The Stacks project

Lemma 37.62.2. 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 (hence an epimorphism of rings). Condition (1) means that for all $x$ the second arrow is flat. Condition (2) is that for all $x$ the composition is flat. These conditions are equivalent by Algebra, Lemma 10.39.4 and More on Algebra, Lemma 15.104.2. $\square$

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