Lemma 76.52.13 (0GFK)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 76: More on Morphisms of Spaces
Section 76.52: Relatively perfect objects
Lemma 76.52.13 (cite)

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Lemma 76.52.13. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$ such that the structure morphism $f : X \to S$ is flat and locally of finite presentation. Let $E$ be a pseudo-coherent object of $D(\mathcal{O}_ X)$. The following are equivalent

$E$ is $S$-perfect, and
$E$ is locally bounded below and for every point $s \in S$ the object $L(X_ s \to X)^*E$ of $D(\mathcal{O}_{X_ s})$ is locally bounded below.

Proof. Since everything is local we immediately reduce to the case that $X$ and $S$ are affine, see Lemma 76.52.3. This case is handled by Derived Categories of Schemes, Lemma 36.35.13. $\square$

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