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\title[Universal test rings]{Universal test rings
for rational Gersten injectivity}
\author{Niels Feld}
\address{IRMAR, University of Rennes, 263 Avenue G\'en\'eral Leclerc,
35000 Rennes, France}
\email{\href{mailto:niels.feld@univ-rennes.fr}{niels.feld@univ-rennes.fr}}
\urladdr{\url{https://nielsfeld.wixsite.com/website}}
\date{3 October 2026}
\subjclass[2020]{19D50, 13F30, 12J20, 03C60}
\keywords{Gersten conjecture, henselian discrete valuation ring, Witt vectors,
Cohen ring, existential closedness, algebraic $K$-theory}
\begin{document}
\begin{abstract}
For each prime $p$, rational Gersten injectivity in a fixed degree
for mixed-characteristic henselian discrete valuation rings of
residue characteristic $p$ is equivalent to injectivity for
$W(\overline{\F_p(t_1,t_2,\ldots)})$. For a finite bound $d$ on residue transcendence degree, the test
ring is $W(\overline{\F_p(t_1,\ldots,t_d)})$.
For these henselian DVRs, rational localization boundary images depend only
on the residue field and are reflected by arbitrary residue-field
extensions. A normalization by the absolute ramification index makes
the resulting comparisons of kernels canonical and functorial.
The proof uses regular approximation, finite transfers, and
perfection. An appendix gives an alternative comparison based on
the existential embedding theorem of Anscombe--Dittmann--Jahnke,
including an example with nonnoetherian valuation rings.
We also describe the additional boundary condition that induction
between consecutive test rings would require.
\end{abstract}
\maketitle
\tableofcontents

\section{Introduction}
\label{sec:introduction}

We use $\N=\{0,1,2,\ldots\}$, and $\Z$ and $\Q$ denote the
integers and rational numbers. Rings are commutative with identity,
and ring homomorphisms preserve the identity. For a local ring $R$, its residue field is denoted by
$\kappa(R)$. For a discrete valuation ring (DVR) $R$, the valuation
$v_R$ is normalized by
$v_R(\pi)=1$ for a uniformizer $\pi$, and $\widehat R$ denotes
completion for the maximal-ideal topology.
If $R$ has mixed characteristic $(0,p)$, its \emph{absolute
ramification index} is $v_R(p)$, the ramification index of
$\Z_{(p)}\subset R$ \cite[\stackstag{09EK}]{Stacks}.

Let $R$ be a DVR and $n\in\Z$. Rational Gersten injectivity in
degree $n$ is the assertion that
\begin{equation}
 K_n(R)_{\Q}\longrightarrow K_n(\Frac R)_{\Q}
 \quad\text{is injective}.
 \label{eq:gersten-map}
\end{equation}
Here $K$ denotes nonconnective algebraic $K$-theory and
$G_{\Q}=G\otimes_{\Z}\Q$ for an abelian group $G$.
For DVRs, the rings in \eqref{eq:gersten-map} are regular and
noetherian. Their negative $K$-groups vanish
\cite[Proposition~6.8]{TT}. We allow
$n\in\Z$ in the statements involving arbitrary spectrum coefficients.
For a discrete valuation ring, injectivity in every degree is
equivalent to the rational Gersten exact sequences; this follows
directly from localization, as recalled in \cref{sec:localization}.

Fix a prime $p$. For every integer $d\geq0$, define
\begin{equation}
 \Omega_{p,d}=\overline{\F_p(t_1,\ldots,t_d)},\qquad
 \Omega_{p,\infty}=\overline{\F_p(t_1,t_2,\ldots)},
 \label{eq:omega}
\end{equation}
where the variables are algebraically independent. The algebraic
closures and the embeddings between them are chosen compatibly.
For example, inside a fixed algebraic closure of
$\F_p(t_1,t_2,\ldots)$, one can take the algebraic closure of
$\F_p(t_1,\ldots,t_d)$. In a bound on residue transcendence degree,
the symbol $\infty$ will mean that no bound is imposed. Thus a bound
by $\infty$ permits arbitrary cardinalities, whereas
$\Omega_{p,\infty}$ has countable transcendence degree.
For a perfect field $k$ of characteristic $p$, $W(k)$ denotes
its ring of $p$-typical Witt vectors. A strict $p$-ring is a
$p$-adically separated complete ring in which $p$ is a
nonzerodivisor and whose quotient by $p$ is perfect of characteristic
$p$. The ring $W(k)$ is the unique strict $p$-ring with prescribed
residue field $k$; it is a complete DVR with uniformizer $p$.
Existence and uniqueness are proved in
\cite[Chapitre~II, \S6]{Serre}; functoriality is
\cite[Chapitre~II, \S6, Proposition~10, p.~48]{Serre}.

For a field $k$ of characteristic $p$, its perfection is
$k^\perf=\bigcup_{m\in\N}k^{1/p^m}$ inside an algebraic closure.
Restriction and transfer in $K$-theory are denoted by $\res$ and
$\Tr$, respectively, with the conventions of \cref{sec:localization}.
For a DVR $R$ with residue field $k$ and $n\in\Z$, the maps
\[
 \partial_R:K_{n+1}(\Frac R)_\Q\longrightarrow K_n(k)_\Q,
 \qquad \Tr_{k/R}:K_n(k)_\Q\longrightarrow K_n(R)_\Q
\]
are the localization boundary and the closed-point Gysin map.
The latter is induced by restriction of scalars on perfect complexes.

The principal comparison is the following theorem, proved in
\cref{sec:residue-extension}.

\begin{theorem}[Residue fields and rational boundary images]
\label{thm:boundary-functoriality}
Let $R$ be a henselian DVR of mixed characteristic $(0,p)$ with
residue field $k$, and let $n\in\Z$.
\begin{enumerate}
\item In $K_n(k)_{\Q}$ one has
\begin{equation}
 \im\partial_R
 =\res_{k^\perf/k}^{-1}\bigl(\im\partial_{W(k^\perf)}\bigr).
 \label{eq:intrinsic-boundary}
\end{equation}
In particular this image depends only on $k$.
\item If $S$ is another henselian DVR of mixed characteristic
$(0,p)$ with residue field $\ell$, every field embedding
$k\subseteq\ell$ makes the square
\[
\begin{tikzcd}[column sep=large,row sep=large]
 \im\partial_R\arrow[r,hook]\arrow[d,"\res_{\ell/k}"']
   &K_n(k)_{\Q}\arrow[d,"\res_{\ell/k}"]\\
 \im\partial_S\arrow[r,hook]&K_n(\ell)_{\Q}
\end{tikzcd}
\]
cartesian.
\item There is a canonical isomorphism
\begin{multline}
 \ker\bigl(K_n(R)_{\Q}\to K_n(\Frac R)_{\Q}\bigr)\\
 \xrightarrow{\ \sim\ }
 \ker\bigl(K_n(W(k^\perf))_{\Q}
          \to K_n(\Frac W(k^\perf))_{\Q}\bigr),
 \label{eq:canonical-witt-kernel}
\end{multline}
characterized, for every $a\in K_n(k)_{\Q}$, by
\begin{equation}
 \Tr_{k/R}(a)\longmapsto
 \frac{1}{v_R(p)}\Tr_{k^\perf/W(k^\perf)}\res_{k^\perf/k}(a).
 \label{eq:canonical-witt-class}
\end{equation}
These isomorphisms are natural for ring homomorphisms
between henselian DVRs of mixed characteristic $(0,p)$.
Every such homomorphism induces an injection on rational kernels.
\end{enumerate}
\end{theorem}

\begin{definition}
An \emph{arithmetic henselian trait} of residue characteristic $p$ is a ring
\begin{equation}
 U=(\widetilde A_{p\widetilde A})^h,
 \label{eq:arithmetic-trait}
\end{equation}
where $\widetilde A$ is a smooth finitely presented
$\Z_{(p)}$-algebra and $\widetilde A/p\widetilde A$ is a nonzero
integral domain. The superscript $h$ denotes henselization.
\end{definition}

Such a ring is a henselian DVR with uniformizer $p$ and residue field
$\Frac(\widetilde A/p\widetilde A)$. Indeed, smoothness implies
flatness and regularity over the regular base $\Z_{(p)}$
\cite[\stackstag{00TA}, \stackstag{07NF}]{Stacks}.
The prime $p\widetilde A$ has height one: its height is at most one
by the principal ideal theorem \cite[\stackstag{00KV}, \textup{(1)}]{Stacks}
and is not zero because $p$ is a
nonzerodivisor in the regular localization. That localization is
therefore a one-dimensional regular local ring. Henselization
preserves its dimension, regularity, residue field, and uniformizer
\cite[\stackstag{0AP3}, \stackstag{04GN}]{Stacks}.

\begin{corollary}[Universal Witt test]
\label{thm:universal-witt-test}
Fix a prime $p$, an integer $n$, and
$d\in\N\cup\{\infty\}$. The following assertions are equivalent.
\begin{enumerate}
\item The map
$K_n(W(\Omega_{p,d}))_{\Q}\to
K_n(\Frac W(\Omega_{p,d}))_{\Q}$ is injective.
\item For every mixed-characteristic henselian DVR $R$ of residue
characteristic $p$ with $\trdeg_{\F_p}\kappa(R)\leq d$, the map
$K_n(R)_{\Q}\to K_n(\Frac R)_{\Q}$ is injective.
\item For every arithmetic henselian trait $U$ of residue
characteristic $p$ with $\trdeg_{\F_p}\kappa(U)\leq d$, the map
$K_n(U)_{\Q}\to K_n(\Frac U)_{\Q}$ is injective.
\end{enumerate}
For $d=\infty$, there is no residue transcendence-degree bound in
\textup{(ii)} or \textup{(iii)}.

More precisely, a nonzero rational kernel class over any such $R$
produces a nonzero kernel class over $W(\Omega_{p,s})$ for a finite
integer $s$, over an arithmetic henselian trait of residue
transcendence degree $s$, and over $W(\Omega_{p,\infty})$.
One can require $s\leq\trdeg_{\F_p}\kappa(R)$ whenever the latter
number is finite. The equivalences also hold when injectivity is
required in every degree simultaneously.
\end{corollary}

\Cref{thm:universal-witt-test} detects every possible failure on a prescribed family
of complete unramified DVRs. It does not supply the vanishing needed
on that family: proving injectivity for the test ring is equivalent
to the original rational henselian DVR problem.

\subsection*{The proof}
The comparison in \cref{thm:boundary-functoriality} has three steps.
Completion preserves the kernel by \cref{thm:completion}.
A Cohen subring of the completion has the same rational boundary
image by \cref{cor:cohen-core,prop:boundary-ramification}.
Adjoining residue $p$-th roots and completing then compares that
subring with $W(k^\perf)$ by \cref{thm:perfection}.
The compatibility with arbitrary residue-field embeddings follows
from \cref{cor:witt-restriction}. The factor $v_R(p)^{-1}$ in
\eqref{eq:canonical-witt-class} compensates for the ramification
multiplicity in \eqref{eq:flat-gysin}.

To deduce \cref{thm:universal-witt-test}, localization represents a
kernel class by $a\in K_n(k)_\Q$. By \cref{lem:continuity},
$a=\res_{k/k_0}(a_0)$ for a finitely generated subfield
$k_0/\F_p$ of $k$. If $s=\trdeg_{\F_p}k_0$, choose an embedding
$k_0\hookrightarrow\Omega_{p,s}$. The field diagram is
\[
\begin{tikzcd}[column sep=large]
 k & k_0\arrow[l,hook]\arrow[r,hook]
   &\Omega_{p,s}\arrow[r,hook]&\Omega_{p,\infty}.
\end{tikzcd}
\]
Part~\textup{(ii)} of \cref{thm:boundary-functoriality} reflects
membership in the boundary image along every arrow. A nonzero
kernel class over $R$ therefore yields one over each indicated
Witt ring. An arithmetic trait with residue field $k_0$ gives the
third equivalent condition. Its construction and the complete
deduction are in \cref{sec:main-proof}.

\subsection*{Approximation and existential embeddings}
There are two routes to the comparison needed in this argument.
In the main text, N\'eron--Popescu desingularization factors a
regular map of DVRs through smooth algebras. A point on the closed
fibre lifts by henselianity; a rational point gives a retraction,
and a point over a finite separable extension gives a transfer.
This is the approximation method of
\cite[Theorem~2.5 and \S3]{Popescu}, implemented in
\cref{prop:regular-comparison}.
For completion it specializes to the classical approximation
theorems of Greenberg and Artin. The passage from finite polynomial
equations to injectivity on $K$-groups already appears in
Banaszak's work \cite{Banaszak}.

The second route uses the existential embedding theorem of
Anscombe--Dittmann--Jahnke
\cite[\arxivhtmlref{2305.12145v2}{S5.Thmtheorem14}{Theorem~5.14}]{ADJ}.
It transfers existential closedness of the residue-field and
value-group embeddings to the valued-field embedding, under equal
finite initial ramification. Finite equations over the valuation
ring can then be reflected to the smaller ring. The finite-presentation
argument in \cref{thm:equational-purity} turns this into pure
monomorphisms on $K$-groups, with arbitrary spectrum coefficients.
\Cref{app:existential} develops it with all the required logical
definitions. \Cref{ex:rank-two} exhibits its application to
nonnoetherian valuation rings of rank two.
For the DVR comparisons in \cref{thm:universal-witt-test}, the
classical approximation argument already suffices; the
model-theoretic proof is an alternative with the broader
valued-field scope stated in \cref{cor:adj-k-theory}.

\subsection*{Relation to the Gersten problem}
The classical obstruction already factors through the part of
Quillen $K$-theory not generated by Milnor symbols. For every DVR
$R$ with residue field $k$, \cref{lem:milnor-obstruction} recalls
the surjection
\[
 \coker\bigl(K_n^M(k)_{\Q}\to K_n(k)_{\Q}\bigr)
 \twoheadrightarrow
 \ker\bigl(K_n(R)_{\Q}\to K_n(\Frac R)_{\Q}\bigr)
 \qquad(n\in\N).
\]
Parshin's conjecture asserts the vanishing of $K_i(X)_{\Q}$ for
$i>0$ and $X$ smooth projective over a finite field. Geisser
\cite[\href{https://www2.rikkyo.ac.jp/web/geisser/Tate_KtheoryJournal.pdf\#page=2}{Theorem~1.3(ii)}]{Geisser}
proves that it implies $K_n^M(k)_{\Q}=K_n(k)_{\Q}$ for every
field $k$ of positive characteristic. Consequently it implies
rational Gersten injectivity for every DVR of positive residue
characteristic. Geisser--Levine
\cite[following Theorem~8.2, p.~490]{GeisserLevine} explain that
this rational generation by Milnor symbols, usually called
Beilinson's conjecture for fields of positive characteristic,
implies rational Gersten injectivity and, together with their
finite-coefficient theorem, integral Gersten injectivity.
The unconditional cases of residue transcendence degree at most one
in \cref{prop:low-transcendence} come from the classical results
of Quillen and Harder.

Skalit reduces the integral Gersten conjecture for unramified
regular local rings to DVRs essentially smooth over $\Z$
\cite[\arxivhtmlref{1710.00303v1}{S5.Thmthm4}{Theorem~5.4}]{Skalit}.
The arithmetic traits used here are their henselizations. A passage
back from a henselization to the original DVR would require an
additional comparison, which is not proved here. In
equicharacteristic, integral Gersten injectivity is known by Panin
\cite[Theorem~A]{Panin}. Hence the rings $W(\Omega_{p,\infty})$,
as $p$ varies, test rational Gersten injectivity for all henselian
DVRs.

A related reduction for mod-$p^r$ Milnor $K$-theory is due to
L\"uders--Morrow \cite[Corollary~1.19]{LuedersMorrow}.
Using left Kan extension from smooth algebras, they reduce their Gersten injectivity
and Galois-symbol assertions for local $p$-henselian ind-smooth
algebras to the mod-$p$ case over absolutely unramified complete
DVRs with sufficiently large residue fields.
The present work concerns rational Quillen $K$-theory and detects
its Gersten kernels on prescribed Witt rings by comparing
localization boundary images.

