The Stacks project

Lemma 29.35.13. Let $f : X \to S$ be a morphism.

  1. If $f$ is unramified, then the diagonal morphism $\Delta : X \to X \times _ S X$ is an open immersion.

  2. If $f$ is locally of finite type and $\Delta $ is an open immersion, then $f$ is unramified.

  3. If $f$ is locally of finite presentation and $\Delta $ is an open immersion, then $f$ is G-unramified.

Proof. The first statement follows from Algebra, Lemma 10.151.4. The second statement from the fact that $\Omega _{X/S}$ is the conormal sheaf of the diagonal morphism (Lemma 29.32.7) and hence clearly zero if $\Delta $ is an open immersion. $\square$

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