The Stacks project

Lemma 37.65.4 (Variant with separable extensions over curves). Let $f : X \to S$ be a flat, finite type morphism of schemes. Assume $S$ is Nagata, integral with function field $K$, and regular of dimension $1$. Assume the field extensions $\kappa (\eta )/K$ are separable for every generic point $\eta $ of an irreducible component of $X$. Then there exists a finite separable extension $L/K$ such that in the diagram

\[ \xymatrix{ Y \ar[rd]_ g \ar[r]_-\nu & X \times _ S T \ar[d] \ar[r] & X \ar[d]_ f \\ & T \ar[r] & S } \]

the morphism $g$ is smooth at all generic points of fibres. Here $T$ is the normalization of $S$ in $\mathop{\mathrm{Spec}}(L)$ and $\nu : Y \to X \times _ S T$ is the normalization.

Proof. This follows from Lemma 37.65.3 in exactly the same manner that Lemma 37.65.2 follows from Theorem 37.65.1. $\square$

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