For DVRs, Gersten injectivity with finite coefficients is classical:
the prime-to-residue-characteristic case is due to Gillet \cite{Gillet}, and the
remaining case to Geisser--Levine; see
\cite[\href{https://www2.rikkyo.ac.jp/web/geisser/p-part.pdf\#page=32}{Theorem~8.2 and its proof}]{GeisserLevine}.
Two companion preprints concern broader henselian settings.
For coefficients invertible in the ring,
\cite[\arxivhtmlref{2608.18789v1}{S6.Thmtheorem1}{Theorem~6.1} and
\arxivhtmlref{2608.18789v1}{S6.Thmtheorem13}{Corollary~6.13}]{FeldFinite}
treats henselian valuation rings and noetherian henselian regular
local rings.
\cite[\arxivhtmlref{2608.05005v2}{S1.Thmtheorem1}{Theorem~1.1}]{FeldCounterexample}
gives a failure with coefficients $\Z/3$ for a complete ramified
regular local surface of residue characteristic $3$.
Neither preprint is used in the proofs below. Integral Gersten
injectivity requires control of torsion in addition to the rational
problem; the higher-dimensional case requires further arguments.

Integral Gersten injectivity for DVRs is announced in
\cite[\arxivhtmlref{math/0702315v1}{S0.Thmthm3}{Theorem~0.3}]{Mochizuki}
and \cite[\arxivhtmlref{1503.07966v8}{Thmthm2}{Corollary~2}]{MochizukiLocal};
\cite[\arxivhtmlref{2512.01923v1}{S4.Thmtheorem5}{Theorem~4.5}]{Druzhinin}
includes henselian DVRs. The author does not regard the cited
versions as complete proofs. In Mochizuki's arguments, the
retraction asserted in
\cite[\arxivhtmlref{math/0702315v1}{S3.Thmthm13}{Theorem~3.13}]{Mochizuki}
and the functoriality asserted in
\cite[\arxivhtmlref{1503.07966v8}{Thmthm13}{Lemma~13(III)}]{MochizukiLocal}
require correction. In Druzhinin's version~1, Theorems~2.1--2.2,
Theorems~2.3--2.4, and the factorization in the proof of
Theorem~3.1 invoke, respectively, references~[9], [8], and~[10],
all listed there as work in preparation.\footnote{All three are
attributed there to A.~E.~Druzhinin: [9], \emph{$\phi$-invariance
and lifting properties}; [8],
\emph{$(\phi,\underline{\overline{\mathbb A}},\mathrm{Bl}_{\beta})$-localisation
and blow-ups and loops functors}; [10], \emph{Blow-ups and
$\rho$-filtrations}.}
Proofs of these inputs are not supplied in that version.
We therefore do not treat these announcements as established
resolutions of the mixed-characteristic Gersten problem.

\subsection*{Organization of the paper}
We include details of several standard arguments to make the use
of the hypotheses and the compatibility of the comparison maps
explicit. The aim is to facilitate verification and a possible
future formalization in Lean.

Expert readers interested in the universal test rings may skip
directly to \cref{sec:main-proof}, where
\cref{thm:universal-witt-test} is deduced from
\cref{thm:boundary-functoriality}.
\Crefrange{sec:localization}{sec:residue-extension}
establish the latter theorem.
\Cref{sec:applications} compares consecutive test rings and
states the additional condition required for induction.

\subsection*{Acknowledgments}
The author is grateful to Fr\'ed\'eric D\'eglise for introducing him
to the Gersten conjecture in January 2018 and for many subsequent
discussions around this problem.
The author thanks Morten Lüders for comments on an earlier draft.

\medskip
\noindent\textbf{Use of AI-assisted tools}.
The main mathematical ideas were found during discussions with chatgpt (GPT-5.5 and then Sol 5.6). 
The paper was mostly written by Sol 5.6 and GPT-6 Astra, under strict supervision. Separate AI-assisted (GPT-6 Astra and Opus 5.5)
referee passes were also used to search for proof gaps and incorrect
hypotheses. The author assumes full
responsibility for the mathematical statements, proofs, and references.

\section{Localization and closed-point support}
\label{sec:localization}

For a ring $R$, a spectrum $M$, and an integer $n$, the notation
\[
 K_n(R;M)=\pi_n(K(R)\wedge M)
\]
denotes algebraic $K$-theory with spectrum coefficients.
Here $\Sph$ is the sphere spectrum and $H\Q$ is the
Eilenberg--Mac Lane spectrum of $\Q$. For a positive integer $m$,
$\Sph[1/m]$ is the localization of $\Sph$ obtained by inverting $m$.
Thus
\[
 K_n(R;\Sph)=K_n(R),\qquad
 K_n(R;H\Q)=K_n(R)_{\Q},\qquad
 K_n(R;\Sph[1/m])=K_n(R)[1/m].
\]
For a ring homomorphism $R\to S$, derived extension of scalars
$P\mapsto S\otimes_R^{\mathbf L}P$ on perfect complexes induces
\[
 \res_{S/R}:K_n(R;M)\longrightarrow K_n(S;M).
\]
If $S$ is perfect as an $R$-module, restriction of scalars
preserves perfect complexes and induces
\[
 \Tr_{S/R}:K_n(S;M)\longrightarrow K_n(R;M).
\]
These are the restriction and transfer conventions used throughout;
see \cite[V, 3.3.2]{Weibel}.
The notation applies to quotient maps as well as to inclusions.
For a DVR $R$ with residue field $k$, the resolution
$0\to R\xrightarrow{\pi}R\to k\to0$ makes $k$ a perfect $R$-module.
Thus $\Tr_{k/R}$ is defined; it is the closed-point Gysin map.
Localization boundaries are denoted by $\partial_R$.

The next proposition is the DVR case of Quillen localization and
d\'evissage \cite[V, Theorems~4.1 and~5.1, and \textup{(6.6)}]{Weibel}.
We recall the argument, including the ramification multiplicity.

\begin{proposition}[Localization and base change]
\label{prop:localization}
Let $R$ be a DVR with residue field $k$.
For every spectrum $M$ and every integer $n$, localization and
d\'evissage give an exact sequence
\begin{equation}
\begin{split}
 K_{n+1}(\Frac R;M)&\xrightarrow{\partial_R}K_n(k;M)
 \xrightarrow{\Tr_{k/R}}K_n(R;M)\\
 &\longrightarrow K_n(\Frac R;M).
\end{split}
\label{eq:localization}
\end{equation}
In particular,
\begin{equation}
 \ker\bigl(K_n(R;M)\to K_n(\Frac R;M)\bigr)
 =\im(\Tr_{k/R}:K_n(k;M)\to K_n(R;M)).
 \label{eq:support-image}
\end{equation}
If $R\to S$ is a local map of DVRs with residue extension
$k\to\ell$, $\pi_R$ is a uniformizer of $R$, and
$0<v_S(\pi_R)=e<\infty$, then it is flat and
\begin{equation}
 \res_{S/R}\Tr_{k/R}=e\,\Tr_{\ell/S}\res_{\ell/k}.
 \label{eq:flat-gysin}
\end{equation}
\end{proposition}
\begin{proof}
Nonconnective localization \cite[Theorem~7.4]{TT} identifies the fibre of
$K(R)\to K(\Frac R)$ with the $K$-theory of perfect complexes
whose cohomology is supported at the closed point.
Every finitely generated $R$-module supported there has finite
length and admits a filtration whose quotients are finite-dimensional
$k$-vector spaces. Since $R$ is regular, these modules are perfect.
The exact functor sending a finite-dimensional $k$-vector space
to its underlying perfect $R$-complex induces the map to this fibre.
Quillen localization and d\'evissage, together with the resolution
theorem for the regular ring $R$, say that this map is an isomorphism
on homotopy groups in degrees at least zero
\cite[V, Theorems~3.1, 4.1, and~5.1]{Weibel}.
The negative groups of
$R$ and $\Frac R$ vanish by regularity, and
$K_0(R)\to K_0(\Frac R)$ is the identity on ranks $\Z$.
The homotopy exact sequence gives zero in degree $-1$ for the
fibre because this rank map is surjective. In every smaller degree,
both adjacent groups in that sequence vanish, so the fibre group
vanishes as well. The field $k$ also has vanishing negative
$K$-groups. Hence the d\'evissage map is an isomorphism on every
homotopy group and is an equivalence of spectra. This gives the
cofiber sequence
\[
 K(k)\xrightarrow{\Tr_{k/R}}K(R)\longrightarrow K(\Frac R).
\]
Smash product with $M$ preserves cofiber sequences. Its homotopy
long exact sequence is \eqref{eq:localization}, which gives
\eqref{eq:support-image}.

For base change, the kernel of $R\to S$ is a prime ideal.
It is not the maximal ideal, since the image of $\pi_R$ is nonzero.
It is therefore zero. The $R$-module $S$ is torsion-free, hence
flat over the DVR $R$ \cite[\stackstag{0539}]{Stacks}.
If $\pi_S$ is a uniformizer of $S$, then
$\pi_RS=\pi_S^eS$. The $(S,k)$-bimodule $S/\pi_RS$ has the
filtration
\[
 S/\pi_S^eS\supset\pi_SS/\pi_S^eS\supset\cdots
 \supset\pi_S^{e-1}S/\pi_S^eS\supset0.
\]
For every integer $j$ with $0\leq j<e$, multiplication by
$\pi_S^j$ induces the $(S,k)$-bimodule isomorphism
\[
 \ell=S/\pi_SS\xrightarrow{\ \sim\ }
 \pi_S^jS/\pi_S^{j+1}S,\qquad
 \overline s\longmapsto \pi_S^js\bmod\pi_S^{j+1}S.
\]
For a finite-dimensional $k$-vector space $V$, flat pullback is
$(S/\pi_RS)\otimes_k V$. Tensoring the filtration with $V$ is
exact; each quotient is $\ell\otimes_k V$. Every term has finite
length over $S$ and is perfect. Additivity
\cite[V, Theorem~1.2]{Weibel} therefore identifies the induced map
with $e$ copies of $\Tr_{\ell/S}\res_{\ell/k}$.
This is an equality of homotopy classes of maps of $K$-theory
spectra \cite[Corollary~1.7.3]{TT}, hence remains valid after
smashing with $M$.
For $e=1$ it gives the commutative square
\[
\begin{tikzcd}[column sep=large,row sep=large]
 K_n(k;M)\arrow[r,"\Tr_{k/R}"]\arrow[d,"\res_{\ell/k}"']
   & K_n(R;M)\arrow[d,"\res_{S/R}"]\\
 K_n(\ell;M)\arrow[r,"\Tr_{\ell/S}"']
   & K_n(S;M).
\end{tikzcd}
\]
\end{proof}

It follows that injectivity of \eqref{eq:gersten-map} is equivalent
to surjectivity of
$\partial_R:K_{n+1}(\Frac R)_{\Q}\to K_n(k)_{\Q}$.
If injectivity holds in all degrees, localization gives
\[
 0\longrightarrow K_n(R)_{\Q}\longrightarrow K_n(\Frac R)_{\Q}
 \xrightarrow{\partial_R}K_{n-1}(k)_{\Q}\longrightarrow0
\]
for every $n$. Conversely these exact sequences include the asserted
injectivity.

For a ring $A$ and $u\in A^\times$, denote by
$\{u\}\in K_1(A)$ the class of the automorphism
$A\to A$, $x\mapsto ux$.
For a field $k$ and $n\in\N$, $K_n^M(k)$ denotes Milnor
$K$-theory: $K_0^M(k)=\Z$, and for $n\geq1$ it is the
degree-$n$ part of the tensor algebra on $k^\times$ modulo the
two-sided ideal generated by $a\otimes(1-a)$ for
$a\in k\setminus\{0,1\}$ \cite[III, Definition~7.1]{Weibel}.
Its symbol $\{a_1,\ldots,a_n\}$ maps to the product of the
classes of $a_1,\ldots,a_n$ in Quillen $K$-theory
\cite[IV, Example~1.10.1]{Weibel}.
The following is the classical symbol-lifting argument; compare
\cite[proof of Theorem~8.2 and the following paragraph]{GeisserLevine}.

\begin{lemma}[Milnor symbols]
\label{lem:milnor-obstruction}
For every DVR $R$ with residue field $k$ and every $n\in\N$,
the transfer $\Tr_{k/R}:K_n(k)\to K_n(R)$ vanishes on the
image of $K_n^M(k)$. It induces a surjection
\[
 \coker\bigl(K_n^M(k)\to K_n(k)\bigr)
 \twoheadrightarrow
 \ker\bigl(K_n(R)\to K_n(\Frac R)\bigr).
\]
\end{lemma}
\begin{proof}
For a uniformizer $\pi$, the resolution
$0\to R\xrightarrow{\pi}R\to k\to0$ gives
$\Tr_{k/R}(1)=[R]-[R]=0$ in $K_0(R)$.
For $n\geq1$ and $a_1,\ldots,a_n\in k^\times$, choose lifts
$u_1,\ldots,u_n\in R^\times$. Their product in $K_n(R)$
restricts to the image of $\{a_1,\ldots,a_n\}$. The projection
formula \cite[V, Example~3.5.3]{Weibel} gives
\[
 \Tr_{k/R}\bigl(\res_{k/R}(\{u_1\}\cdots\{u_n\})\bigr)
 =\{u_1\}\cdots\{u_n\}\,\Tr_{k/R}(1)=0.
\]
Symbols generate $K_n^M(k)$, and the calculation for $1$ treats
$n=0$. Localization \eqref{eq:support-image} gives the surjection.
\end{proof}

\begin{lemma}[Degrees at most two]
\label{lem:low-degrees}
For every DVR $R$ and every integer $n\leq2$, the map
$K_n(R)\to K_n(\Frac R)$ is injective.
\end{lemma}
\begin{proof}
For $n<0$, both groups vanish by regularity
\cite[Proposition~6.8]{TT}. For $n=0$, rank identifies the map
with the identity of $\Z$. For $n=1$, the determinant identifies
it with $R^\times\subseteq(\Frac R)^\times$.
For $n=2$, Matsumoto's theorem identifies $K_2^M(k)$ with
$K_2(k)$ \cite[III, Theorem~6.1]{Weibel}; apply
\cref{lem:milnor-obstruction}. This is the degree-two theorem of
Dennis--Stein \cite{DennisStein}, in the form \cite[V, Corollary~6.6.2]{Weibel}.
\end{proof}

\begin{lemma}[Filtered colimits]
\label{lem:continuity}
For every spectrum $M$ and every integer $n$, the functor
$K_n(-;M)$ preserves filtered colimits of rings.
If $(R_\lambda)_\lambda$ is a filtered system of domains with
injective transition maps, then
\begin{multline}
 \varinjlim_\lambda
 \ker\bigl(K_n(R_\lambda;M)\to K_n(\Frac R_\lambda;M)\bigr)\\
 \simeq
 \ker\bigl(K_n(\varinjlim_\lambda R_\lambda;M)
 \to K_n(\Frac(\varinjlim_\lambda R_\lambda);M)\bigr).
 \label{eq:kernel-continuity}
\end{multline}
\end{lemma}
\begin{proof}
Nonconnective $K$-theory commutes with filtered colimits of rings
\cite[Theorem~7.2]{TT}; see also
\cite[the proof of \arxivhtmlref{1001.2282v4}{S9.E8}{Theorem~9.8}]{BGT}.
Smash product with $M$ preserves colimits. Homotopy groups of
spectra preserve filtered colimits, by commutation of filtered
colimits of spaces with finite limits
\cite[\kerodontag{05XS}]{Kerodon}.
This proves the first assertion.
For the second, the injective transition maps identify
$\Frac(\varinjlim R_\lambda)$ with $\varinjlim\Frac R_\lambda$:
each fraction has its numerator and nonzero denominator in one
stage. Filtered colimits of abelian groups are exact
\cite[\stackstag{00DB}]{Stacks}; applying this to the two systems
of $K$-groups gives \eqref{eq:kernel-continuity}.
\end{proof}

