A root of the uniformizer kills the prime-to-residue-characteristic ramification required in specialization

The ring $A_n$ is a DVR. It is finite free over $A$, with unique maximal ideal generated by $\theta$: reduction modulo $t$ has unique prime $(\theta )$, and integrality makes every maximal ideal lie over that of $A$. It has dimension one, and its maximal ideal has one generator, so it is regular local and hence a domain and DVR. By [F1] the normalization $B_n$ in the indicated generic algebra is finite and a product of normal domain factors. We may complete $A_n$ faithfully flatly by [F2]. This also completes each local factor of $B_n$: its maximal-adic topology is cofinal with the $t$-adic topology, and [F3]'s Chinese remainder decomposition expresses the completion as a product of completed local DVRs. By regularity of completion in [F2], that product is normal and is exactly the integral closure in its generic algebra. Thus it is enough to prove the assertion after completing $A$, then to descend étaleness using [F3].

Let $A$ now be complete and let $E\mathrm{/}F$ be any finite field extension. Its maximal separable subextension $F_s\mathrm{/}F$ is finite, and $E\mathrm{/}{F}_s$ is purely inseparable by [F4]. The normalization $A_s$ of $A$ in $F_s$ is a finite normal domain by [F1], and is $t$-adically complete by [F2]. The quotient $A_s\mathrm{/}t{A}_s$ is Artinian. If it had several local factors, [F3] would give a nontrivial idempotent; this would lift to $A_s$ by the complete-pair and idempotent assertions in [F2], contradicting that $A_s$ is a domain. Thus $A_s$ is local, since all its maximal ideals lie over $(t)$. It is one-dimensional and hence a DVR by [F1], complete for its maximal ideal because that topology is cofinal with the $t$-adic topology. Its fraction field $F_s$ is complete for $\mathrm{\mid }a\mathrm{\mid }=2^{-v_s(a)}$: a Cauchy sequence is eventually contained in a fixed uniformizer multiple of $A_s$, where completeness applies.

Over complete $A$, construct an unramified local extension $A^{\mathrm{u}\mathrm{r}}$ with residue field a separable closure of $\kappa (A)$. For every finite separable residue extension use [F2] to lift its field algebra to a finite étale local $A$-algebra. It is a DVR with the same uniformizer (a one-dimensional regular local ring is a DVR by one dimensional regular local rings are dvrs, and is a domain by regular local rings are domains and cohen macaulay): its residue ring is a field, its maximal ideal is generated by $t$, and its dimension is one. Maps and composita lift uniquely by [F2], so choosing compatible residue embeddings gives a directed union $A^{\mathrm{u}\mathrm{r}}$. Every nonzero element has integral valuation and is a uniformizer power times a unit in some stage; consequently the union is a DVR, with value group $\mathbb{Z}$. It is flat and faithfully flat over $A$, as a filtered union of finite free local extensions. It is henselian: a polynomial and a simple residue root occur in some finite residue stage, and the simple root lifts in that complete stage by [F2]. Its residue field is separably closed. Normalization commutes with this extension by [F1], first at each finite étale stage and then in the union, because every integral equation involves finitely many coefficients. Finally complete this DVR, retaining the notation $A^{\mathrm{u}\mathrm{r}}$. Its residue field remains separably closed and the completion is faithfully flat by [F2]. Normalization after completion is the product of the completed local DVR factors by the argument of step 1.1.

