Surface completed polynomial generic fibre

The field $\kappa (\mathfrak{r})$ is finite over $K=\mathrm{Frac}(A)$; for every finite purely inseparable extension $L\mathrm{/}\kappa (\mathfrak{r})$ choose a finite $A\subseteq A^{\mathrm{\prime }}\subseteq L$ with $\mathrm{Frac}(A^{\mathrm{\prime }})=L$. Such an $A^{\mathrm{\prime }}$ is a complete local domain, and the finite-completion-factor lemma reduces the problem to the case $\kappa (\mathfrak{r})=\mathrm{Frac}(A)$.

In that case $\mathfrak{r}K[t]=(t-f)$ for some $f\in K$, and with $T=\widehat{A[t{]}_{\mathfrak{q}}}{\otimes }_{A}K$ one reduces first to a finite regular power-series subring $A_0\subseteq A$ and to factors of the finite base change of its completed polynomial local ring $B_0$; $B_0$ is regular because $A_0[t{]}_{{\mathfrak{q}}_0}$ is.

Here is the needed regularity of $T$, including the finite-base reduction. Put $K_0=\mathrm{Frac}{A}_0$ and $T_0=\widehat{A_0[t{]}_{{\mathfrak{q}}_0}}{\otimes }_{A_0}{K}_0$, which is regular by localization and completion of regular rings. A finite separable extension of $K_0$ is monogenic with invertible polynomial derivative, so its tensor with $T_0$ is a standard smooth algebra with zero-dimensional regular fibres and is regular. Follow this by degree-$p$ purely inseparable steps. For a finite $A_0$-subalgebra $B$ of the preceding field $M$, completion factors express $T_0{\otimes }_{K_0}M$ as the product of the completed polynomial localizations of $B[t]$, localized to $M$. After multiplying the next defining element by a $p$th power from $K_0^{\times }$, choose $B$ to contain it; this clears denominators without changing the field extension. A derivation of $B$ detecting that non-$p$th-power element extends fixing $t$ and then through localization and completion. It preserves the product factors, since derivations kill idempotents, and its value on that element is a unit after localization to $M$. Each preceding factor is regular by induction, so the monogenic derivative-unit criterion makes its next field tensor regular. Every finite extension has a separable subextension followed by a purely inseparable one. The finite-factor formula of step 2.1 therefore makes $T$ regular.

In the reduced case of step 1.1, the fibre is $T\mathrm{/}(t-f)$, with $f\in K$. The derivation $\mathrm{\partial }\mathrm{/}\mathrm{\partial }t$ fixes $K$, extends through localization and completion, and takes the value $1$ on $t-f$. The general derivative-unit quotient criterion therefore makes this fibre regular. The same calculation after the finite reduction works for every finite purely inseparable extension of the original residue field, so the field test proves geometric regularity. In characteristic zero all finite extensions in the coefficient reduction are separable. AC is inherited from the suppliers.