Lemma 37.61.9. Let $f : X \to Y$ be a morphism of schemes. Then $X \to Y$ is weakly étale in each of the following cases

$X \to Y$ is a flat monomorphism,

$X \to Y$ is an open immersion,

$X \to Y$ is flat and unramified,

$X \to Y$ is étale.

Lemma 37.61.9. Let $f : X \to Y$ be a morphism of schemes. Then $X \to Y$ is weakly étale in each of the following cases

$X \to Y$ is a flat monomorphism,

$X \to Y$ is an open immersion,

$X \to Y$ is flat and unramified,

$X \to Y$ is étale.

**Proof.**
If (1) holds, then $\Delta _{X/Y}$ is an isomorphism (Schemes, Lemma 26.23.2), hence certainly $f$ is weakly étale. Case (2) is a special case of (1). The diagonal of an unramified morphism is an open immersion (Morphisms, Lemma 29.35.13), hence flat. Thus a flat unramified morphism is weakly étale. An étale morphism is flat and unramified (Morphisms, Lemma 29.36.5), hence (4) follows from (3).
$\square$

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