Lemma 37.62.4 (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

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.

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