Lemma 37.61.5 (068A)—The Stacks project

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Part 2: Schemes
Chapter 37: More on Morphisms
Section 37.61: Perfect morphisms
Lemma 37.61.5 (cite)

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Lemma 37.61.5. Let $f : X \to S$ be a morphism of schemes. The following are equivalent

$f$ is flat and perfect, and
$f$ is flat and locally of finite presentation.

Proof. The implication (2) $\Rightarrow$ (1) is More on Algebra, Lemma 15.82.4. The converse follows from the fact that a pseudo-coherent morphism is locally of finite presentation, see Lemma 37.60.5. $\square$

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