Lemma 103.17.9 (0GRD)—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 103.17.9 (cite)

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Lemma 103.17.9. Let $\mathcal{X}$ be a locally Noetherian algebraic stack. Let $\mathcal{F}$ and $\mathcal{G}$ be coherent be $\mathcal{O}_\mathcal {X}$ -modules. Then the internal hom $hom(\mathcal{F}, \mathcal{G})$ constructed in Lemma 103.10.8 is a coherent $\mathcal{O}_\mathcal {X}$ -module.

Proof. Let $U \to \mathcal{X}$ be a smooth surjective morphism from a scheme. By item (12) in Section 103.12 we see that the restriction of $hom(\mathcal{F}, \mathcal{G})$ to $U$ is the Hom sheaf of the restrictions. Hence this lemma follows from the case of algebraic spaces, see Cohomology of Spaces, Lemma 69.12.5. $\square$

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