Lemma 37.65.2 (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 37.65.1 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 37.65.1 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
Comments (0)
There are also: