The Stacks project

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.

  1. 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 $\mathcal{G}$.

  2. 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.

Proof. Proof of (1). Let $Z$, $Z'$ be the support of $\mathcal{F}$ and $\mathcal{G}$. Then all the sheaves mentioned in (1) have support contained in $Z \cup Z'$. Thus the assertion itself is clear from Lemmas 30.26.3 and 30.26.6 provided we check that these sheaves are finite type and quasi-coherent. For quasi-coherence we refer the reader to Schemes, Section 26.24. For “finite type” we suggest the reader take a look at Modules, Section 17.9.

Proof of (2). The proof is the same as the proof of (1). Note that the assertions make sense as $X$ is locally Noetherian by Morphisms, Lemma 29.15.6 and by the description of the category of coherent modules in Section 30.9. $\square$

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