The Stacks project

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

  1. $f$ is smooth,

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

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

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

Proof. Assume (1). Then $f$ is flat by Morphisms, Lemma 29.34.9. The projections $X \times _ Y X \to X$ are smooth by Morphisms, Lemma 29.34.5. Hence the diagonal is a section to a smooth morphism and hence a regular immersion, see Divisors, Lemma 31.22.8. Hence (1) $\Rightarrow $ (2). The implication (2) $\Rightarrow $ (3) is Lemma 37.59.9. The implication (3) $\Rightarrow $ (4) is Lemma 37.59.4. The interesting implication (4) $\Rightarrow $ (1) follows immediately from Divided Power Algebra, Lemma 23.10.2. $\square$


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