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

Lemma 37.55.9. 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. $\square$

Comment #915 by Matthieu Romagny on

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

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