The Stacks project

Lemma 103.17.7. Let $\mathcal{X}$ be a locally Noetherian algebraic stack. Then $\textit{Coh}(\mathcal{O}_\mathcal {X})$ is a Serre subcategory of $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a map of quasi-coherent $\mathcal{O}_\mathcal {X}$-modules. We have

  1. if $\mathcal{F}$ is coherent and $\varphi $ surjective, then $\mathcal{G}$ is coherent,

  2. if $\mathcal{F}$ is coherent, then $\mathop{\mathrm{Im}}(\varphi )$ is coherent, and

  3. if $\mathcal{G}$ coherent and $\mathop{\mathrm{Ker}}(\varphi )$ parasitic, then $\mathcal{F}$ is coherent.

Proof. Choose a scheme $U$ and a surjective smooth morphism $f : U \to \mathcal{X}$. Then the functor $f^* : \mathit{QCoh}(\mathcal{O}_\mathcal {X}) \to \mathit{QCoh}(\mathcal{O}_ U)$ is exact (Lemma 103.4.1) and moreover by definition $\textit{Coh}(\mathcal{O}_\mathcal {X})$ is the full subcategory of $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$ consisting of objects $\mathcal{F}$ such that $f^*\mathcal{F}$ is in $\textit{Coh}(\mathcal{O}_ U)$. The statement that $\textit{Coh}(\mathcal{O}_\mathcal {X})$ is a Serre subcategory of $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$ follows immediately from this and the corresponding fact for $U$, see Cohomology of Spaces, Lemmas 69.12.3 and 69.12.4. We omit the proof of (1), (2), and (3). Hint: compare with the proof of Lemma 103.17.5. $\square$

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