Lemma 37.58.9 (09TE)—The Stacks project

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A direct summand of a module inherits the property of being finitely presented relative to a base.

Lemma 37.58.9. Let $X \to S$ be a morphism of schemes which is locally of finite type. Let $\mathcal{F}, \mathcal{F}'$ be quasi-coherent $\mathcal{O}_ X$ -modules. If $\mathcal{F} \oplus \mathcal{F}'$ is finitely presented relative to $S$ , then so are $\mathcal{F}$ and $\mathcal{F}'$ .

Proof. Translation of the result of More on Algebra, Lemma 15.80.10 into the language of schemes. $\square$

Comments (1)

Comment #915 by Matthieu Romagny on August 11, 2014 at 20:44

Suggested slogan: Direct summands of quasi-coherent modules are quasi-coherent

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