The Stacks project

Lemma 20.47.10. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_ X$-modules.

  1. $\mathcal{F}$ viewed as an object of $D(\mathcal{O}_ X)$ is $0$-pseudo-coherent if and only if $\mathcal{F}$ is a finite type $\mathcal{O}_ X$-module, and

  2. $\mathcal{F}$ viewed as an object of $D(\mathcal{O}_ X)$ is $(-1)$-pseudo-coherent if and only if $\mathcal{F}$ is an $\mathcal{O}_ X$-module of finite presentation.

Proof. Use Lemma 20.47.9 to prove the implications in one direction and Lemma 20.47.8 for the other. $\square$

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