Let $C$ be one local factor of the normalization of $A^{\mathrm{u}\mathrm{r}}$ in $L{\otimes }_F\mathrm{Frac}(A^{\mathrm{u}\mathrm{r}})$. Its generic extension is Galois of degree $e_0$ dividing $n$: scalar extension of a finite Galois algebra is a product of Galois field extensions with subgroup Galois groups, as seen by the action on its embeddings. By [F1]–[F2], $C$ is a DVR finite over the complete $A^{\mathrm{u}\mathrm{r}}$. It is complete by the finite-module completion theorem in [F2], and its $t$-adic and maximal-adic topologies are cofinal. Hence it is henselian by the complete-pair criterion in [F2]. Write its ramification index as $e$ and its residue degree as $f$. It is torsion-free over the base DVR, hence flat by Over a principal ideal domain flatness is equivalent to torsion-freeness, and finite over its Noetherian base, hence finitely presented. The local freeness proof in Finite étale algebras have finite locally free underlying modules therefore makes it finite free. Consequently reduction modulo $t$ and the filtration by its uniformizer give $e_0=ef$. The residue extension of a separably closed field is purely inseparable, hence $f$ is a power of $p$ if $p>0$ by A finite purely inseparable extension in characteristic $p$ has degree a power of $p$, and is $1$ in characteristic zero. As $e_0\mid n$ is prime to $p$, we get $f=1$ and $e=e_0$. Write $t=u{\pi }^e$ for its uniformizer $\pi$ and unit $u$. The equation $T^e-u$ has a residue root and invertible derivative, so henselianity lifts an $e$th root $v$ of $u$ to $C$. Then $(v\pi {)}^e=t$. The powers $1,v\pi ,\dots ,(v\pi {)}^{e-1}$ are linearly independent over $\mathrm{Frac}(A^{\mathrm{u}\mathrm{r}})$: with the valuation of $C$ normalized by $\mathrm{val}(\pi )=1$, nonzero terms in a linear relation have distinct valuations modulo $e$, so a unique term would have smallest valuation and the sum could not vanish. The root therefore has degree $e$, and the generic extension is exactly $\mathrm{Frac}(A^{\mathrm{u}\mathrm{r}})(t^{1\mathrm{/}e})$. The base contains every $e$th root of unity by the same simple-root lifting, and the extension is cyclic. This proves the necessary tame-inertia description explicitly.

Since $e\mid n$, adjoining $\theta =t^{1\mathrm{/}n}$ contains $t^{1\mathrm{/}e}={\theta }^{n\mathrm{/}e}$. Thus every field factor in step 3.1 becomes split after this root extension, and its normalized algebra is a product of copies of the base DVR $A^{\mathrm{u}\mathrm{r}}[\theta ]$. The normalization of $A_n$ therefore becomes finite étale after the faithfully flat unramified extension and completion used in steps 1.1 and 2.1. Descent in [F3] makes $B_n\mathrm{/}{A}_n$ finite étale. For a smooth trait family the special fibre is reduced, so its base uniformizer $t$ has valuation one at each vertical codimension-one DVR; the common base-root extension therefore has this effect simultaneously at all such DVRs.

If $E=F_s$, step 1.2 suffices. Otherwise $\mathrm{char}{F}_s=p>0$ and finite pure inseparability gives a common $q=p^r$ with $x^q\in F_s$ for all $x\in E$. Define $w(x)=v_s(x^q)\mathrm{/}q$ for $x\mathrm{\ne }0$ and $w(0)=+\mathrm{\infty }$. Frobenius and the valuation axioms make $w$ a valuation extending $v_s$, with value group a subgroup of $q^{-1}\mathbb{Z}$ containing $\mathbb{Z}$. Its valuation ring $C=\{x:w(x)\ge 0\}$ is exactly the integral closure of $A_s$ in $E$: $x\in C$ satisfies the monic equation $T^q-x^q$ over $A_s$, while integral $x$ has $x^q\in A_s$ because $A_s$ is integrally closed. The discrete value group has a least positive element, and each nonzero ideal of $C$ is generated by an element of its least value, so $C$ is a DVR. The absolute value $2^{-w}$ is an $F_s$-vector-space norm on $E$. For an $F_s$-basis $b_i$, [F4] bounds the coefficients of every $x\in C$ by a fixed constant. Thus $C\subseteq {\sum }_i{s}^{-N}{A}_s{b}_i$ for some $N$, where $s$ is a uniformizer of $A_s$. Being an $A_s$-submodule of this finite lattice over the Noetherian ring $A_s$, $C$ is finite over $A_s$, hence over $A$. It is complete by [F2] and cofinality of the uniformizer-adic topologies. Transitivity of integrality identifies $C$ with the normalization of $A$ in $E$. Its residue field is finite over that of $A$, hence unchanged if the latter is algebraically closed. All clauses hold with the stated AC and characteristic hypotheses.