Lemma 30.26.9. Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $\mathcal{F}$, $\mathcal{G}$ be finite type, quasi-coherent $\mathcal{O}_ X$-module.

If the supports of $\mathcal{F}$, $\mathcal{G}$ are proper over $S$, then the same is true for $\mathcal{F} \oplus \mathcal{G}$, for any extension of $\mathcal{G}$ by $\mathcal{F}$, for $\mathop{\mathrm{Im}}(u)$ and $\mathop{\mathrm{Coker}}(u)$ given any $\mathcal{O}_ X$-module map $u : \mathcal{F} \to \mathcal{G}$, and for any quasi-coherent quotient of $\mathcal{F}$ or $\mathcal{G}$.

If $S$ is locally Noetherian, then the category of coherent $\mathcal{O}_ X$-modules with support proper over $S$ is a Serre subcategory (Homology, Definition 12.10.1) of the abelian category of coherent $\mathcal{O}_ X$-modules.

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