\section{Algebraic purity and regular approximation}
\label{sec:purity}

\begin{definition}
\label{def:purity}
An inclusion of rings $R\subseteq S$ is \emph{algebraically pure}
if, for all integers $r,s\in\N$ and all
$f_i\in R[X_1,\ldots,X_r]$, $1\leq i\leq s$,
\[
\begin{gathered}
 \exists b\in S^r,\quad f_i(b)=0\quad(1\leq i\leq s)\\
 \Longrightarrow\\
 \exists a\in R^r,\quad f_i(a)=0\quad(1\leq i\leq s).
\end{gathered}
\]
An inclusion of abelian groups $A\subseteq B$ is \emph{pure}
if, for all integers $r,s\in\N$, all integers
$c_{ij}$, $1\leq i\leq s$, $1\leq j\leq r$, and all
$a_i\in A$, $1\leq i\leq s$,
\[
\begin{gathered}
 \exists (b_1,\ldots,b_r)\in B^r,\quad
       \sum_{j=1}^r c_{ij}b_j=a_i\quad(1\leq i\leq s)\\
 \Longrightarrow\\
 \exists (x_1,\ldots,x_r)\in A^r,\quad
       \sum_{j=1}^r c_{ij}x_j=a_i\quad(1\leq i\leq s).
\end{gathered}
\]
Empty conjunctions are true, and empty sums are zero.
\end{definition}

For rings, this is Popescu's definition of algebraic purity
\cite[\S3, p.~91]{Popescu}. The corresponding categorical notion
is recalled in \cite[\S2, p.~18]{BR}. The following argument expresses $R\to S$
as a filtered colimit of maps $R\to Q$, each admitting an
$R$-algebra retraction. This is the approximation by split
monomorphisms in \cite[\S2, property~\textup{(5)}]{BR}, proved
below directly from polynomial equations.

\begin{lemma}[Finite presentations and retractions]
\label{lem:finite-presentation-retractions}
Let $R\subseteq S$ be an inclusion of rings.
\begin{enumerate}
\item The category of finitely presented $R$-algebras $Q$ equipped
with an $R$-algebra map $Q\to S$ is filtered, and its canonical
colimit map
\[
 \varinjlim_{(Q\to S)}Q\longrightarrow S
\]
is an isomorphism.
\item The inclusion $R\subseteq S$ is algebraically pure if and
only if every object $Q\to S$ of this category admits an
$R$-algebra homomorphism $\rho:Q\to R$. Such a homomorphism
satisfies $\rho\circ(R\to Q)=\mathrm{id}_R$.
\end{enumerate}
\end{lemma}
\begin{proof}
Part \textup{(i)} is \cite[\stackstag{0BUF}]{Stacks}, applied to
$R$-algebras; the morphisms are the $R$-algebra maps over $S$.
For \textup{(ii)}, suppose first that $R\subseteq S$ is
algebraically pure. A presentation
\[
 Q=R[X_1,\ldots,X_r]/(f_1,\ldots,f_s)
\]
and a map $Q\to S$ give elements $b_j\in S$,
$1\leq j\leq r$, with $f_i(b_1,\ldots,b_r)=0$ for
$1\leq i\leq s$.
Choose $a_j\in R$, $1\leq j\leq r$, satisfying the same
equations. Evaluation $X_j\mapsto a_j$ induces an
$R$-algebra map $Q\to R$.
Conversely, a solution in $S$ of a finite system
$f_i(X_1,\ldots,X_r)=0$, $1\leq i\leq s$, defines a map
from the displayed algebra $Q$ to $S$. An $R$-algebra map
$Q\to R$ sends the classes of the $X_j$ to a solution in $R$.
\end{proof}

Finite-equation hypotheses were used for injectivity in $K$-theory
by Banaszak \cite{Banaszak}. We give a variant
that records purity for arbitrary rings and spectrum coefficients.

\begin{theorem}[Purity on $K$-groups]
\label{thm:equational-purity}
If $R\subseteq S$ is algebraically pure, then
\[
 \res_{S/R}:K_n(R;M)\longrightarrow K_n(S;M)
\]
is a pure monomorphism for every spectrum $M$ and every integer $n$.
\end{theorem}
\begin{proof}
By \cref{lem:finite-presentation-retractions,lem:continuity},
\[
 K_n(S;M)=\varinjlim_{(Q\to S)}K_n(Q;M),
\]
and every structural map $R\to Q$ in this diagram has an
$R$-algebra retraction.
Fix integers $r,s\in\N$, integers $c_{ij}$ for
$1\leq i\leq s$, $1\leq j\leq r$, elements
$a_i\in K_n(R;M)$ for $1\leq i\leq s$, and elements
$b_j\in K_n(S;M)$ for $1\leq j\leq r$, satisfying
\[
 \sum_{j=1}^r c_{ij}b_j=\res_{S/R}(a_i)
 \qquad(1\leq i\leq s).
\]
Filteredness gives an object $\sigma:Q\to S$ and elements
$\widetilde b_j\in K_n(Q;M)$ such that
$K_n(\sigma;M)(\widetilde b_j)=b_j$ for
$1\leq j\leq r$. For each $i\in\{1,\ldots,s\}$, the
description of equality in a filtered colimit
\cite[\stackstag{04AX}]{Stacks}, applied to the $i$-th equality, gives
an arrow $u_i:Q\to Q_i$ over $S$ with
\[
 K_n(u_i;M)
 \left(\sum_{j=1}^r c_{ij}\widetilde b_j
                    -\res_{Q/R}(a_i)\right)=0.
\]
A cocone on the finite diagram consisting of $Q$, the $Q_i$,
and the arrows $u_i$ consists of an object $T\to S$ and arrows
\[
 u:Q\to T,\qquad v_i:Q_i\to T,\qquad
 v_i u_i=u\quad(1\leq i\leq s).
\]
Such a cocone exists by filteredness. Applying $K_n(v_i;M)$
to the preceding equality gives
\[
 \sum_{j=1}^r c_{ij}K_n(u;M)(\widetilde b_j)
       =\res_{T/R}(a_i)
 \qquad(1\leq i\leq s).
\]
For an $R$-algebra retraction $\rho:T\to R$, the elements
\[
 x_j=K_n(\rho u;M)(\widetilde b_j)
       \in K_n(R;M)\qquad(1\leq j\leq r)
\]
therefore satisfy
\[
 \sum_{j=1}^r c_{ij}x_j
   =K_n(\rho;M)\res_{T/R}(a_i)=a_i
 \qquad(1\leq i\leq s).
\]
For $r=0$ and $s=1$, this calculation says that
$\res_{S/R}(a_1)=0$ implies $a_1=0$, proving injectivity.
For arbitrary $r,s$, it proves purity as in \cref{def:purity}.
\end{proof}

\begin{remark}
The proof applies to every functor from commutative rings to
abelian groups which preserves filtered colimits.
If $A\subseteq B$ is pure and $m\geq1$, then $A/mA\to B/mB$
is injective: for $a\in A\cap mB$, solvability of $mX=a$
in $B$ implies $a\in mA$.
\end{remark}

The following comparison is the usual approximation argument:
factor through a smooth algebra, lift a point by henselianity, and
use a retraction or a finite free transfer. For completion, the
approximation input goes back to Greenberg
\cite[\href{https://www.numdam.org/item/PMIHES_1966__31__59_0.pdf\#page=2}{Theorem~1 and Corollary~2}]{Greenberg};
compare also Artin
\cite[\href{https://www.numdam.org/item/PMIHES_1969__36__23_0.pdf\#page=5}{Theorem~1.10}]{Artin}.
We use N\'eron--Popescu desingularization \cite[Theorem~2.5]{Popescu}, in the form
\cite[\stackstag{07GC}]{Stacks}.

\begin{proposition}[Comparison by regular approximation]
\label{prop:regular-comparison}
Let $R\to S$ be a local homomorphism of DVRs with
$\operatorname{char}(\Frac R)=0$. Assume that $R$ is henselian,
that a uniformizer $\pi$ of $R$ maps to a uniformizer of $S$, and
that the residue extension
$k=R/\pi R\subseteq\ell=S/\pi S$ is separable.
For every spectrum $M$ and every $n\in\Z$, the maps
\[
 K_n(R;M)\longrightarrow K_n(S;M),\qquad
 K_n(\Frac R;M)\longrightarrow K_n(\Frac S;M)
\]
have torsion kernels. If $k\subseteq\ell$ is algebraically pure,
both maps are pure monomorphisms.
\end{proposition}
\begin{proof}
The kernel of $R\to S$ is prime and does not contain $\pi$;
it is therefore zero. The $R$-module $S$ is torsion-free, hence
flat \cite[\stackstag{0539}]{Stacks}. Its fibres over the two
primes of $R$ are
\[
 S\otimes_R\Frac R=\Frac S,
 \qquad S\otimes_R k=\ell.
\]
The generic fibre is geometrically regular since its base field
has characteristic zero. The same argument applies to the closed
fibre if $\operatorname{char}k=0$; in positive characteristic, for every finite purely inseparable
extension $k'/k$, the ring $\ell\otimes_k k'$ is reduced by
\cite[\stackstag{030W}]{Stacks}. It is Artinian, being
finite-dimensional over $\ell$, and hence is a product of fields.
It is therefore regular. The criterion in
\cite[\stackstag{0381}]{Stacks} proves geometric regularity.
Thus $R\to S$ is a regular homomorphism of noetherian rings
\cite[\stackstag{07BZ}]{Stacks}. Popescu's theorem gives a filtered
system of smooth $R$-algebras $B_\lambda$ equipped with compatible
maps $B_\lambda\to S$ such that
\begin{equation}
 S=\varinjlim_\lambda B_\lambda,
 \qquad \Frac S=\varinjlim_\lambda B_\lambda[1/\pi].
 \label{eq:smooth-approximation}
\end{equation}

For each $\lambda$, the smooth $k$-algebra $B_\lambda/\pi B_\lambda$
is nonzero, since it maps to the field $\ell$. It has a point with
values in a finite separable extension $k'/k$
\cite[\stackstag{056U}]{Stacks}. The equivalence for finite \'etale
algebras over a henselian local ring supplies a finite \'etale
$R$-algebra $R'$ with $R'/\pi R'=k'$
\cite[\stackstag{04GK}]{Stacks}.
Every maximal ideal of $R'$ lies over $(\pi)$, and the quotient
by $\pi$ is a field. Hence $R'$ is local. It is henselian
\cite[\stackstag{04GH}]{Stacks}, and it is free as an $R$-module
of rank $[k':k]$. Smooth lifting over the henselian pair
$(R',\pi R')$ \cite[\stackstag{0H74}]{Stacks} gives a lift
\[
\begin{tikzcd}[column sep=large,row sep=large]
 B_\lambda\arrow[r]\arrow[dr,dashed]&k'\\
 R\arrow[u]\arrow[r]&R'\arrow[u,two heads].
\end{tikzcd}
\]

Fix $x\in K_n(R;M)$ whose image in $K_n(S;M)$ is zero.
By \cref{lem:continuity} and \eqref{eq:smooth-approximation},
there is $\lambda$ with $\res_{B_\lambda/R}(x)=0$.
The displayed lift gives $\res_{R'/R}(x)=0$. Finite freeness
and additivity give
\[
 [k':k]x=\Tr_{R'/R}\res_{R'/R}(x)=0.
\]
For $y\in K_n(\Frac R;M)$ with zero image in
$K_n(\Frac S;M)$, choose $\lambda$ with
$\res_{B_\lambda[1/\pi]/\Frac R}(y)=0$ instead.
Localizing the displayed lift and applying the transfer for the
finite free $\Frac R$-algebra $R'[1/\pi]$ gives
$[k':k]y=0$. This proves the two torsion assertions.

Suppose now that $k\subseteq\ell$ is algebraically pure.
A finite presentation of $B_\lambda/\pi B_\lambda$ and its
map to $\ell$ give a finite polynomial system over $k$ with a
solution in $\ell$. Purity gives a solution in $k$, hence a
$k$-algebra map $B_\lambda/\pi B_\lambda\to k$.
We may therefore take $k'=k$ and $R'=R$ in the preceding lift.
Every map $R\to B_\lambda$ admits an $R$-algebra retraction.
Given a finitely presented $R$-algebra $Q$ and a map $Q\to S$,
finite presentation factors it through some $B_\lambda$;
composing with its retraction gives $Q\to R$.
By \cref{lem:finite-presentation-retractions}, $R\subseteq S$
is algebraically pure. Every finitely presented $\Frac R$-algebra mapping to
$\Frac S=\varinjlim_\lambda B_\lambda[1/\pi]$ factors through
some $B_\lambda[1/\pi]$. Composing with the localized retraction
$B_\lambda[1/\pi]\to\Frac R$ proves algebraic purity of
$\Frac R\subseteq\Frac S$ in the same way.
Apply \cref{thm:equational-purity} to both inclusions.
\end{proof}

\begin{theorem}[Completion]
\label{thm:completion}
Let $R$ be a henselian DVR with residue field $k$ and
$\operatorname{char}(\Frac R)=0$.
For every spectrum $M$ and every integer $n$, completion induces
pure monomorphisms
\[
 K_n(R;M)\longrightarrow K_n(\widehat R;M),\qquad
 K_n(\Frac R;M)\longrightarrow K_n(\Frac\widehat R;M)
\]
and an isomorphism
\begin{multline}
 \ker\bigl(K_n(R;M)\to K_n(\Frac R;M)\bigr)\\
 \xrightarrow{\ \sim\ }
 \ker\bigl(K_n(\widehat R;M)\to K_n(\Frac\widehat R;M)\bigr).
 \label{eq:completion-kernel}
\end{multline}
\end{theorem}
\begin{proof}
The map from $R$ to its completion is injective because
$\bigcap_{m\geq1}\pi^mR=0$ for a uniformizer $\pi$.
By \cite[\stackstag{0AP1}]{Stacks}, $\widehat R$ is a complete DVR.
\cite[\stackstag{05GG}]{Stacks} gives
\[
 \widehat R/\pi\widehat R=R/\pi R.
\]
Thus $\pi$ generates the maximal ideal of $\widehat R$, and
the residue map is the identity under this identification.
The residue extension is the identity, hence is separable and
algebraically pure. \Cref{prop:regular-comparison} gives the two
pure monomorphisms.

