Lemma 69.12.7. Let S be a scheme. Let X be a locally Noetherian algebraic space over S. Let \mathcal{F} be a coherent \mathcal{O}_ X-module. Let i : Z \to X be the scheme theoretic support of \mathcal{F} and \mathcal{G} the quasi-coherent \mathcal{O}_ Z-module such that i_*\mathcal{G} = \mathcal{F}, see Morphisms of Spaces, Definition 67.15.4. Then \mathcal{G} is a coherent \mathcal{O}_ Z-module.
Proof. The statement of the lemma makes sense as a coherent module is in particular of finite type. Moreover, as Z \to X is a closed immersion it is locally of finite type and hence Z is locally Noetherian, see Morphisms of Spaces, Lemmas 67.23.7 and 67.23.5. Finally, as \mathcal{G} is of finite type it is a coherent \mathcal{O}_ Z-module by Lemma 69.12.2 \square
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