Lemma 103.17.6 (0GRA)—The Stacks project

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Table of contents
Part 7: Algebraic Stacks
Chapter 103: Cohomology of Algebraic Stacks
Section 103.17: Coherent sheaves on locally Noetherian stacks
Lemma 103.17.6 (cite)

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Lemma 103.17.6. Let $\mathcal{X}$ be a locally Noetherian algebraic stack. 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 $\mathcal{F}_3$ coherent, then $\mathcal{F}_2$ is coherent.

Proof. By Sheaves on Stacks, Lemma 96.15.1 part (7) we see that $\mathcal{F}_2$ is quasi-coherent. Then we can check that $\mathcal{F}_2$ is coherent by restricting to $U_{\acute{e}tale}$ for some $U \to \mathcal{X}$ surjective and smooth. This follows from Cohomology of Spaces, Lemma 69.12.3. Some details omitted. $\square$

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