The map $R\to\widehat R$ preserves the uniformizer and induces
the identity on $k$. By \cref{prop:localization} and naturality
of restriction, the following diagram commutes and its rows
are exact at their middle terms:
\[
\begin{tikzcd}[column sep=large,row sep=large]
 K_n(k;M)\arrow[r,"\Tr_{k/R}"]\arrow[d,equal]
 & K_n(R;M)\arrow[r]\arrow[d,hook,"\res_{\widehat R/R}"]
 & K_n(\Frac R;M)\arrow[d,hook]\\
 K_n(k;M)\arrow[r,"\Tr_{k/\widehat R}"']
 & K_n(\widehat R;M)\arrow[r]
 & K_n(\Frac\widehat R;M).
\end{tikzcd}
\]
The middle vertical arrow is injective, so its restriction to
the kernel of the top rightward arrow is injective.
For
$\widehat x\in\ker(K_n(\widehat R;M)\to K_n(\Frac\widehat R;M))$,
exactness gives $a\in K_n(k;M)$ with
$\Tr_{k/\widehat R}(a)=\widehat x$. Exactness of the top row and
commutativity of the left square give, respectively,
\[
 \Tr_{k/R}(a)\in
 \ker(K_n(R;M)\to K_n(\Frac R;M)),\qquad
 \res_{\widehat R/R}\Tr_{k/R}(a)=\widehat x.
\]
Thus the map in \eqref{eq:completion-kernel} is surjective.
\end{proof}

\begin{remark}
For a positive integer $m$, the Moore spectrum $\Sph/m$ is the
cofiber of multiplication by $m$ on $\Sph$.
For $M=\Sph/m$, \cref{thm:completion} concerns finite-coefficient
$K$-theory $\pi_n(K(R)\wedge\Sph/m)$.
The coefficient cofiber sequence gives
\[
 0\longrightarrow K_n(R)/m\longrightarrow K_n(R;\Sph/m)
 \longrightarrow K_{n-1}(R)[m]\longrightarrow0,
\]
where, for any abelian group $G$ and any integer $m\geq1$,
$G[m]=\{g\in G:mg=0\}$.
Separately, purity for $M=\Sph$ implies that
$K_n(R)/m\to K_n(\widehat R)/m$ is injective:
if the image of $a\in K_n(R)$ is $mb$ in the target, the
equation $mX=a$ has a solution in $K_n(R)$.
\end{remark}

\section{Finite extensions and Cohen subrings}
\label{sec:cohen}

The identities below express the compatibility of transfer
and restriction with the maps from residue fields
\cite[V, 3.3.2 and \textup{(6.6.3)--(6.6.4)}]{Weibel}.
We record their effect on kernels of restriction to fraction fields, keeping the degree
and ramification factors explicit.

\begin{lemma}[Transfer and ramification]
\label{lem:finite-gysin}
Let $A\subseteq B$ be a finite extension of DVRs, with residue
extension $k\subseteq\ell$ and ramification index $e$.
The $A$-module $B$ is free of rank $e[\ell:k]$.
For every integer $n$, the following identities hold on integral $K$-groups:
\begin{equation}
 \Tr_{B/A}\Tr_{\ell/B}=\Tr_{k/A}\Tr_{\ell/k},\qquad
 \res_{B/A}\Tr_{k/A}=e\,\Tr_{\ell/B}\res_{\ell/k}.
 \label{eq:finite-gysin}
\end{equation}
The maps $\res_{B/A}$ and $\Tr_{B/A}$ preserve the kernels of
restriction to the fraction fields.
\end{lemma}
\begin{proof}
The $A$-module $B$ is torsion-free, hence flat; being finite over
the local ring $A$, it is finite free.
The map is local: the maximal ideal of the integral extension $B$
contracts to the unique maximal ideal of $A$.
Choose uniformizers $\pi_A,\pi_B$. By definition of $e$,
$\pi_A=u\pi_B^e$ for some $u\in B^\times$.
The quotient $B/\pi_AB$ has the filtration
\[
 B/\pi_B^eB\supset\pi_BB/\pi_B^eB\supset\cdots
 \supset\pi_B^{e-1}B/\pi_B^eB\supset0.
\]
Multiplication by $\pi_B^j$ identifies the $j$-th successive
quotient with $B/\pi_BB=\ell$, for $0\leq j<e$.
Its $k$-action is the one induced by $A\to B\to\ell$.
Taking $k$-dimensions gives
\[
 \operatorname{rank}_A B
 =\dim_k(B/\pi_AB)=e[\ell:k].
\]

For the first identity in \eqref{eq:finite-gysin}, an $\ell$-vector
space regarded successively as a $B$-module and an $A$-module
is the same $A$-module as the one obtained by first restricting
scalars from $\ell$ to $k$ and then applying $\Tr_{k/A}$.
The natural isomorphism of these exact functors proves the identity.

The second identity is \eqref{eq:flat-gysin}.

Finally,
$B\otimes_A\Frac A=B[1/\pi_A]=B[1/\pi_B]=\Frac B$.
The extension-of-scalars square and the restriction-of-scalars
square therefore commute with passage to fraction fields.
Consequently both maps preserve the kernels of restriction
to fraction fields.
\end{proof}

\begin{proposition}[Purely inseparable residue extension]
\label{prop:inseparable-kernel}
Under the hypotheses of \cref{lem:finite-gysin}, assume that
$\ell/k$ is purely inseparable. For every integer $n$,
restriction induces an isomorphism
\begin{multline}
 \ker\bigl(K_n(A)\to K_n(\Frac A)\bigr)[1/(e[\ell:k])]\\
 \xrightarrow{\ \res_{B/A}\ }
 \ker\bigl(K_n(B)\to K_n(\Frac B)\bigr)[1/(e[\ell:k])].
 \label{eq:inseparable-kernel}
\end{multline}
Its inverse is $(e[\ell:k])^{-1}\Tr_{B/A}$.
\end{proposition}
\begin{proof}
On $K_n(A)$ the composite $\Tr_{B/A}\res_{B/A}$ is multiplication by
$e[\ell:k]$. Indeed, after a choice of an $A$-basis of $B$,
the underlying functor is naturally isomorphic to the direct sum
of $e[\ell:k]$ copies of the identity functor. Additivity gives
the asserted equality of maps on $K$-groups.

The purely inseparable field identity
\cite[IV, Exercise~6.13]{Weibel} is
\begin{equation}
 \res_{\ell/k}\Tr_{\ell/k}=[\ell:k]\,\mathrm{id}_{K_n(\ell)}.
 \label{eq:inseparable-field-transfer}
\end{equation}
We include its bimodule proof.
If $\ell=k$, this is the identity. Otherwise the characteristic
is a positive prime $p$. Since the purely inseparable extension
is finite, there is an integer $s\geq0$ with $b^{p^s}\in k$ for
every $b\in\ell$. For an element
$z=\sum_i a_i\otimes b_i$ in the kernel of multiplication
$\ell\otimes_k\ell\to\ell$, one has
\[
 z^{p^s}=\sum_i a_i^{p^s}\otimes b_i^{p^s}
 =\left(\sum_i a_ib_i\right)^{p^s}\otimes1=0.
\]
Thus this kernel is nil, and the quotient is the field $\ell$.
The finite-dimensional $\ell$-algebra $\ell\otimes_k\ell$
is therefore local Artinian with residue field $\ell$.
Every factor of a composition series by ideals is that residue
field. The number of factors is $[\ell:k]$, since the
$\ell$-dimension of $\ell\otimes_k\ell$ is $[\ell:k]$
and each factor has dimension one.
On each factor, the two $\ell$-actions agree through multiplication.
The functor inducing the left side of
\eqref{eq:inseparable-field-transfer} is tensor product with
this bimodule. Its composition series gives, by additivity,
$[\ell:k]$ copies of the identity functor, proving the equality.

Take $x$ in the integral kernel for $B$. By localization,
$x=\Tr_{\ell/B}(a)$ for some $a\in K_n(\ell)$.
The two identities of \cref{lem:finite-gysin} and
\eqref{eq:inseparable-field-transfer} give
\[
\begin{aligned}
 \res_{B/A}\Tr_{B/A}x
 &=\res_{B/A}\Tr_{k/A}\Tr_{\ell/k}(a)\\
 &=e\,\Tr_{\ell/B}\res_{\ell/k}\Tr_{\ell/k}(a)\\
 &=e[\ell:k]\,\Tr_{\ell/B}(a)=e[\ell:k]\,x.
\end{aligned}
\]
Both $\res_{B/A}$ and $\Tr_{B/A}$ preserve the kernels. Their two composites
on those kernels are multiplication by $e[\ell:k]$.
After this integer is inverted, the displayed inverse follows.
\end{proof}

A \emph{Cohen ring} with residue field $k$ of characteristic $p$
means a complete DVR with uniformizer $p$ and residue field $k$
\cite[\stackstag{0327}]{Stacks}.
A Cohen subring of a complete mixed-characteristic DVR is such
a subring inducing an isomorphism on residue fields.

\begin{corollary}[Comparison with a Cohen subring]
\label{cor:cohen-core}
Let $A$ be a complete DVR of mixed characteristic $(0,p)$ with residue field
$k$ and absolute ramification index $e=v_A(p)$.
There is a Cohen subring $C\subseteq A$ inducing the identity
on $k$. For every such Cohen subring, $A$ is finite free of
rank $e$ over $C$.
For every integer $n$, restriction induces an isomorphism
\begin{multline}
 \ker\bigl(K_n(C)\to K_n(\Frac C)\bigr)[1/e]\\
 \xrightarrow{\ \res_{A/C}\ }
 \ker\bigl(K_n(A)\to K_n(\Frac A)\bigr)[1/e],
 \qquad (\res_{A/C})^{-1}=e^{-1}\Tr_{A/C}.
 \label{eq:cohen-kernel}
\end{multline}
Integrally, $\Tr_{A/C}$ is surjective on these kernels. If the kernel
for $C$ is zero, the kernel for $A$ is killed by $e$; if the
kernel for $A$ is zero, the kernel for $C$ is zero.
\end{corollary}
\begin{proof}
The Cohen structure theorem for a complete noetherian local domain
gives a complete regular local subring of $A$ with the same residue
field, over which $A$ is finite \cite[\stackstag{032D}]{Stacks}.
In mixed characteristic this subring is a formal power series ring
over a Cohen ring. Finite integral extensions preserve dimension;
hence that subring has dimension one. A formal power series ring
in $r$ variables over a Cohen DVR has dimension $r+1$, so $r=0$.
This subring is therefore a Cohen ring $C$.

For any Cohen embedding $C\subseteq A$ inducing the identity
on $k$, choose a uniformizer $\pi$ of $A$. Since $pA=\pi^eA$,
the filtration by powers of $\pi$ gives
\[
 \dim_k(A/pA)
 =\sum_{i=0}^{e-1}\dim_k(\pi^iA/\pi^{i+1}A)=e.
\]
Here the $C$-action on each quotient factors through $C/pC=k$,
and multiplication by $\pi^i$ identifies that quotient with $k$.
The ring $C$ is $p$-adically complete and
$\bigcap_{m\geq1}p^mA=0$. Hence $A$ is a finite $C$-module by
\cite[\stackstag{031D}]{Stacks}. It is torsion-free over the DVR
$C$, so it is finite free; its rank is $\dim_k(A/pA)=e$.
Apply \cref{prop:inseparable-kernel} with $\ell=k$ to obtain
\eqref{eq:cohen-kernel}.

For integral surjectivity, every element of the kernel for $C$
is $\Tr_{k/C}(a)$ for some $a\in K_n(k)$. The first identity in
\eqref{eq:finite-gysin} gives
$\Tr_{A/C}\Tr_{k/A}(a)=\Tr_{k/C}(a)$.
If the kernel for $A$ is zero, its surjective image is zero.
If the kernel for $C$ is zero and $x$ lies in the kernel for $A$,
then $\Tr_{A/C}x=0$ and the calculation in the preceding proposition
gives $ex=\res_{A/C}\Tr_{A/C}x=0$.
\end{proof}

\begin{proposition}[Boundary images and total ramification]
\label{prop:boundary-ramification}
Let $A\subseteq B$ be a finite extension of DVRs inducing the
identity on their common residue field $k$, and let $e$ be their
ramification index. For every integer $n$, one has
\begin{equation}
\begin{aligned}
 e\,\im\bigl(\partial_A:K_{n+1}(\Frac A)\to K_n(k)\bigr)
 &\subseteq\im\bigl(\partial_B:K_{n+1}(\Frac B)\to K_n(k)\bigr)\\
 &\subseteq\im\bigl(\partial_A:K_{n+1}(\Frac A)\to K_n(k)\bigr).
\end{aligned}
\label{eq:integral-boundary-images}
\end{equation}
The images are therefore equal after inverting $e$; in particular,
\begin{multline}
 \im\bigl(K_{n+1}(\Frac A)_{\Q}\xrightarrow{\partial_A}K_n(k)_{\Q}\bigr)\\
 =\im\bigl(K_{n+1}(\Frac B)_{\Q}\xrightarrow{\partial_B}K_n(k)_{\Q}\bigr).
 \label{eq:boundary-images}
\end{multline}
\end{proposition}
\begin{proof}
By \cref{prop:localization},
\[
 \im\partial_A=\ker\Tr_{k/A},\qquad
 \im\partial_B=\ker\Tr_{k/B}.
\]
The two identities in \cref{lem:finite-gysin}, with residue degree
one, are
\[
 \Tr_{B/A}\Tr_{k/B}=\Tr_{k/A},\qquad
 \res_{B/A}\Tr_{k/A}=e\,\Tr_{k/B}.
\]
For $a\in K_n(k)$, the first identity gives
$\Tr_{k/B}(a)=0\Rightarrow\Tr_{k/A}(a)=0$; the second gives
$\Tr_{k/A}(a)=0\Rightarrow\Tr_{k/B}(ea)=0$.
These are exactly the right and left inclusions in
\eqref{eq:integral-boundary-images}. Inverting $e$ makes the
outer subgroups equal. Tensoring with $\Q$ gives
\eqref{eq:boundary-images}.
\end{proof}

\section{Perfection of the residue field}
\label{sec:perfection}

At limit stages we use the following valuation argument.
The preservation of henselianity is \cite[\stackstag{04GI}]{Stacks}.

\begin{lemma}[Directed colimits with compatible uniformizers]
\label{lem:union-dvr}
Let $I$ be a nonempty directed partially ordered set and
$(C_i,\iota_{ij})_{i\leq j}$ a directed system of henselian DVRs.
Assume that each $\iota_{ij}:C_i\to C_j$ is an injective local
homomorphism. Suppose that uniformizers $\pi_i\in C_i$ have been
chosen with
\[
 \iota_{ij}(\pi_i)=\pi_j\qquad(i\leq j).
\]
For
\[
 C=\varinjlim_{i\in I}C_i,\qquad \iota_i:C_i\longrightarrow C,
\]
every $\iota_i$ is injective and local. The ring $C$ is a henselian
DVR, and the common image $\pi=\iota_i(\pi_i)$ is a uniformizer.
The canonical maps induce isomorphisms
\[
 \varinjlim_{i\in I}C_i/\pi_iC_i\xrightarrow{\ \sim\ }C/\pi C,
 \qquad
 \varinjlim_{i\in I}\Frac C_i\xrightarrow{\ \sim\ }\Frac C.
\]
Its normalized valuation satisfies
\[
 v_C(\iota_i(x))=v_{C_i}(x)
 \qquad(i\in I,\ x\in C_i),
\]
where every valuation takes the value $\infty$ at zero.
\end{lemma}
\begin{proof}
For $i\leq j$ and $x\in C_i\setminus\{0\}$, express
$x=\pi_i^m u$ with $m\in\N$ and $u\in C_i^\times$.
Since $\iota_{ij}(u)\in C_j^\times$, one has
\begin{equation}
 \iota_{ij}(x)=\pi_j^m\iota_{ij}(u),\qquad
 v_{C_j}(\iota_{ij}(x))=m=v_{C_i}(x).
 \label{eq:valuation-transition}
\end{equation}
In a directed colimit of rings,
$\iota_i(x)=\iota_j(y)$ if and only if some $h\geq i,j$
satisfies $\iota_{ih}(x)=\iota_{jh}(y)$.
In particular, injectivity of the transition maps implies
injectivity of each $\iota_i$.
If two nonzero elements of $C$ are represented in one $C_i$,
their product is the image of a nonzero product in that domain.
Thus $C$ is a domain.

Equation \eqref{eq:valuation-transition} and the equality criterion show that
\[
 v_C(0)=\infty,\qquad
 v_C(\iota_i(x))=v_{C_i}(x)\quad(x\ne0)
\]
defines a function on $C$. For all $a,b\in C$, computation in
a common stage gives
\[
 v_C(ab)=v_C(a)+v_C(b),\qquad
 v_C(a+b)\geq\min\{v_C(a),v_C(b)\}.
\]
For $a\ne0$, its expression at one stage gives
$a=\pi^{v_C(a)}u$ with $u\in C^\times$.
The element $\pi$ is not invertible: an equation $\pi b=1$
would hold at a stage and make its uniformizer invertible.
It follows that
\[
 C^\times=\{a\in C:v_C(a)=0\},\qquad
 C\setminus C^\times=\pi C.
\]
Consequently $C$ is local with maximal ideal $\pi C$.
Equation \eqref{eq:valuation-transition} also gives
$\iota_i^{-1}(\pi C)=\pi_iC_i$, so $\iota_i$ is local.

For a nonzero ideal $J\subseteq C$, the nonempty subset
$\{v_C(a):a\in J\setminus\{0\}\}$ of $\N$
has a least element $m$. Choose $a\in J$ with $v_C(a)=m$.
Then $a=\pi^m u$ with $u\in C^\times$, whence $\pi^mC\subseteq J$.
Every $b\in J\setminus\{0\}$ has the form
$b=\pi^{v_C(b)}v$ with $v_C(b)\geq m$ and $v\in C^\times$.
Thus $b\in\pi^mC$ and
\[
 J=\pi^mC.
\]
Hence $C$ is noetherian, is a local domain, and has nonzero
principal maximal ideal. It is a DVR
\cite[\stackstag{00PD}]{Stacks}, with the stated valuation.

