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Let $\pi: X \to S$ be a projective morphism of schemes of relative dimension $\le 1$. Furthermore, we know that $S$ is affine, Noetherian and integral and $X$ is reduced.

Let $a$ be the generic point of $S$ and $X_a$ be the generic fiber of $S$. Then there exists a purely inseparable extension $L/k(a)$ such that the normalization $Y$ of $(X \times_{\operatorname{spec} k(a)} \operatorname{spec} L)_{red}$ is smooth over $L$. This result is EGA IV 17.15.14

I would like to extend this to a (surjective radicial) map $S' \to S$ inducing $\operatorname{Spec} \to \operatorname{Spec}k(a)$ on generic points. Moreover I would like to extend $Y$ to a smooth projective scheme $\overline{X} \to S'$ equipped with a finite surjective morphism $\overline{X} \to X \times_{S} S'=:X'$ which is an isomorphism outside a proper closed subset.

In all of this, I don't care about shrinking $S$ to an open subset.

All of this comes from page 6 of [SGA IV Exposé XIV] (http://www.normalesup.org/~forgogozo/SGA4/14/14.pdf) and it is related to proving that the higher direct images of constructible sheaves are again constructible.

What I have tried: Let $S=\operatorname{Spec} R$ and let $R'$ be the integral closure of $R$ in $L$. We know that $Y$ is projective since it is the normalization of a projective scheme over a field. This means that $Y$ is cut out by homogeneous equations $f_1, \cdots, f_n$ with coefficients in $L$. Clearing denominators, this gives us equations over $R'$.

Now we know that the Jacobian of those equations is invertible in the generic point, hence we can find a nonempty open subset $S' \subset \operatorname{Spec} R'$ where it is invertible. Then define $\overline{X}$ to be the projective scheme defined by $f_1, \cdots, f_n$, it is smooth by the Jacobian criterion.

Then I guess that the map $Y \to (X \times_{\operatorname{spec} k(a)} \operatorname{spec} L)$ extends to a map $\overline{X} \to X'$ which is finite and surjective (possibly after shrinking $S$ again). Here, my source refers to EGA IV 9.6.1 which says that all these properties are constructible on $S$.

Question: Is this train of thought correct? How do I show that $\overline{X} \to X'$ is an isomorphism outside a proper closed subset? How does one define the morphism $\overline{X} \to X'$?

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