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

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

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

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

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