Lemma 37.58.5. 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.

Lemma 37.58.5. 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 37.57.5.
$\square$

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