Lemma 33.39.3. Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme. Let $K/k$ be an extension of fields. Let $y \in X_ K$ be a point with image $x$ in $X$. Then $X$ is geometrically unibranch at $x$ if and only if $X_ K$ is geometrically unibranch at $y$.

Proof. Immediate from Lemma 33.39.2 and More on Algebra, Lemma 15.105.7. $\square$

Comment #4079 by Matthieu Romagny on

Typo in last sentence of the statement of the Lemma: "Then X is geometrically unibranch at x if and only if $X_K$ (and not $Y$) is geometrically unibranch at $y$ (and not $Y$)".

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