Lemma 37.37.1. Let $f : X \to Y$ be a morphism of schemes. Let $x \in X$ with image $y \in Y$. Assume
$Y$ is integral and geometrically unibranch at $y$,
$f$ is locally of finite type,
there is a specialization $x' \leadsto x$ such that $f(x')$ is the generic point of $Y$,
$f$ is unramified at $x$.
Then $f$ is étale at $x$.
Proof. We may replace $X$ and $Y$ by suitable affine open neighbourhoods of $x$ and $y$. Then $Y$ is the spectrum of a domain $A$ and $X$ is the spectrum of a finite type $A$-algebra $B$. Let $\mathfrak q \subset B$ be the prime ideal corresponding to $x$ and $\mathfrak p \subset A$ the prime ideal corresponding to $y$. The local ring $A_\mathfrak p = \mathcal{O}_{Y, y}$ is geometrically unibranch. The ring map $A \to B$ is unramified at $\mathfrak q$. Also, the point $x'$ in (3) corresponds to a prime ideal $\mathfrak q' \subset \mathfrak q$ such that $A \cap \mathfrak q' = (0)$. It follows that $A_\mathfrak p \to B_\mathfrak q$ is injective. We conclude by More on Algebra, Lemma 15.107.2. $\square$
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