The Stacks project

Lemma 75.7.9. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ 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 $Y$, 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 $Y$ is locally Noetherian, then the category of coherent $\mathcal{O}_ X$-modules with support proper over $Y$ is a Serre subcategory (Homology, Definition 12.10.1) of the abelian category of coherent $\mathcal{O}_ X$-modules.

Proof. Proof of (1). Let $T$, $T'$ be the support of $\mathcal{F}$ and $\mathcal{G}$. Then all the sheaves mentioned in (1) have support contained in $T \cup T'$. Thus the assertion itself is clear from Lemmas 75.7.3 and 75.7.6 provided we check that these sheaves are finite type and quasi-coherent. For quasi-coherence we refer the reader to Properties of Spaces, Section 66.29. For “finite type” we refer the reader to Properties of Spaces, Section 66.30.

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 of Spaces, Lemma 67.23.5 and by the description of the category of coherent modules in Cohomology of Spaces, Section 69.12. $\square$

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