finite presentation on $\mathcal{X}_{fppf}$ 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 $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$.

Proof. Properties (3), (4), and (5) were proven in Lemma 94.11.9. Part (1) is Proposition 94.13.1. Part (2) follows from Groupoids in Spaces, Lemma 76.12.5 as discussed above. $\square$


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