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.
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
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