The Stacks project

Lemma 37.37.3. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of schemes. Let $x \in X$ with image $y \in Y$. Assume

  1. $Y$ is integral and geometrically unibranch at $y$,

  2. $g \circ f$ is étale at $x$,

  3. there is a specialization $x' \leadsto x$ such that $f(x')$ is the generic point of $Y$.

Then $f$ is étale at $x$ and $g$ is étale at $y$.

Proof. After replacing $X$ by an open neighbourhood of $x$ we may assume $g \circ f$ is étale. Then we find $f$ is unramified by Morphisms, Lemmas 29.35.16 and 29.36.5. Hence $f$ is étale at $x$ by Lemma 37.37.1. Then by étale descent we see that $g$ is étale at $y$, see for example Descent, Lemma 35.14.4. $\square$

Comments (2)

Comment #8619 by on

Condition (2) isn't necessary in the presence of (3): in a neighbourhood of the morphism is of finite type by (3) combined with Lemma 29.15.8 (and the trivial fact that etale morphisms are of finite type).

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