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".



reference for "curves over S are locally the base change of a curve over S' which is finite type over R"
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 $S\_\lambda$.


