Lemma 37.64.2. Let $f : X \to Y$ be a morphism of schemes. The following are equivalent
$X \to Y$ is weakly étale, and
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
Comment #9127 by Stacks project on