Lemma 37.64.2 (094Q)—The Stacks project

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 (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. $\square$

Comments (1)

Comment #8542 by Michael Barz on June 30, 2023 at 11:24

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 $\mathcal{O}_{X,x}$ is flat as a $\mathcal{O}_{X\times_Y X, \Delta_{X/Y}(x)}$ -module, so we should take $B = \mathcal{O}_{X\times_Y X, \Delta_{X/Y}(x)}$ in lemma 092C, but then I'm not sure if lemma 092C actually applies.

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