The Stacks project

Lemma 37.62.2 (Variant 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 $1$. Then there exists a finite 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. Choose a finite affine open covering $S = \bigcup \mathop{\mathrm{Spec}}(A_ i)$. Then $K$ is equal to the fraction field of $A_ i$ for all $i$. Let $X_ i = X \times _ S \mathop{\mathrm{Spec}}(A_ i)$. Choose $L_ i/K$ as in Theorem 37.62.1 for the morphism $X_ i \to \mathop{\mathrm{Spec}}(A_ i)$. Let $B_ i \subset L_ i$ be the integral closure of $A_ i$ and let $Y_ i$ be the normalized base change of $X$ to $B_ i$. Let $L/K$ be a finite extension dominating each $L_ i$. Let $T_ i \subset T$ be the inverse image of $\mathop{\mathrm{Spec}}(A_ i)$. For each $i$ we get a commutative diagram

\[ \xymatrix{ g^{-1}(T_ i) \ar[r] \ar[d] & Y_ i \ar[r] \ar[d] & X \times _ S \mathop{\mathrm{Spec}}(A_ i) \ar[d] \\ T_ i \ar[r] & \mathop{\mathrm{Spec}}(B_ i) \ar[r] & \mathop{\mathrm{Spec}}(A_ i) } \]

and in fact the left hand square is a normalized base change as discussed at the beginning of the section. In the proof of Theorem 37.62.1 we have seen that the smooth locus of $Y \to T$ contains the inverse image in $g^{-1}(T_ i)$ of the set of points where $Y_ i$ is smooth over $B_ i$. This proves the lemma. $\square$

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