Lemma 37.59.18. Let $f : X \to Y$ be a finite type morphism of locally Noetherian schemes. Denote $\Delta : X \to X \times _ Y X$ the diagonal morphism. The following are equivalent

$f$ is smooth,

$f$ is flat and $\Delta : X \to X \times _ Y X$ is a regular immersion,

$f$ is flat and $\Delta : X \to X \times _ Y X$ is a local complete intersection morphism,

$f$ is flat and $\Delta : X \to X \times _ Y X$ is perfect.

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