A specialization is represented by a complete DVR trait

Choose a Noetherian affine neighbourhood $\mathrm{Spec}A$ of $s_0$. It contains $s_1$, since an open set contains every generalization of each of its points. Write their primes as $\mathfrak{p}\subseteq \mathfrak{q}$ and set $B=(A\mathrm{/}\mathfrak{p}{)}_{\mathfrak{q}\mathrm{/}\mathfrak{p}}$. This is a local Noetherian domain whose generic and closed points map to $s_1,s_0$. If they coincide, take $R=\kappa (s_0)\mathrm{⟦}t\mathrm{⟧}$ and the constant map. Otherwise $B$ is not a field. Let $V$ be a valuation ring of $\mathrm{Frac}B$ dominating $B$, by [F1]. For generators $a_1,\dots ,a_r$ of its maximal ideal choose $a_r$ of smallest valuation. Then $C=B[a_1\mathrm{/}{a}_r,\dots ,a_{r-1}\mathrm{/}{a}_r]\subset V$ is Noetherian and ${\mathfrak{m}}_{B}C=a_{r}C$ is proper. A prime $\mathfrak{n}$ minimal over it has height one by [F2] and contracts to ${\mathfrak{m}}_B$. Thus $D=C_{\mathfrak{n}}$ is a one-dimensional Noetherian local domain dominating $B$, with the same fraction field.

We give the needed normalization argument even when $D$ is not excellent. Put $F=\mathrm{Frac}D$ and let $M\subset F$ be any $D$-submodule. For $0\mathrm{\ne }x\in {\mathfrak{m}}_D$, put $\mathrm{\ell }={\mathrm{length}}_D(D\mathrm{/}xD)<\mathrm{\infty }$; finiteness follows since its only prime is maximal and the quotient is Noetherian of dimension zero. If $N\subset F$ is nonzero and finite over $D$, clear denominators so $N\subset D$. The torsion finite module $D\mathrm{/}N$ has finite length and is killed by some power $x^c$, so $x^{c}D\subset N\subset D$. Since multiplication by $x$ is injective, $\mathrm{length}(N\mathrm{/}{x}^{n}N)=n\mathrm{length}(N\mathrm{/}xN)$. For $n\ge c$ the inclusions imply $x^{n+c}D\subset x^{n}N\subset N\subset D$ and $x^{n}N\subset x^{c}D\subset N$, giving $(n-c)\mathrm{\ell }\le \mathrm{length}(N\mathrm{/}{x}^{n}N)\le (n+c)\mathrm{\ell }$. Divide by $n$ and let $n$ increase to obtain $\mathrm{length}(N\mathrm{/}xN)=\mathrm{\ell }$. Any finite strict chain in $M\mathrm{/}xM$ can be witnessed by finitely many elements of $M$; the submodule $N$ they generate has a chain at least as long in $N\mathrm{/}xN$. Thus $\mathrm{length}(M\mathrm{/}xM)\le \mathrm{\ell }$.

Let $E$ be the integral closure of $D$ in $F$. For a nonzero ideal $I\subset E$, a nonzero $y\in I$ can be written $a\mathrm{/}b$ with $a,b\in D$ nonzero; hence $0\mathrm{\ne }a=by\in I\cap D$. Step 2.1 with $M=E$ says $E\mathrm{/}aE$ has finite length (the unit case gives zero). Therefore $I\mathrm{/}aE$ is finite over $D$, and lifts of its generators together with $a$ generate $I$ over $E$. Every ideal of $E$ is finite, so $E$ is Noetherian. By lying over choose a prime $\mathfrak{r}$ over ${\mathfrak{m}}_D$. Its height is one: after inverting $D\setminus \{0\}$ the integral closure is $F$, so the only prime contracting to zero is zero; primes above the maximal ideal cannot be strictly comparable, since localizing and then quotienting by the lower prime gives an integral algebraic domain over a field, hence a field. Thus $R=E_{\mathfrak{r}}$ is normal local Noetherian of dimension one and is a DVR by [F3]. Its local inclusion $B\subset R\subset F$ gives the required two point images.

Let $\pi$ be a uniformizer of $R$. By [F4], $\widehat{R}$ is Noetherian local with maximal ideal $(\pi )$ and the same residue field; flatness makes $\pi$ a nonzerodivisor. Each nonzero element lies in a largest power $({\pi }^n)$, because the completion is separated, and is ${\pi }^n$ times a unit. Products of two such elements are nonzero, so $\widehat{R}$ is a domain, and this description is the DVR property. The map $R\to \widehat{R}$ is injective by [F4]; its generic point contracts to zero and its closed point to $(\pi )$. Both images in $S$ are therefore unchanged.

To make the residue field algebraically closed, fix an algebraic closure $\bar{k}$ of $k=R\mathrm{/}(\pi )$ and well order its elements, using AC. At a successor stage, given a DVR $T$ with uniformizer $\pi$ and residue subfield $k_T\subset \bar{k}$, take the monic minimal polynomial of the next element over $k_T$, lift its coefficients to $T$, and form $T^{\mathrm{\prime }}=T[z]\mathrm{/}(P)$. This is finite free over $T$, and its reduction modulo $\pi$ is the required residue field extension. Every maximal ideal lies over $(\pi )$, so $T^{\mathrm{\prime }}$ is local with maximal ideal $(\pi )$. It is Noetherian; $\pi$ is a nonzerodivisor by freeness and the $\pi$-adic intersection is zero by Krull intersection. As in step 4.1 every nonzero element is a unit times a power of $\pi$, proving that $T^{\mathrm{\prime }}$ is a DVR and that $T\to T^{\mathrm{\prime }}$ is injective. At limit stages take unions. Every nonzero element is still a unit times a power of the same $\pi$, and every nonzero ideal has an element of minimal exponent, so is principal. The union is therefore a Noetherian DVR with residue field $\bar{k}$. Its completion is again a DVR by step 4.1, with residue field $\bar{k}$. All extensions are local and injective, preserving the point images. AC is used for the valuation overring, algebraic closure and the transfinite choices, and is inherited from the listed suppliers.