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The Stacks project

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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