Strict henselization of a DVR and smooth sections

Consider the directed system of pairs $(R^{\mathrm{\prime }},\alpha )$, where $R^{\mathrm{\prime }}$ is a local $R$-algebra which is etale over $R$ and whose residue field embeds as an $R$-subalgebra $k\to R^{\mathrm{\prime }}\mathrm{/}{\mathfrak{m}}_{R^{\mathrm{\prime }}}\hookrightarrow k^s$ over those already chosen, with transition maps the local $R$-algebra maps over $k^s$. Each such $R^{\mathrm{\prime }}$ is flat over $R$ because etale maps are flat; its residue field is a finite separable extension of $k$, so ${\mathfrak{m}}_{R^{\mathrm{\prime }}}=\pi R^{\mathrm{\prime }}$ and $\dim R^{\mathrm{\prime }}=1$; being regular of dimension one it is a DVR with uniformizer $\pi$ by [F4]. The transition maps are injective local maps preserving $\pi$. In the union $R^{\mathrm{s}\mathrm{h}}=\mathrm{colim}{R}^{\mathrm{\prime }}$ over the directed system, every nonzero element lies in a finite stage as ${\pi }^n$ times a unit, and every nonzero ideal has a least such exponent, so it is principal; hence $R^{\mathrm{s}\mathrm{h}}$ is a DVR with uniformizer $\pi$ and residue field $k^s$, realized as a filtered colimit of local etale $R$-algebras.

The ring $R^{\mathrm{s}\mathrm{h}}$ embeds into a separable closure of $K$ and hence is $R$-torsion-free, so it is $R$-flat by [F5] and $R$-faithfully flat because it is local with nonzero closed fibre. Its strict henselianity follows from the colimit description: an etale neighbourhood of a point with residual coefficients involves finitely many elements of $R^{\mathrm{s}\mathrm{h}}$, hence is defined at a finite stage, and adjoining that pointed neighbourhood to the directed system exhibits its residual lift in the limit; this proves the henselian neighbourhood-lifting criterion of [F1] without assuming the individual stages are henselian.

Let $V$ be smooth over $\mathrm{\Lambda }$ and let $x\in V_{\kappa }(\kappa )$. By [F3] choose a standard smooth chart $U=\mathrm{Spec}C$ around $x$ with $C=(\mathrm{\Lambda }[t_1,\dots ,t_n]\mathrm{/}(f_1,\dots ,f_m){)}_g$ and a unit $m\times m$ Jacobian minor. Cutting the chart by the $n-m$ coordinate differences $t_i-\widetilde{t_i(x)}$ with chosen lifts of the residue coordinates to $\mathrm{\Lambda }$ that are not involved in that minor produces an etale $\mathrm{\Lambda }$-scheme through $x$: the original $m$ equations and the $n-m$ coordinate differences have a unit $n\times n$ Jacobian minor. Its special fibre has the same $\kappa$-point $x$. The henselian lifting property of [F1] gives a $\mathrm{\Lambda }$-section of this etale neighbourhood, hence a section of $V$ through $x$.

Every nonempty open of the smooth special fibre contains a $\kappa$-rational point by [F6], since $\kappa$ is separably closed; step 3.1 lifts that point to a section of $V$. Thus the specializations of sections meet every nonempty open and are dense in $V_{\kappa }$. Empty special fibre makes the density assertion vacuous.

Now suppose $\mathrm{\Lambda }$ is a strictly henselian DVR with fraction field $F$, $V$ is smooth integral of finite type with nonempty special fibre, and $U\subset V_F$ is dense. Give $V_F\setminus U$ its reduced closed structure and let $Z$ be its schematic closure. Its ideal is saturated under multiplication by $\pi$, since it contracts an ideal after inverting $\pi$. At a generic point $\xi$ of a special-fibre component, the local ring $B={\mathcal{O}}_{V,\xi }$ has maximal ideal $(\pi )$, because the smooth special fibre is reduced and its local ring there is a field. Since $V$ is integral and flat, $\pi$ is a nonzero nonunit; the principal ideal theorem (Krull's principal ideal theorem) gives $\dim B=1$, and $B$ is regular, hence a DVR by [F4]. The localized ideal of $Z$ is nonzero (the generic complement is proper in the integral $V$) and $\pi$-saturated, so it is all of $B$: every nonzero proper DVR ideal is $({\pi }^n)$ and fails saturation. Thus $Z$ misses every special-fibre generic point. A rational point of the nonempty smooth open $V_{\kappa }\setminus Z$ exists by [F6] and lifts to a section by step 3.1. Its generic point cannot lie in the closed $Z$, since its specialization does not. This gives a section with generic point in $U$.