The maps $C_i/\pi_iC_i\to C_j/\pi_jC_j$ and
$C_i/\pi_iC_i\to C/\pi C$ are injective by \eqref{eq:valuation-transition}.
Every element of $C/\pi C$ is represented at one stage; the
first asserted isomorphism follows.
The injections $\iota_{ij}$ and $\iota_i$ extend to fraction fields.
Every fraction in $\Frac C$ has numerator and nonzero denominator
represented in a common $C_i$. The induced map
$\varinjlim_i\Frac C_i\to\Frac C$ is therefore surjective.
It is injective because all its maps from the fields $\Frac C_i$
are injective.

Finally, $C$ is henselian by \cite[\stackstag{04GI}]{Stacks},
since it is a filtered colimit of henselian local rings under local maps.
\end{proof}

\begin{lemma}[Adjoining one residue $p$-th root]
\label{lem:one-residue-root}
Let $C$ be a henselian DVR with uniformizer $p$ and residue field
$k$ of characteristic $p$. For $a\in k\setminus k^p$ and a lift
$\widetilde a\in C$, consider
\[
 D=C[X]/(X^p-\widetilde a).
\]
The canonical map $C\to D$ is injective and local, and $D$ is a
henselian DVR with uniformizer $p$. It is free of rank $p$ over $C$,
and the residue identification is
\[
 D/pD\xrightarrow{\ \sim\ }k(a^{1/p}),\qquad X\bmod pD\longmapsto a^{1/p}.
\]
For every $n\in\Z$, the maps
\begin{gather*}
 K_n(C)[1/p]\xrightarrow{\res_{D/C}}K_n(D)[1/p],\\
 K_n(\Frac C)[1/p]\xrightarrow{\res_{\Frac D/\Frac C}}
 K_n(\Frac D)[1/p]
\end{gather*}
are injective, and restriction induces an isomorphism
\begin{multline}
 \ker\bigl(K_n(C)\to K_n(\Frac C)\bigr)[1/p]\\
 \xrightarrow{\ \sim\ }
 \ker\bigl(K_n(D)\to K_n(\Frac D)\bigr)[1/p].
 \label{eq:one-residue-root-kernel}
\end{multline}
Its inverse on these kernels is $p^{-1}\Tr_{D/C}$.
\end{lemma}
\begin{proof}
Division by the monic polynomial gives the $C$-basis
$1,X,\ldots,X^{p-1}$ of $D$. The map $C\to D$ is therefore
injective, and multiplication by $p$ on $D$ is injective.
Reduction modulo $p$ gives
\[
 D/pD=k[X]/(X^p-a)=k(a^{1/p}).
\]
Here $X^p-a$ is irreducible because $a\notin k^p$
\cite[\stackstag{09HF}]{Stacks}.
Every maximal ideal of the finite integral $C$-algebra $D$
contracts to the maximal ideal $pC$.
Since $D/pD$ is a field, $pD$ is the unique maximal ideal of $D$.
The ring $D$ is noetherian and $\dim D=\dim C=1$, by injectivity
and integrality of $C\to D$.
Its maximal ideal is generated by $p$, so it is regular of
dimension one and hence a DVR
\cite[\stackstag{00PD}]{Stacks}.
Since it is finite and local over $C$, it is henselian
\cite[\stackstag{04GH}]{Stacks}.

Because $p$ is a uniformizer in both rings,
\[
 D\otimes_C\Frac C=D[1/p]=\Frac D.
\]
Thus $\Frac D$ has degree $p$ over $\Frac C$.
The free bases over the rings and fields, together with
additivity, give, for every $n\in\Z$,
\[
 \Tr_{D/C}\res_{D/C}
   =p\,\mathrm{id}_{K_n(C)},\qquad
 \Tr_{\Frac D/\Frac C}\res_{\Frac D/\Frac C}
   =p\,\mathrm{id}_{K_n(\Frac C)}.
\]
After inverting $p$, each restriction consequently has a left
inverse and is injective.
The residue extension $k(a^{1/p})/k$ is purely inseparable of
degree $p$, and the ramification index is one.
\Cref{prop:inseparable-kernel} gives
\eqref{eq:one-residue-root-kernel}.
\end{proof}

\begin{theorem}[Perfection]
\label{thm:perfection}
Let $C$ be a henselian DVR of mixed characteristic $(0,p)$ with
uniformizer $p$ and residue field $k$.
For a fixed perfection $k^\perf$ of $k$, there is an injective
local map $C\to W(k^\perf)$ inducing the prescribed embedding
$k\to k^\perf$.
For every integer $n$, it induces an isomorphism
\begin{multline}
 \ker\bigl(K_n(C)\to K_n(\Frac C)\bigr)[1/p]\\
 \xrightarrow{\ \sim\ }
 \ker\bigl(K_n(W(k^\perf))\to K_n(\Frac W(k^\perf))\bigr)[1/p].
 \label{eq:perfection-kernel}
\end{multline}
For every integer $n$, the two restriction maps
\begin{gather*}
 K_n(C)[1/p]\longrightarrow K_n(W(k^\perf))[1/p],\\
 K_n(\Frac C)[1/p]\longrightarrow K_n(\Frac W(k^\perf))[1/p]
\end{gather*}
are injective.
\end{theorem}
\begin{proof}
We construct rings $C_\alpha$, subfields $k_\alpha\subseteq k^\perf$,
and injective local maps
$\iota_{\alpha\beta}:C_\alpha\to C_\beta$ for $\alpha\leq\beta$.
The required identities and residue identifications are
\begin{gather}
 \iota_{\alpha\alpha}=\mathrm{id},\qquad
 \iota_{\beta\gamma}\iota_{\alpha\beta}=\iota_{\alpha\gamma},
 \qquad \iota_{\alpha\beta}(p)=p,
 \label{eq:perfection-transition}\\
 C_\alpha/pC_\alpha\simeq k_\alpha,\qquad
 \overline{\iota_{\alpha\beta}}:
 k_\alpha\hookrightarrow k_\beta\subseteq k^\perf.
 \label{eq:perfection-residue-transition}
\end{gather}
Each $C_\alpha$ will be henselian with uniformizer $p$.
Choose an infinite cardinal $\mu>|k^\perf|$, viewed as an ordinal.
The recursion is performed for ordinals less than $\mu$ and stops
at the first stage whose residue field is perfect.

Start with $C_0=C$ and $k_0=k$.
Suppose that $C_\alpha$ has been constructed and $k_\alpha$ is
imperfect. Choose
$a_\alpha\in k_\alpha\setminus k_\alpha^p$ and a lift
$\widetilde a_\alpha\in C_\alpha$. Define
\[
 C_{\alpha+1}=C_\alpha[X]/(X^p-\widetilde a_\alpha),
 \qquad
 k_{\alpha+1}=k_\alpha(a_\alpha^{1/p})\subseteq k^\perf,
\]
where $a_\alpha^{1/p}$ is the unique $p$-th root in $k^\perf$.
Take the canonical map $\iota_{\alpha,\alpha+1}$ and define
$\iota_{\beta,\alpha+1}
=\iota_{\alpha,\alpha+1}\iota_{\beta\alpha}$ for $\beta\leq\alpha$.
The residue class of $X$ maps to $a_\alpha^{1/p}$.
\Cref{lem:one-residue-root} proves all the required ring properties
and the identities \eqref{eq:perfection-transition}--%
\eqref{eq:perfection-residue-transition} at this successor stage.

For a nonzero limit ordinal $\beta<\mu$, provided the construction
has not stopped earlier, define
\[
 C_\beta=\varinjlim_{\alpha<\beta}C_\alpha,
 \qquad
 k_\beta=\bigcup_{\alpha<\beta}k_\alpha\subseteq k^\perf,
\]
and take the canonical colimit maps for $\iota_{\alpha\beta}$.
\Cref{lem:union-dvr} proves that $C_\beta$ is a henselian DVR
with uniformizer $p$ and gives the compatible identifications
\[
 C_\beta/pC_\beta
   \simeq\varinjlim_{\alpha<\beta}C_\alpha/pC_\alpha
   \simeq k_\beta,\qquad
 \Frac C_\beta\simeq\varinjlim_{\alpha<\beta}\Frac C_\alpha.
\]
The colimit identities give \eqref{eq:perfection-transition}.

If no stage less than $\mu$ had perfect residue field, the elements
$a_\alpha^{1/p}$, for $\alpha<\mu$, would all be defined and distinct.
Indeed, for $\alpha<\beta<\mu$,
\[
 a_\alpha^{1/p}\in k_{\alpha+1}\subseteq k_\beta,
 \qquad
 a_\beta^{1/p}\notin k_\beta.
\]
This would give an injection $\mu\to k^\perf$, contrary to
$\mu>|k^\perf|$. Thus a first terminal ordinal $\lambda<\mu$
exists. Since $k_\lambda$ is perfect and contains $k$, induction
on $r\geq0$ shows that it contains the unique $p^r$-th root of
every element of $k$. Every element of $k^\perf$ is such a root.
The inclusion $k_\lambda\subseteq k^\perf$ is therefore an equality.

Fix $n\in\Z$. We next prove, by transfinite induction on
$\beta\leq\lambda$, that for every $\alpha\leq\beta$ the maps
\begin{gather*}
 K_n(C_\alpha)[1/p]\longrightarrow K_n(C_\beta)[1/p],\\
 K_n(\Frac C_\alpha)[1/p]\longrightarrow K_n(\Frac C_\beta)[1/p]
\end{gather*}
are injective, and that
\begin{multline*}
 \ker\bigl(K_n(C_\alpha)\to K_n(\Frac C_\alpha)\bigr)[1/p]
 \longrightarrow\\
 \ker\bigl(K_n(C_\beta)\to K_n(\Frac C_\beta)\bigr)[1/p]
\end{multline*}
is an isomorphism.
For $\alpha=\beta$ these are identity maps.
At a successor stage, compose the induction assertions for the
preceding stage with \cref{lem:one-residue-root}.

Suppose that $\beta$ is a nonzero limit ordinal and $\alpha<\beta$.
The subset $\{\gamma:\alpha\leq\gamma<\beta\}$ is cofinal in $\beta$,
since $\max(\alpha,\eta)<\beta$ for every $\eta<\beta$.
By \cref{lem:continuity} and the induction hypothesis,
\begin{align*}
 &\ker\bigl(K_n(C_\alpha)[1/p]\to K_n(C_\beta)[1/p]\bigr)\\
 &\hspace{1cm}=
 \bigcup_{\alpha\leq\gamma<\beta}
 \ker\bigl(K_n(C_\alpha)[1/p]\to K_n(C_\gamma)[1/p]\bigr)=0.
\end{align*}
Replacing every $C_\gamma$ by $\Frac C_\gamma$ gives the
fraction-field assertion by the same equality.
Exactness of localization of abelian groups and
\cref{lem:continuity} give
\begin{multline*}
 \ker\bigl(K_n(C_\beta)\to K_n(\Frac C_\beta)\bigr)[1/p]
 \simeq\\
 \varinjlim_{\alpha\leq\gamma<\beta}
 \ker\bigl(K_n(C_\gamma)\to K_n(\Frac C_\gamma)\bigr)[1/p].
\end{multline*}
On these kernels, for $\alpha\leq\gamma\leq\delta<\beta$,
functoriality and the induction hypothesis give
\[
 (\res_{C_\delta/C_\alpha})^{-1}\res_{C_\delta/C_\gamma}
   =(\res_{C_\gamma/C_\alpha})^{-1}.
\]
Hence the inverses on the right define a map from the displayed
colimit to the kernel over $C_\alpha$. Its composites with the
canonical map from that kernel are identities, since this holds
at each stage. This completes the induction.

The completion $\widehat{C_\lambda}$ is a complete DVR with
uniformizer $p$ and residue field $k^\perf$.
The map $C_\lambda\to\widehat{C_\lambda}$ is injective because
\[
 \ker(C_\lambda\to\widehat{C_\lambda})
   =\bigcap_{m\geq1}p^mC_\lambda=0;
\]
the last equality follows from the finite valuation of every
nonzero element of $C_\lambda$.
This completion is a strict $p$-ring. By
\cite[Chapitre~II, \S6, Proposition~10]{Serre}, there is an isomorphism
\[
 \widehat{C_\lambda}\xrightarrow{\ \sim\ }W(k^\perf)
\]
inducing the identity on $k^\perf$.
The composite
\[
 C=C_0\xrightarrow{\iota_{0\lambda}}C_\lambda
 \longrightarrow\widehat{C_\lambda}
 \xrightarrow{\ \sim\ }W(k^\perf)
\]
is an injective local map with the prescribed residue embedding.

Apply \cref{thm:completion} to $C_\lambda$ with
$M=\Sph[1/p]$.
It gives injectivity on the two $K$-groups of the ring and
fraction field, and an isomorphism on the kernels of restriction
to fraction fields.
For every domain $R$, flatness of $\Z[1/p]$ over $\Z$ identifies
\[
 \ker\bigl(K_n(R)[1/p]\to K_n(\Frac R)[1/p]\bigr)
 =
 \ker\bigl(K_n(R)\to K_n(\Frac R)\bigr)[1/p].
\]
Combining the completion assertions with the induction assertions
for $\alpha=0$ and $\beta=\lambda$ proves
\eqref{eq:perfection-kernel} and both injectivity statements.
The integer $n$ was arbitrary.
\end{proof}

\section{Witt vectors and rational Gersten kernels}
\label{sec:residue-extension}

We first recall the finite extension property of Witt rings.
It follows from the equivalence for finite \'etale algebras over
a henselian local ring \cite[\stackstag{04GK}]{Stacks} and
\cite[Chapitre~II, \S6, Proposition~10]{Serre}.

\begin{lemma}[Finite extensions of Witt rings]
\label{lem:finite-witt-extension}
For every finite extension $k\subseteq\ell$ of perfect fields of
characteristic $p$, the extension $W(k)\to W(\ell)$ is finite
\'etale and free of rank $[\ell:k]$.
\end{lemma}
\begin{proof}
The extension $\ell/k$ is separable. By
\cite[\stackstag{04GK}]{Stacks}, it lifts to a finite \'etale
$W(k)$-algebra $B$ with $B/pB=\ell$.
A finite \'etale algebra is finite projective as a module;
over the local ring $W(k)$ it is free. Its rank is
$\dim_k(B/pB)=[\ell:k]$. Finite freeness also makes $B$
$p$-adically separated, complete, and $p$-torsion-free.
Thus $B$ is a strict $p$-ring with residue field $\ell$.
\cite[Chapitre~II, \S6, Proposition~10]{Serre} gives a unique isomorphism
$B\simeq W(\ell)$ inducing the identity on $\ell$.
Its composite with $W(k)\to B$ equals $W(k)\to W(\ell)$,
since both maps induce the prescribed embedding $k\subseteq\ell$
and their source is a strict $p$-ring. This proves the
assertion for the prescribed Witt homomorphism.
\end{proof}

\begin{theorem}[Extension with fixed ramification]
\label{thm:perfect-extension}
Let $A$ be a complete DVR of mixed characteristic $(0,p)$ with perfect residue
field $k$, and fix a Cohen embedding $W(k)\hookrightarrow A$
inducing the identity on $k$.
For an extension $k\subseteq\ell$ of perfect fields, define
\[
 A_\ell=A\otimes_{W(k)}W(\ell).
\]
Then the following assertions hold.
\begin{enumerate}
\item The ring $A_\ell$ is a complete DVR with residue field
$\ell$. Every uniformizer of $A$ is a uniformizer of $A_\ell$,
and their absolute ramification indices are equal.
\item For every integer $n$, the maps
\[
 K_n(A)_{\Q}\longrightarrow K_n(A_\ell)_{\Q},\qquad
 K_n(\Frac A)_{\Q}\longrightarrow K_n(\Frac A_\ell)_{\Q}
\]
are injective.
\item If $k\subseteq\ell$ is algebraically pure, then, for every integer $n$ and every spectrum $M$,
\[
 K_n(A;M)\longrightarrow K_n(A_\ell;M),\qquad
 K_n(\Frac A;M)\longrightarrow K_n(\Frac A_\ell;M)
\]
are pure monomorphisms. In particular this applies to an embedding
with $k$ algebraically closed.
\end{enumerate}
\end{theorem}
\begin{proof}
\textup{(i)} Choose a uniformizer $\pi$ of $A$.
For a unit $u\in A^\times$ and the positive integer $e=v_A(p)$,
one has $p=u\pi^e$.
By \cref{cor:cohen-core}, $A$ is finite free of rank $e$ over
$W(k)$. Functoriality of Witt vectors gives an injection
$W(k)\to W(\ell)$: in Witt coordinates it is the coordinatewise
injection. The target is a domain and is torsion-free over the
DVR $W(k)$, so this map is flat.
The ring $A_\ell$ is finite free of rank $e$ over $W(\ell)$.
A $W(k)$-basis of $A$ also shows that $A\to A_\ell$ is injective.

