The Stacks project

Lemma 37.56.11. Let $X \to S$ be a morphism of schemes which is locally of finite type. Let $\mathcal{F}$ be an $\mathcal{O}_ X$-module. Then

  1. $\mathcal{F}$ is $m$-pseudo-coherent relative to $S$ for all $m > 0$,

  2. $\mathcal{F}$ is $0$-pseudo-coherent relative to $S$ if and only if $\mathcal{F}$ is a finite type $\mathcal{O}_ X$-module,

  3. $\mathcal{F}$ is $(-1)$-pseudo-coherent relative to $S$ if and only if $\mathcal{F}$ is quasi-coherent and finitely presented relative to $S$.

Proof. Part (1) is immediate from the definition. To see part (3) we may work locally on $X$ (both properties are local). Thus we may assume $X$ and $S$ are affine. Choose a closed immersion $i : X \to \mathbf{A}^ n_ S$. Then we see that $\mathcal{F}$ is $(-1)$-pseudo-coherent relative to $S$ if and only if $i_*\mathcal{F}$ is $(-1)$-pseudo-coherent, which is true if and only if $i_*\mathcal{F}$ is an $\mathcal{O}_{\mathbf{A}^ n_ S}$-module of finite presentation, see Cohomology, Lemma 20.44.9. A module of finite presentation is quasi-coherent, see Modules, Lemma 17.11.2. By Morphisms, Lemma 29.4.1 we see that $\mathcal{F}$ is quasi-coherent if and only if $i_*\mathcal{F}$ is quasi-coherent. Having said this part (3) follows. The proof of (2) is similar but less involved. $\square$

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