The Stacks project

Remark 111.34.9. In Exercises 111.34.7 and 111.34.8 we are given a morphism $f : X \to S$ where $S$ is the spectrum of a Dedekind ring $A$ with $f$ flat and surjective with geometrically irreducible generic fibre. In both cases the outcome is that there exists a finite surjective morphism $S' \to S$ such that the base change $X_{S'} \to S'$ does have a section. It turns out this holds if $A$ is excellent, its residue fields at maximal ideals are algebraic extensions of finite fields, and $\mathop{\mathrm{Pic}}\nolimits (A')$ is torsion when $A'$ is the integral closure of $A$ in a finite extension of its fraction field, see [MB1]. For example, if $A = \mathbf{Z}$ or $A = \mathbf{F}_ p[x]$ or a finite extension of these. However, it turns out that there exists a Dedekind ring $A$ with finite residue fields at maximal primes whose Picard group has a nontorsion element, see [Goldman], and the result is false for the spectrum of such a ring. Exercise 111.34.10 gives a geometric example where it doesn't work.