Every maximal ideal of $A_\ell$ contracts to $(p)$ in $W(\ell)$,
since the extension is finite integral. It contains $p$ and hence
$\pi$, because $p=u\pi^e$. Moreover
\[
 A_\ell/\pi A_\ell
 =(A/\pi A)\otimes_{W(k)}W(\ell)
 =k\otimes_{W(k)}W(\ell)=\ell.
\]
Thus $A_\ell$ is local with maximal ideal $(\pi)$.
Multiplication by $\pi$ on $A$ is injective and remains so after
the flat base change from $W(k)$ to $W(\ell)$.
The ring $A_\ell$ is noetherian and has dimension one, since
it is finite integral over $W(\ell)$.
It is a one-dimensional regular local ring, hence a DVR.
It is $p$-adically complete because it is a finite free module
over the complete ring $W(\ell)$.
The equality $p=u\pi^e$ identifies the $p$-adic and $\pi$-adic
topologies and gives $v_{A_\ell}(p)=e$.

\smallskip
\noindent\textup{(ii), (iii).}
By \textup{(i)}, $A\to A_\ell$ is a local map of DVRs preserving
a uniformizer. The field $k$ is perfect, so $\ell/k$ is separable.
\Cref{prop:regular-comparison} gives torsion kernels for the two
integral maps, hence the injectivity in \textup{(ii)} after
tensoring with $\Q$. Under the purity hypothesis, it gives
\textup{(iii)} directly.
If $k$ is algebraically closed, every nonzero finitely presented
$k$-algebra has a $k$-point by the weak Nullstellensatz
\cite[\stackstag{00FV}]{Stacks}. Applying this to the algebras
in \cref{lem:finite-presentation-retractions} proves that
$k\subseteq\ell$ is algebraically pure.
\end{proof}

\begin{samepage}
\begin{corollary}[Restriction between Witt rings]
\label{cor:witt-restriction}
For every extension $k\subseteq\ell$ of perfect fields of
characteristic $p$ and every integer $n$, restriction induces injections
\begin{gather*}
 \res_{W(\ell)/W(k)}:K_n(W(k))_{\Q}
       \lhook\joinrel\longrightarrow K_n(W(\ell))_{\Q},\\
 \res_{\Frac W(\ell)/\Frac W(k)}:K_n(\Frac W(k))_{\Q}
       \lhook\joinrel\longrightarrow K_n(\Frac W(\ell))_{\Q}.
\end{gather*}
In particular restriction induces an injection
\begin{multline}
 \ker\bigl(K_n(W(k))_{\Q}\to K_n(\Frac W(k))_{\Q}\bigr)\\
 \lhook\joinrel\longrightarrow
 \ker\bigl(K_n(W(\ell))_{\Q}\to K_n(\Frac W(\ell))_{\Q}\bigr).
 \label{eq:witt-kernel-injection}
\end{multline}
If the residue embedding is algebraically pure, then, for every
integer $n$ and every spectrum $M$, the maps
\[
 K_n(W(k);M)\longrightarrow K_n(W(\ell);M),\qquad
 K_n(\Frac W(k);M)\longrightarrow K_n(\Frac W(\ell);M)
\]
are pure monomorphisms.
\end{corollary}
\end{samepage}
\begin{proof}
Apply \cref{thm:perfect-extension} with $A=W(k)$.
The square
\[
\begin{tikzcd}[column sep=large,row sep=large]
 K_n(W(k))_{\Q}\arrow[r]\arrow[d,hook,"\res"]
 &K_n(\Frac W(k))_{\Q}\arrow[d,hook,"\res"]\\
 K_n(W(\ell))_{\Q}\arrow[r]
 &K_n(\Frac W(\ell))_{\Q}
\end{tikzcd}
\]
commutes. It sends the upper kernel to the lower kernel, and
injectivity of the left vertical arrow proves
\eqref{eq:witt-kernel-injection}.
\end{proof}

\begin{proof}[Proof of \cref{thm:boundary-functoriality}]

Choose a Cohen subring $C\subseteq\widehat R$ inducing the
identity on $k$, and a map $C\to W(k^\perf)$ furnished by
\cref{thm:perfection}. For $a\in K_n(k)_{\Q}$, localization,
completion, total ramification, and perfection give, respectively,
\begin{align*}
 a\in\im\partial_R
 &\ \Longleftrightarrow\ \Tr_{k/R}(a)=0\\
 &\ \Longleftrightarrow\ \Tr_{k/\widehat R}(a)=0\\
 &\ \Longleftrightarrow\ \Tr_{k/C}(a)=0\\
 &\ \Longleftrightarrow\
   \Tr_{k^\perf/W(k^\perf)}\res_{k^\perf/k}(a)=0\\
 &\ \Longleftrightarrow\
   \res_{k^\perf/k}(a)\in\im\partial_{W(k^\perf)}.
\end{align*}
The second equivalence uses the injectivity in
\cref{thm:completion}; the third uses the finiteness of $\widehat R$ over $C$ from
\cref{cor:cohen-core} and then \cref{prop:boundary-ramification}. For the fourth,
\eqref{eq:flat-gysin} identifies the two transfers after restriction,
and \cref{thm:perfection} gives injectivity on their images.
The final expression contains no choice of Cohen subring or lift,
which proves \textup{(i)}.

For perfect fields $k^\perf\subseteq\ell^\perf$, flat base change
and \cref{cor:witt-restriction} give, for every
$b\in K_n(k^\perf)_{\Q}$,
\[
 \Tr_{k^\perf/W(k^\perf)}(b)=0
 \quad\Longleftrightarrow\quad
 \Tr_{\ell^\perf/W(\ell^\perf)}\res_{\ell^\perf/k^\perf}(b)=0.
\]
Apply this with $b=\res_{k^\perf/k}(a)$ and use
\textup{(i)} for $R$ and $S$. One obtains
$a\in\im\partial_R$ if and only if
$\res_{\ell/k}(a)\in\im\partial_S$, which is precisely
the cartesian assertion in \textup{(ii)}.

Restriction $K_n(k)_{\Q}\to K_n(k^\perf)_{\Q}$ is surjective by \eqref{eq:inseparable-field-transfer} and
\cref{lem:continuity}. Equation \eqref{eq:intrinsic-boundary}
therefore induces an isomorphism of the quotients by the boundary
images. Localization identifies these quotients with the kernels
in \eqref{eq:canonical-witt-kernel}. Multiplication by
$v_R(p)^{-1}$ then gives exactly \eqref{eq:canonical-witt-class}.
This proves existence, uniqueness, and choice independence.

Every ring map $R\to S$ between DVRs of mixed characteristic
$(0,p)$ is injective and local: its kernel is a prime not containing
$p$, hence is zero; the inverse image of the maximal ideal is a
prime containing $p$, hence is the maximal ideal of $R$.
For such a map of ramification index $e$, one has $v_S(p)=e\,v_R(p)$ and
\[
 \res_{S/R}\Tr_{k/R}(a)=e\,\Tr_{\ell/S}\res_{\ell/k}(a)
 \qquad(a\in K_n(k)_{\Q}).
\]
Applying \eqref{eq:canonical-witt-class} for $S$ to this equality
gives
\[
 \frac{e}{v_S(p)}
 \Tr_{\ell^\perf/W(\ell^\perf)}\res_{\ell^\perf/k}(a)
 =\frac{1}{v_R(p)}
 \Tr_{\ell^\perf/W(\ell^\perf)}\res_{\ell^\perf/k}(a).
\]
By flat base change this is the image under
$\res_{W(\ell^\perf)/W(k^\perf)}$ of the right side of
\eqref{eq:canonical-witt-class}. The classes $\Tr_{k/R}(a)$
exhaust the source kernel, so naturality follows.
The Witt restriction is injective by \cref{cor:witt-restriction};
the two canonical isomorphisms give the last assertion.
\end{proof}

\section{The universal test rings}
\label{sec:finite-origin}
\label{sec:main-proof}

\begin{proposition}[Arithmetic realization of a residue field]
\label{prop:arithmetic-realization}
Every finitely generated field $k_0/\F_p$ is the residue field
of an arithmetic henselian trait.
\end{proposition}
\begin{proof}
Since $\F_p$ is perfect, a finitely generated extension admits
a separating transcendence basis
\cite[\stackstag{030Q}, \stackstag{030Z}]{Stacks}.
Choose such a basis
$t_1,\ldots,t_s$ for $k_0/\F_p$.
Then $k_0/\F_p(t_1,\ldots,t_s)$ is finite separable.
Choose a primitive element $\alpha$
\cite[\stackstag{030N}]{Stacks}, with monic minimal polynomial
$\overline f\in\F_p(t_1,\ldots,t_s)[X]$.
This includes the trivial extension, for which a degree-one
polynomial may be used.

Choose a nonzero polynomial $\overline g$ in the variables
$T_1,\ldots,T_s$ over $\F_p$ so that all coefficients of
$\overline f$ belong, under $T_i\mapsto t_i$, to the localization
\[
 \F_p[T_1,\ldots,T_s,1/\overline g].
\]
Lift $\overline g$ to $g\in\Z_{(p)}[T_1,\ldots,T_s]$.
Lifting the numerators of the coefficients of $\overline f$ and
replacing their denominators by the corresponding powers of $g$
gives a monic polynomial of the same degree
\[
 f\in\Z_{(p)}[T_1,\ldots,T_s,1/g][X].
\]
Define
\begin{equation}
 \widetilde A=
 \bigl(\Z_{(p)}[T_1,\ldots,T_s,1/g,1/\disc(f)]\bigr)[X]/(f).
 \label{eq:explicit-smooth-ring}
\end{equation}
The discriminant of $\overline f$ is nonzero by separability,
so the reduction of the base ring in this formula is nonzero.
The monic algebra is finite free; its discriminant in the power
basis is $\disc(f)$. Since this element has been inverted,
the algebra is finite \'etale
\cite[\stackstag{0BJF}]{Stacks}.
The base is a localization of a polynomial
algebra by finitely many elements, and is smooth and finitely
presented over $\Z_{(p)}$. These properties are preserved by a
finite \'etale extension. Thus $\widetilde A$ is smooth and
finitely presented over $\Z_{(p)}$.

We check the special fibre.
The quotient by the monic polynomial $\overline f$ is free,
with basis $1,X,\ldots,X^{\deg f-1}$, over
$\F_p[T_1,\ldots,T_s,1/\overline g,1/\disc(\overline f)]$.
It therefore injects into its localization at the nonzero elements
of this domain. That localization is
\[
 \F_p(t_1,\ldots,t_s)[X]/(\overline f)=k_0.
\]
Hence $\widetilde A/p\widetilde A$ is a domain. Its fraction
field is $k_0$, since its image contains the $t_i$ and $\alpha$.
The ring $U=(\widetilde A_{p\widetilde A})^h$ is consequently
an arithmetic henselian trait with residue field $k_0$.

\end{proof}

\begin{proof}[Proof of \cref{thm:universal-witt-test}]
Fix $R$ as in \textup{(ii)}, with residue field $k$, and a nonzero
class
\[
 x\in\ker\bigl(K_n(R)_\Q\to K_n(\Frac R)_\Q\bigr).
\]
By \cref{prop:localization,lem:continuity}, there are a finitely
generated subfield $k_0/\F_p$ of $k$ and $a_0\in K_n(k_0)_\Q$
such that
\[
 x=\Tr_{k/R}\res_{k/k_0}(a_0),\qquad
 \res_{k/k_0}(a_0)\notin\im\partial_R.
\]
Choose an arithmetic henselian trait $U$ with residue field $k_0$
by \cref{prop:arithmetic-realization}. The cartesian square in
\cref{thm:boundary-functoriality}\textup{(ii)}, for $U$ and $R$,
gives $a_0\notin\im\partial_U$. Thus
\[
 0\ne\Tr_{k_0/U}(a_0)\in
 \ker\bigl(K_n(U)_\Q\to K_n(\Frac U)_\Q\bigr).
\]

The integer $s=\trdeg_{\F_p}k_0$ is finite. A transcendence basis
of $k_0/\F_p$ identifies its rational function subfield with
$\F_p(t_1,\ldots,t_s)$; extending to algebraic closures gives an
embedding $k_0\hookrightarrow\Omega_{p,s}$.
For each $b\in\N\cup\{\infty\}$ with $b\geq s$, compose this
with the fixed embedding $\Omega_{p,s}\hookrightarrow\Omega_{p,b}$.
Applying \cref{thm:boundary-functoriality}\textup{(ii)} to $U$ and
$W(\Omega_{p,b})$ gives
\[
 a_0\notin\im\partial_U
 \quad\Longleftrightarrow\quad
 \res_{\Omega_{p,b}/k_0}(a_0)
       \notin\im\partial_{W(\Omega_{p,b})}.
\]
Consequently
\[
 0\ne\Tr_{\Omega_{p,b}/W(\Omega_{p,b})}
          \res_{\Omega_{p,b}/k_0}(a_0)
 \in\ker\bigl(K_n(W(\Omega_{p,b}))_\Q
             \to K_n(\Frac W(\Omega_{p,b}))_\Q\bigr).
\]
For finite $d$, the inclusion $k_0\subseteq k$ gives $s\leq d$.
For $d=\infty$, $s$ is still finite, regardless of the cardinality
of $k$. Taking $b=d$ proves \textup{(i)}$\Rightarrow$\textup{(ii)}
by contraposition. The class over $U$ proves
\textup{(iii)}$\Rightarrow$\textup{(ii)} by contraposition.
Conversely, \textup{(ii)} implies \textup{(i)} and \textup{(iii)}
because all their rings occur in \textup{(ii)}.
Taking $b=s$ and $b=\infty$ proves the more precise assertion.
Applying the equivalence separately for every $n$ proves the
simultaneous assertion.
\end{proof}

\begin{remark}
In condition~\textup{(iii)} of \cref{thm:universal-witt-test},
one may choose just one arithmetic henselian trait for each finitely
generated residue field: the proof uses only its residue field.
Continuity was applied to $K_n(k)$, not to $W(k)$; full Witt vectors
need not commute with filtered unions of perfect fields.
\end{remark}

\section{Known vanishing and the residue field}
\label{sec:thresholds}

The low-dimensional vanishing below is classical.
For finitely generated residue fields, the Quillen--Harder argument
appears in \cite[\arxivhtmlref{1710.00303v1}{S5.Thmthm12}{Corollary~5.12(i)} and the proof of
\arxivhtmlref{1710.00303v1}{S5.Thmthm13}{Proposition~5.13}]{Skalit}. We give its rational form and pass to
arbitrary residue fields by continuity. The vanishing used here is
the finite-field and curve case of the consequence of Parshin's
conjecture in \cite[Theorem~1.3(i)]{Geisser}.
The subsequent comparisons follow from
\cref{thm:boundary-functoriality,thm:universal-witt-test}.

\begin{samepage}
\begin{proposition}[Residue transcendence degree at most one]
\label{prop:low-transcendence}
Let $R$ be a DVR whose residue field $k$ has characteristic
$p>0$ and satisfies $\trdeg_{\F_p}k\leq1$.
Then $K_n(R)_{\Q}\to K_n(\Frac R)_{\Q}$ is injective for every $n$.
\end{proposition}
\end{samepage}
\begin{proof}
If $k/\F_p$ is algebraic, it is the union of its finite subfields.
Quillen's finite-field computation \cite[Theorem~8(i)]{Quillen} gives
$K_m(l)_{\Q}=0$ for each such field $l$ and each $m\geq1$.
Filtered-colimit continuity gives $K_m(k)_{\Q}=0$ for $m\geq1$.

