Lemma 76.45.4 (0DII)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 76: More on Morphisms of Spaces
Section 76.45: Relative pseudo-coherence
Lemma 76.45.4 (cite)

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Lemma 76.45.4. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $E$ in $D_\mathit{QCoh}(\mathcal{O}_ X)$. If $f$ is flat and locally of finite presentation, then the following are equivalent

$E$ is pseudo-coherent relative to $Y$, and
$E$ is pseudo-coherent on $X$.

Proof. By étale localization and the definitions we may assume $X$ and $Y$ are schemes. For the case of schemes this follows from More on Morphisms, Lemma 37.59.18. $\square$

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