The Stacks project

Remark 37.38.3. The second condition in Lemma 37.38.2 is necessary even if $x$ is a closed point of a positive dimensional fibre. An example is the following: Let $k$ be a field of characteristic $p > 0$ which is imperfect. Let $a \in k$ be an element which is not a $p$th power. Let $\mathfrak m = (x, y^ p - a) \subset k[x, y]$. This corresponds to a closed point $w$ of $X = \mathbf{A}^2_ k$. Set $S = \mathbf{A}^1_ k$ and let $f : X \to S$ be the morphism corresponding to $k[x] \to k[x, y]$. Then there does not exist any commutative diagram

\[ \xymatrix{ S' \ar[rr]_ h \ar[rd]_ g & & X \ar[ld]^ f \\ & S } \]

with $g$ ├ętale and $w$ in the image of $h$. This is clear as the residue field extension $\kappa (w)/\kappa (f(w))$ is purely inseparable, but for any $s' \in S'$ with $g(s') = f(w)$ the extension $\kappa (s')/\kappa (f(w))$ would be separable.

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