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The Stacks project

Lemma 70.14.4. Let S be a scheme. Let f : X \to Y be a quasi-compact and quasi-separated morphism of algebraic spaces over S. Let \mathcal{I} \subset \mathcal{O}_ Y be a quasi-coherent sheaf of ideals of finite type. Let \mathcal{F} be a quasi-coherent \mathcal{O}_ X-module. Let \mathcal{F}' \subset \mathcal{F} be the subsheaf of sections annihilated by f^{-1}\mathcal{I}\mathcal{O}_ X. Then f_*\mathcal{F}' \subset f_*\mathcal{F} is the subsheaf of sections annihilated by \mathcal{I}.

Proof. Omitted. Hint: The assumption that f is quasi-compact and quasi-separated implies that f_*\mathcal{F} is quasi-coherent (Morphisms of Spaces, Lemma 67.11.2) so that Lemma 70.14.2 applies to \mathcal{I} and f_*\mathcal{F}. \square


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