Suppose $\trdeg_{\F_p}k=1$ and choose an element $t\in k$
transcendental over $\F_p$.
Every element of $k$ is algebraic over $\F_p(t)$, and any
finite collection of such elements generates a finite extension.
Consequently $k$ is the filtered union of the finite extensions
$L/\F_p(t)$ contained in it.
Each $L$ is the function field of a smooth projective curve $X$
over a finite field. Harder's theorem \cite{Harder}, in the
$K$-theoretic form \cite[VI, Theorem~6.1]{Weibel}, gives
$K_m(X)_{\Q}=0$ for $m\geq1$.
Localization, followed by the filtered colimit over the nonempty
open subsets of $X$, gives the exact sequence
\[
 K_m(X)_{\Q}\longrightarrow K_m(L)_{\Q}
 \longrightarrow\bigoplus_{x\in X^{(1)}}K_{m-1}(\kappa(x))_{\Q}.
\]
For $m\geq2$ the first group is zero by Harder's theorem, and
each summand on the right is zero by Quillen's computation, since
$\kappa(x)$ is finite and $m-1\geq1$.
Thus $K_m(L)_{\Q}=0$ for $m\geq2$.
This is also \cite[\arxivhtmlref{1710.00303v1}{S5.Thmthm12}{Corollary~5.12(i)}]{Skalit}.
Continuity now gives $K_m(k)_{\Q}=0$ for $m\geq2$.

In either case, for $n\geq2$ the source of
$\Tr_{k/R}:K_n(k)_{\Q}\to K_n(R)_{\Q}$ is zero, so its image
is zero. Equation \eqref{eq:support-image} proves injectivity
in those degrees. For $n\leq1$, use \cref{lem:low-degrees}.
\end{proof}

\begin{remark}
Restriction between $W(\Omega_{p,a})$ and $W(\Omega_{p,b})$ is
injective on rational $K$-groups for $a\leq b$, by
\cref{cor:witt-restriction}. Thus a failure of rational Gersten
injectivity at finite transcendence degree $a$ persists at every
$b\geq a$. By \cref{thm:universal-witt-test,prop:low-transcendence},
any failure occurs at some finite transcendence degree at least two.
\end{remark}

\begin{remark}
By \cref{thm:boundary-functoriality,thm:universal-witt-test},
$W(\Omega_{p,d})$ in condition~\textup{(i)} may be replaced by
any henselian DVR of mixed characteristic $(0,p)$ with residue
field $\Omega_{p,d}$. The comparison of its rational kernel with
that of $W(\Omega_{p,d})$ is \eqref{eq:canonical-witt-class}.
\end{remark}

\section{Comparing consecutive test rings}
\label{sec:applications}

Fix a prime $p$. \Cref{thm:universal-witt-test} turns induction on the
residue transcendence degree into a comparison of successive Witt rings.
For each integer $d\geq0$, we factor that comparison through the
perfection of a rational function field:
\[
\begin{tikzcd}[column sep=large]
 W(\Omega_{p,d})\arrow[r]
 &W(\Omega_{p,d}(t)^{\perf})\arrow[r]
 &W(\Omega_{p,d+1}).
\end{tikzcd}
\]
Here $t$ is transcendental over $\Omega_{p,d}$, and the last residue
field is an algebraic closure of $\Omega_{p,d}(t)$. The first arrow
is controlled by localization on the affine line. For the second,
\cref{thm:cyclic-descent} gives a criterion involving cyclic extensions
of prime degree over all finite intermediate fields.
Combining the two statements with \cref{thm:universal-witt-test}
gives \cref{thm:dimension-step,cor:bounded-induction}.

The affine-line calculation, Galois transfer identities, and
normalized transfers below use classical localization, homotopy
invariance, and additivity
\cite[IV, Exercise~6.13; V, Theorem~6.3 and Corollary~6.7.1]{Weibel}.
We apply these identities to the kernels defined in
\eqref{eq:witt-kernel-notation}. Injectivity under extension of
perfect residue fields supplies the descent argument. \Cref{thm:universal-witt-test} then gives the statements for henselian DVRs
in \cref{thm:dimension-step,cor:bounded-induction}.

For every perfect field $E$ of characteristic $p$ and every integer
$n\geq0$, define
\begin{equation}
 M_n(E)=\ker\!\left(K_n(W(E))_{\Q}
             \longrightarrow K_n(\Frac W(E))_{\Q}\right).
 \label{eq:witt-kernel-notation}
\end{equation}
By \cref{prop:localization},
\begin{equation}
 M_n(E)=\im\!\left(\Tr_{E/W(E)}:K_n(E)_{\Q}
                          \longrightarrow K_n(W(E))_{\Q}\right).
 \label{eq:witt-kernel-source}
\end{equation}
Restriction sends $M_n(E)$ injectively to $M_n(E')$ for every
extension $E\subseteq E'$ of perfect fields, by
\cref{cor:witt-restriction}.

\begin{lemma}[Unions of perfect residue fields]
\label{lem:witt-kernel-union}
Let $E$ be the union of a filtered family $(E_i)_{i\in I}$ of perfect
subfields of characteristic $p$. For every $n\geq0$, restriction
induces an isomorphism
\[
 \varinjlim_{i\in I} M_n(E_i)\xrightarrow{\ \sim\ }M_n(E).
\]
\end{lemma}
\begin{proof}
Let $x\in M_n(E)$. By \eqref{eq:witt-kernel-source}, there is
$a\in K_n(E)_{\Q}$ with $\Tr_{E/W(E)}(a)=x$.
By \cref{lem:continuity}, there are $i\in I$ and
$a_i\in K_n(E_i)_{\Q}$ with $\res_{E/E_i}(a_i)=a$.
The element $x_i=\Tr_{E_i/W(E_i)}(a_i)$ belongs to $M_n(E_i)$.
Both Witt rings have uniformizer $p$, so \eqref{eq:flat-gysin} gives
\[
 \res_{W(E)/W(E_i)}(x_i)
 =\Tr_{E/W(E)}\res_{E/E_i}(a_i)=x.
\]
This proves surjectivity. If an element of the colimit represented
by $y_i\in M_n(E_i)$ maps to zero, then
$\res_{W(E)/W(E_i)}(y_i)=0$. Injectivity in
\cref{cor:witt-restriction} gives $y_i=0$, which proves injectivity.
\end{proof}

\begin{lemma}[The rational function field]
\label{lem:rational-line-source}
Let $k$ be an algebraically closed field of characteristic $p$, let
$t$ be transcendental over $k$, and let $n\geq1$.
The homomorphism
\begin{equation}
\begin{split}
 K_n(k)_{\Q}\oplus\bigoplus_{a\in k}K_{n-1}(k)_{\Q}
 &\longrightarrow K_n(k(t))_{\Q},\\
 (b,(c_a)_{a\in k})
 &\longmapsto \res_{k(t)/k}(b)
       +\sum_{a\in k}\{t-a\}\cdot\res_{k(t)/k}(c_a)
\end{split}
\label{eq:rational-line-source}
\end{equation}
is an isomorphism; the sum is finite because $(c_a)_{a\in k}$
has finite support. Moreover, restriction gives an isomorphism
$K_n(k(t))_{\Q}\xrightarrow{\sim}K_n(k(t)^{\perf})_{\Q}$.
\end{lemma}
\begin{proof}
Quillen's calculation
\cite[V, Corollary~6.7.1]{Weibel} gives the split exact sequence
\begin{equation}
 0\longrightarrow K_n(k)_{\Q}
 \longrightarrow K_n(k(t))_{\Q}
 \xrightarrow{(\partial_a)_{a\in k}}
 \bigoplus_{a\in k}K_{n-1}(k)_{\Q}\longrightarrow0.
 \label{eq:rational-line-residues}
\end{equation}
Every closed point has the form $(t-a)$ because $k$ is algebraically
closed. With $\partial_a\{t-a\}=1$, the product formula for
boundaries \cite[V, \textup{(6.1.1)} and Corollary~6.3.1]{Weibel}
gives, for $a,b\in k$ and $c\in K_{n-1}(k)_{\Q}$,
\[
 \partial_b\bigl(\{t-a\}\cdot\res_{k(t)/k}(c)\bigr)
 =\begin{cases}c&b=a,\\0&b\ne a.\end{cases}
\]
These identities identify its splitting with
\eqref{eq:rational-line-source}.

For completeness, the assertion about perfection is the field case
of \cite[\arxivhtmlref{1810.12203v1}{S4.Thmtheorem1}{Lemma~4.1}]{KM}.
It also follows from the field transfer identity already proved in
\eqref{eq:inseparable-field-transfer}. For every finite purely
inseparable extension $H/k(t)$, both composites of restriction and
transfer are multiplication by $[H:k(t)]$.
The degree is a power of $p$, so restriction is an isomorphism after
tensoring with $\Q$. The finite subextensions $H$ of
$k(t)^{\perf}/k(t)$ form a filtered system with union $k(t)^{\perf}$.
Apply \cref{lem:continuity} to conclude.
\end{proof}

\begin{proposition}[Adjoining a variable]
\label{prop:rational-line-kernel}
Let $k$ be an algebraically closed field of characteristic $p$, let
$t$ be transcendental over $k$, and let $F=k(t)^{\perf}$.
For $a\in k$, denote the Teichm\"uller lift of $t-a$ by
$[t-a]\in W(F)^\times$. For every integer $n\geq1$, one has the equality
of subspaces of $K_n(W(F))_{\Q}$
\begin{equation}
 M_n(F)=\res_{W(F)/W(k)}M_n(k)
 +\sum_{a\in k}\{[t-a]\}\cdot
                  \res_{W(F)/W(k)}M_{n-1}(k).
 \label{eq:rational-line-kernel}
\end{equation}
In particular,
\begin{equation}
 M_n(k)=M_{n-1}(k)=0\quad\Longrightarrow\quad M_n(F)=0.
 \label{eq:rational-line-propagation}
\end{equation}
More generally, if $M_{n-1}(k)=0$, restriction induces an
isomorphism $M_n(k)\xrightarrow{\sim}M_n(F)$. In particular,
$M_3(k)\xrightarrow{\sim}M_3(F)$ for every such $k$.
\end{proposition}
\begin{proof}
Let $x\in M_n(F)$. Choose $a\in K_n(F)_{\Q}$ with
$\Tr_{F/W(F)}(a)=x$, by \eqref{eq:witt-kernel-source}.
By \cref{lem:rational-line-source}, there are $b\in K_n(k)_{\Q}$,
a finite subset $S\subset k$, and $c_s\in K_{n-1}(k)_{\Q}$ for
each $s\in S$, such that
\[
 a=\res_{F/k}(b)+\sum_{s\in S}\{t-s\}\cdot\res_{F/k}(c_s).
\]
Flat base change and the projection formula give
\begin{align*}
 \Tr_{F/W(F)}\res_{F/k}(b)
 &=\res_{W(F)/W(k)}\Tr_{k/W(k)}(b),\\
 \Tr_{F/W(F)}\bigl(\{t-s\}\cdot\res_{F/k}(c_s)\bigr)
 &=\{[t-s]\}\cdot\Tr_{F/W(F)}\res_{F/k}(c_s)\\
 &=\{[t-s]\}\cdot\res_{W(F)/W(k)}\Tr_{k/W(k)}(c_s).
\end{align*}
The projection formula is \cite[V, Example~3.5.3]{Weibel}.
These identities put $x$ in the right side of
\eqref{eq:rational-line-kernel}.
Conversely, restriction preserves the kernels in
\eqref{eq:witt-kernel-notation}. For
$u\in K_1(W(F))_{\Q}$ and $y\in M_{n-1}(F)$, one has
\[
 \res_{\Frac W(F)/W(F)}(u\cdot y)
 =\res_{\Frac W(F)/W(F)}(u)\cdot
   \res_{\Frac W(F)/W(F)}(y)=0.
\]
Every summand on the right therefore belongs to $M_n(F)$.
This proves the equality and \eqref{eq:rational-line-propagation}.
If $M_{n-1}(k)=0$, \eqref{eq:rational-line-kernel} makes
restriction $M_n(k)\to M_n(F)$ surjective. Its injectivity is
\cref{cor:witt-restriction}. For $n=3$, the hypothesis holds
because $M_2(k)=0$ by \cref{lem:low-degrees}.
\end{proof}

\begin{remark}
Retain $k,t,F,n$ from \cref{prop:rational-line-kernel}.
The sum in \eqref{eq:rational-line-kernel} has not been proved
direct. A sufficient additional compatibility is the following:
for every $b\in K_n(k(t))_\Q$,
\[
 \res_{F/k(t)}(b)\in\im\partial_{W(F)}
 \quad\Longrightarrow\quad
 \partial_a(b)\in
 \im\bigl(\partial_{W(k)}:K_n(\Frac W(k))_\Q
                        \to K_{n-1}(k)_\Q\bigr)
 \quad(a\in k).
\]
To check this, fix $b_0\in K_n(k)_\Q$, a finite subset $S\subset k$,
and $c_a\in K_{n-1}(k)_\Q$ for $a\in S$, and suppose
\[
 \res_{W(F)/W(k)}\Tr_{k/W(k)}(b_0)
 +\sum_{a\in S}\{[t-a]\}\cdot
      \res_{W(F)/W(k)}\Tr_{k/W(k)}(c_a)=0.
\]
By the projection formula and localization this is equivalent to
\[
 \res_{F/k(t)}\left(\res_{k(t)/k}(b_0)
       +\sum_{a\in S}\{t-a\}\cdot\res_{k(t)/k}(c_a)\right)
       \in\im\partial_{W(F)}.
\]
The displayed compatibility and \eqref{eq:rational-line-residues}
give $c_a\in\im\partial_{W(k)}$, and hence
$\Tr_{k/W(k)}(c_a)=0$, for every $a\in S$.
The original relation reduces to
$\res_{W(F)/W(k)}\Tr_{k/W(k)}(b_0)=0$.
By \cref{cor:witt-restriction}, $\Tr_{k/W(k)}(b_0)=0$.
Thus every component of the relation is zero.
\Cref{thm:boundary-functoriality} concerns field embeddings;
the displayed compatibility with tame boundaries is an additional
condition.
\end{remark}

\begin{remark}[The degree-three comparison]
\label{rem:classical-degree-three}
The invariance in degree three has a classical antecedent stronger
than the calculation above. Levine proves
\[
 K_3^{\mathrm{ind}}(k)\xrightarrow{\ \sim\ }
 K_3^{\mathrm{ind}}(k(t)),\qquad
 K_3^{\mathrm{ind}}(k)=\coker\bigl(K_3^M(k)\to K_3(k)\bigr),
\]
for every field $k$
\cite[\href{https://numdam.org/item/10.24033/asens.1585.pdf\#page=74}{\S4.2, pp.~327--328}]{Levine}.
Thus $M_3(k)\to M_3(k(t)^\perf)$ is an isomorphism for every
perfect field $k$ of characteristic $p$. Indeed, given
$a\in K_3(k(t)^\perf)_{\Q}$, rational invariance under perfection
gives $a_0\in K_3(k(t))_{\Q}$ restricting to $a$.
Levine's isomorphism gives $b\in K_3(k)_{\Q}$ and
$c\in K_3^M(k(t))_{\Q}$ such that
\[
 a_0=\res_{k(t)/k}(b)+\operatorname{image}(c).
\]
After restriction to $k(t)^\perf$, \cref{lem:milnor-obstruction}
annihilates the last summand under the closed-point Gysin map.
Flat base change therefore gives
\[
 \Tr_{k(t)^\perf/W(k(t)^\perf)}(a)
 =\res_{W(k(t)^\perf)/W(k)}\Tr_{k/W(k)}(b).
\]
This proves surjectivity on $M_3$; injectivity is
\cref{cor:witt-restriction}.
\end{remark}

