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So recently I heard someone claiming that if $X\rightarrow S$ is a smooth curve (not necessarily proper?) and $S$ is an arbitrary scheme over $\text{Spec }R$ (for $R$ sufficiently nice), then there is an fpqc(fppf? can we go as far as etale?) cover $T\rightarrow S$ and a scheme $S'$ of finite type over $R$ factoring $S\rightarrow\text{Spec }R$ with a curve $Y/S'$ such that $X\times_S T \cong Y\times_{S'} T$.

Does anyone have a reference for a precise statement of this form?

I feel like any attempt to look this up would result in a lot of "noise".

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  • $\begingroup$ Are you assuming that $S$ is quasi-compact? I think you can make a counterexample if $S$ is not quasi-compact. $\endgroup$
    – Jason Starr
    Commented Feb 23, 2015 at 22:15
  • $\begingroup$ @JasonStarr I wasn't, what's your counterexample? $\endgroup$
    – Will Chen
    Commented Feb 23, 2015 at 22:29
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    $\begingroup$ "... what's your counterexample?" Let $A$ be $\mathbb{Q}[x_1,x_2,x_3,\dots]$, let $\mathfrak \subset A$ be $\langle x_1,x_3,\dots \rangle$. Let $S$ be $\text{Spec}(A)\setminus \{ \mathfrak{m} \}$. This is the union of the countably many open subsets $U_n = D(x_1)\cup \dots \cup D(x_n)$. Over each open subset $U_n$, $X\times_S U_n$ will be the blowing up of $\mathbb{P}^1\times U_n$ along a particular closed subscheme. For $U_2$, blowup $s(U_2) = \{[1,0]\}\times U_2$ over $Z(x_1)$. Then, over $U_3$, blowup further the strict transform of $s(U_3)$ over $Z(x_1,x_2)$, etc. $\endgroup$
    – Jason Starr
    Commented Feb 23, 2015 at 22:53
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    $\begingroup$ More trivially, you can take for $S$ a disjoint union of points $s_n=\mathrm{Spec}(k_n)$ ($n\in\mathbb{N}$ and $k_n$ a field) and $X=\coprod_n X_n$ where $X_n$ is smooth projective of genus $n$ over $k_n$. $\endgroup$
    – Laurent Moret-Bailly
    Commented Feb 24, 2015 at 8:01

1 Answer 1

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If $S$ is quasicompact and quasiseparated you can even take $T=S$. By Thomason's approximation theorem (in The Grothendieck Festschrift, vol. III), $S$ is a projective limit of $R$-schemes $(S_\lambda)$ of finite presentation, with affine transition maps. By general results from EGA IV.8, $X$ can be obtained by base change from some $S_\lambda$.

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