The Stacks project

Lemma 37.64.2. Let $f : X \to Y$ be a morphism of schemes. The following are equivalent

  1. $X \to Y$ is weakly étale, and

  2. for every $x \in X$ the ring map $\mathcal{O}_{Y, f(x)} \to \mathcal{O}_{X, x}$ is weakly étale.

Proof. Observe that under both assumptions (1) and (2) the morphism $f$ is flat. Thus we may assume $f$ is flat. Let $x \in X$ with image $y = f(x)$ in $Y$. There are canonical maps of rings

\[ \mathcal{O}_{X, x} \otimes _{\mathcal{O}_{Y, y}} \mathcal{O}_{X, x} \longrightarrow \mathcal{O}_{X \times _ Y X, \Delta _{X/Y}(x)} \longrightarrow \mathcal{O}_{X, x} \]

where the first map is a localization (hence flat) and the second map is a surjection. Condition (1) means that the second arrow is flat for all $x$. Condition (2) is that the composition is flat for all $x$. Thus the equivalence by Algebra, Lemma 10.39.18 part (2). $\square$

Comments (2)

Comment #8542 by Michael Barz on

I'm a little confused by the proof of lemma 094Q -- how does lemma 092C imply that condition (2) implies condition (1)? It seems like we want to prove that is flat as a -module, so we should take in lemma 092C, but then I'm not sure if lemma 092C actually applies.

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