\begin{lemma}[Finite extensions and fixed subfields]
\label{lem:witt-galois-kernels}
Let $K\subseteq L$ be a finite extension of perfect fields of
characteristic $p$, and let $n\geq0$.
Transfer between the Witt rings induces a homomorphism
$\Tr_{W(L)/W(K)}:M_n(L)\to M_n(K)$, and
\begin{equation}
 M_n(L)=\res_{W(L)/W(K)}M_n(K)
          \oplus\ker\bigl(\Tr_{W(L)/W(K)}:M_n(L)\to M_n(K)\bigr).
 \label{eq:finite-kernel-splitting}
\end{equation}
If $L/K$ is Galois with group $G$, then every $g\in G$ acts
on $M_n(L)$ by the homomorphism induced by $W(g):W(L)\to W(L)$,
and
\begin{equation}
 \res_{W(L)/W(K)}M_n(K)=M_n(L)^G.
 \label{eq:galois-kernel-invariants}
\end{equation}
If $G=\langle\sigma\rangle$ is cyclic of finite order $m\geq1$, then
\begin{equation}
 \ker\bigl(\Tr_{W(L)/W(K)}:M_n(L)\to M_n(K)\bigr)
       =(\sigma-1)M_n(L).
 \label{eq:cyclic-transfer-kernel}
\end{equation}
\end{lemma}
\begin{proof}
By \cref{lem:finite-witt-extension}, $W(L)/W(K)$ is finite
\'etale and free of rank $[L:K]$.
Transfer commutes with localization at $p$, by finite flat base change
\cite[V, \textup{(6.6.3)--(6.6.4)}]{Weibel}.
Hence it sends $M_n(L)$ to $M_n(K)$.
The restriction--transfer identity is
\begin{equation}
 \Tr_{W(L)/W(K)}\res_{W(L)/W(K)}
       =[L:K]\,\mathrm{id}_{M_n(K)}.
 \label{eq:witt-transfer-degree}
\end{equation}
Indeed, $W(L)$ is free of rank $[L:K]$ over $W(K)$, so the
composite exact functor is a direct sum of $[L:K]$ copies of the
identity. For every $x\in M_n(L)$ the decomposition
\begin{align*}
 x={}&\frac{1}{[L:K]}\res_{W(L)/W(K)}\Tr_{W(L)/W(K)}(x)\\
 &+\left(x-\frac{1}{[L:K]}\res_{W(L)/W(K)}\Tr_{W(L)/W(K)}(x)\right)
\end{align*}
has its first summand in the restriction image and its second in the
transfer kernel. If $\res_{W(L)/W(K)}(y)$ is in both, then
$[L:K]y=0$ by \eqref{eq:witt-transfer-degree}, and therefore $y=0$.
This proves \eqref{eq:finite-kernel-splitting}.

Suppose $L/K$ is Galois. The homomorphism
\[
 W(L)\otimes_{W(K)}W(L)\longrightarrow\prod_{g\in G}W(L),
 \qquad a\otimes b\longmapsto(a\,W(g)(b))_{g\in G},
\]
is an isomorphism: modulo $p$ it is the Galois isomorphism
$L\otimes_KL\simeq\prod_{g\in G}L$, and both sides are free
$W(L)$-modules of rank $[L:K]$. A matrix of this homomorphism has
invertible determinant modulo $p$, hence invertible determinant in
$W(L)$. Applying additivity to its bimodule decomposition gives
the usual Galois transfer identity
\cite[IV, Exercise~6.13]{Weibel}:
\begin{equation}
 \res_{W(L)/W(K)}\Tr_{W(L)/W(K)}(x)
       =\sum_{g\in G}g(x)\qquad(x\in M_n(L)).
 \label{eq:witt-galois-norm}
\end{equation}
Every restricted class is fixed by $G$. Conversely, if $x\in M_n(L)^G$,
then
\[
 x=\res_{W(L)/W(K)}\left(\frac{1}{|G|}
                  \Tr_{W(L)/W(K)}(x)\right),
\]
which proves \eqref{eq:galois-kernel-invariants}.

In the cyclic case, \eqref{eq:witt-galois-norm} and the injectivity
of restriction identify the transfer kernel with the kernel of
$1+\sigma+\cdots+\sigma^{m-1}$ on $M_n(L)$.
This operator annihilates $(\sigma-1)M_n(L)$.
For an element $x$ in its kernel, the element
\[
 y=\frac1m\sum_{j=0}^{m-1}j\sigma^j(x)\in M_n(L)
\]
satisfies
\[
 (\sigma-1)y=x-\frac1m\sum_{j=0}^{m-1}\sigma^j(x)=x.
\]
This proves the reverse inclusion in \eqref{eq:cyclic-transfer-kernel}.
\end{proof}

\begin{corollary}[Normalized transfer]
\label{cor:normalized-transfer}
Let $F\subseteq E$ be an algebraic extension of perfect fields
of characteristic $p$, and let $n\geq0$.
Restriction $M_n(F)\to M_n(E)$ has a canonical retraction
determined by
\[
 \res_{W(E)/W(L)}(x)\longmapsto
       \frac{1}{[L:F]}\Tr_{W(L)/W(F)}(x)
 \quad\bigl(F\subseteq L\subseteq E,\ [L:F]<\infty,\ x\in M_n(L)\bigr).
\]
\end{corollary}
\begin{proof}
For finite intermediate fields $F\subseteq L\subseteq L'\subseteq E$,
transitivity of transfer and \eqref{eq:witt-transfer-degree} give
\begin{align*}
 \frac{1}{[L':F]}\Tr_{W(L')/W(F)}\res_{W(L')/W(L)}
 &=\frac{1}{[L':F]}\Tr_{W(L)/W(F)}
          \Tr_{W(L')/W(L)}\res_{W(L')/W(L)}\\
 &=\frac{1}{[L:F]}\Tr_{W(L)/W(F)}.
\end{align*}
Thus these homomorphisms are compatible with the restriction maps.
The finite intermediate fields form a filtered family with union $E$.
By \cref{lem:witt-kernel-union}, the displayed rule defines a unique
homomorphism on $M_n(E)$. Its composite with restriction from
$M_n(F)$ is the identity, by taking $L=F$ in the rule.
\end{proof}

\begin{proposition}[Descent from an algebraic closure]
\label{thm:cyclic-descent}
Let $F$ be a perfect field of characteristic $p$, choose an algebraic
closure $\overline F$, and let $n\geq0$. The following are equivalent.
\begin{enumerate}
\item Restriction induces an isomorphism $M_n(F)\xrightarrow{\sim}M_n(\overline F)$.
\item For every tower $F\subseteq K\subsetneq L\subseteq\overline F$
with $[K:F]<\infty$ and $L/K$ cyclic Galois of prime degree,
$\Tr_{W(L)/W(K)}:M_n(L)\to M_n(K)$ is injective.
\item For every tower in \textup{(ii)}, the group
$\operatorname{Gal}(L/K)$ acts trivially on $M_n(L)$.
\item For every tower in \textup{(ii)} and every generator
$\sigma\in\operatorname{Gal}(L/K)$,
\begin{equation}
 (\sigma-1)K_n(L)_{\Q}
 \subseteq\im\!\left(
 K_{n+1}(\Frac W(L))_{\Q}\xrightarrow{\partial_{W(L)}}K_n(L)_{\Q}
 \right).
 \label{eq:cyclic-boundary-condition}
\end{equation}
Here $\sigma$ acts on $K_n(L)_{\Q}$ by restriction along
$\sigma:L\to L$.
\end{enumerate}
If $M_n(F)=0$, these conditions are equivalent to $M_n(\overline F)=0$.
\end{proposition}
\begin{proof}
The equivalence of \textup{(ii)} and \textup{(iii)} is
\eqref{eq:cyclic-transfer-kernel}.
For a tower as in \textup{(ii)}, functoriality of the closed-point Gysin map gives
\[
 (\sigma-1)\Tr_{L/W(L)}(a)
       =\Tr_{L/W(L)}((\sigma-1)a)\qquad(a\in K_n(L)_{\Q}).
\]
The map $\Tr_{L/W(L)}$ surjects onto $M_n(L)$, and its kernel is
the boundary image in \eqref{eq:cyclic-boundary-condition}, by
\cref{prop:localization}. Thus \textup{(iii)} and \textup{(iv)} are
equivalent.

Assume \textup{(i)}. For $x\in M_n(L)$, choose $y\in M_n(F)$ such that
\[
 \res_{W(\overline F)/W(L)}(x)
       =\res_{W(\overline F)/W(F)}(y).
\]
Injectivity of restriction from $W(L)$ to $W(\overline F)$ gives
$x=\res_{W(L)/W(F)}(y)$. Every $\sigma\in\operatorname{Gal}(L/K)$
fixes $F$, so $\sigma(x)=x$. This proves \textup{(iii)}.

Assume \textup{(iii)} and let $E/F$ be a finite Galois extension
inside $\overline F$, with group $G$.
The action of $G$ on $M_n(E)$ has a normal kernel
\[
 N=\{g\in G:g(x)=x\text{ for every }x\in M_n(E)\}.
\]
Suppose $N\ne G$. Choose an element of $G/N$ of order $r>1$ and
a prime divisor $\ell$ of $r$. Its $(r/\ell)$-th power has order
$\ell$; let $C\subseteq G$ be the inverse image of the subgroup
it generates. Then $N\subsetneq C$, $C/N$ is cyclic of order
$\ell$, and
\[
 F\subseteq E^C\subsetneq E^N\subseteq\overline F,
 \qquad \operatorname{Gal}(E^N/E^C)=C/N,
\]
by the finite Galois correspondence
\cite[\stackstag{09DW}, \stackstag{0BMH}]{Stacks}.
By \eqref{eq:galois-kernel-invariants}, restriction identifies
$M_n(E^N)$ with $M_n(E)^N=M_n(E)$.
This identification respects the $C/N$-actions: an element of $C$
restricts to its corresponding automorphism of $E^N$.
By \textup{(iii)}, $C/N$ acts trivially on $M_n(E^N)$, hence on
$M_n(E)$. This contradicts the definition of $N$ and $C\ne N$.
Therefore $N=G$, and \eqref{eq:galois-kernel-invariants} yields
\[
 M_n(E)=M_n(E)^G=\res_{W(E)/W(F)}M_n(F).
\]
Because $F$ is perfect, every finite extension inside $\overline F$
is separable. Its normal closure is Galois
\cite[\stackstag{0EXM}]{Stacks} and finite: it is generated by the
finitely many roots of the minimal polynomials of a finite set of
field generators. Composita of finite Galois extensions are finite
Galois, since the union of the corresponding finite sets of separable
polynomials has that compositum as its splitting field.
Thus the finite Galois subextensions form a filtered family with
union $\overline F$. For $x\in M_n(\overline F)$,
\cref{lem:witt-kernel-union} gives such an extension $E/F$ and
$x_E\in M_n(E)$ satisfying
$\res_{W(\overline F)/W(E)}(x_E)=x$.
The preceding equality gives $y\in M_n(F)$ with
$x_E=\res_{W(E)/W(F)}(y)$. Therefore
\[
 x=\res_{W(\overline F)/W(E)}\res_{W(E)/W(F)}(y)
   =\res_{W(\overline F)/W(F)}(y),
\]
proving surjectivity in \textup{(i)}.
Its injectivity is \cref{cor:witt-restriction}.
The final assertion follows by substituting $M_n(F)=0$ in
\textup{(i)}.
\end{proof}

\begin{corollary}[Consecutive test rings]
\label{cor:successive-tests}
Fix a prime $p$ and integers $d\geq0$, $n\geq1$.
Choose $t$ transcendental over $\Omega_{p,d}$ and identify
$\Omega_{p,d+1}$ with an algebraic closure of $\Omega_{p,d}(t)$.
Assume $M_{n-1}(\Omega_{p,d})=0$. The composite
\[
 M_n(\Omega_{p,d+1})
 \xrightarrow{\text{normalized transfer}}
 M_n(\Omega_{p,d}(t)^{\perf})
 \xrightarrow{\ \res^{-1}\ }M_n(\Omega_{p,d})
\]
is a retraction of restriction
$M_n(\Omega_{p,d})\to M_n(\Omega_{p,d+1})$.
The first arrow is the map in \cref{cor:normalized-transfer};
the second is the inverse of restriction from $W(\Omega_{p,d})$
to $W(\Omega_{p,d}(t)^{\perf})$.
Restriction between the two kernels is an isomorphism if and only if the
equivalent conditions \textup{(ii)--(iv)} of
\cref{thm:cyclic-descent} hold with
$F=\Omega_{p,d}(t)^{\perf}$ and
$\overline F=\Omega_{p,d+1}$.
In particular, this construction gives a retraction of
$M_3(\Omega_{p,d})\to M_3(\Omega_{p,d+1})$ for every $d\geq0$.
\end{corollary}
\begin{proof}
By \cref{prop:rational-line-kernel} and
\cref{cor:witt-restriction}, the left arrow in the diagram
\[
\begin{tikzcd}[column sep=large,row sep=large]
 M_n(\Omega_{p,d})\arrow[r,hook,"\res"]
      \arrow[d,"\res"',"\sim"]
 &M_n(\Omega_{p,d+1})
       \arrow[dl,"\text{normalized transfer}"]\\
 M_n(\Omega_{p,d}(t)^{\perf})
\end{tikzcd}
\]
is an isomorphism. The diagonal arrow is
\cref{cor:normalized-transfer}, applied to the algebraic extension
$\Omega_{p,d}(t)^{\perf}\subseteq\Omega_{p,d+1}$.
The triangle commutes because normalized transfer is a retraction
of restriction from its base field.
Its diagonal arrow followed by the inverse of the left arrow
is therefore a retraction of the top arrow.
The top arrow is an isomorphism if and only if restriction
\[
 M_n(\Omega_{p,d}(t)^{\perf})\longrightarrow M_n(\Omega_{p,d+1})
\]
is an isomorphism. Apply \cref{thm:cyclic-descent} for the asserted
criterion. For $n=3$, the hypothesis is automatic because
$M_2(\Omega_{p,d})=0$ by \cref{lem:low-degrees}.
\end{proof}

\begin{corollary}[One step in residue transcendence degree]
\label{thm:dimension-step}
Fix a prime $p$ and integers $d\geq0$, $n\geq1$. Assume
\begin{equation}
 M_n(\Omega_{p,d})=M_{n-1}(\Omega_{p,d})=0.
 \label{eq:dimension-step-hypothesis}
\end{equation}
Let $t$ be transcendental over $\Omega_{p,d}$, and identify
$\Omega_{p,d+1}$ with an algebraic closure of
$F=\Omega_{p,d}(t)^{\perf}$.
The following are equivalent.
\begin{enumerate}
\item Rational Gersten injectivity in degree $n$ holds for every
mixed-characteristic henselian DVR $R$ of residue characteristic $p$
with $\trdeg_{\F_p}\kappa(R)\leq d+1$.
\item $M_n(\Omega_{p,d+1})=0$.
\item For every tower $F\subseteq K\subsetneq L\subseteq\Omega_{p,d+1}$
with $[K:F]<\infty$ and $L/K$ cyclic Galois of prime degree,
$\Tr_{W(L)/W(K)}:M_n(L)\to M_n(K)$ is injective.
\item For every tower in \textup{(iii)} and every generator
$\sigma\in\operatorname{Gal}(L/K)$, the inclusion
\eqref{eq:cyclic-boundary-condition} holds.
\end{enumerate}
\end{corollary}
\begin{proof}
By \cref{prop:rational-line-kernel} and
\eqref{eq:dimension-step-hypothesis}, $M_n(F)=0$.
Since $\overline F=\Omega_{p,d+1}$, \cref{thm:cyclic-descent}
gives the equivalence of \textup{(ii)}, \textup{(iii)}, and
\textup{(iv)}. The equivalence of \textup{(i)} and \textup{(ii)}
is \cref{thm:universal-witt-test}.
\end{proof}

\begin{corollary}[A finite induction criterion]
\label{cor:bounded-induction}
Fix a prime $p$ and $N,d\in\N$. The following are equivalent.
\begin{enumerate}
\item For every mixed-characteristic henselian DVR $R$ with
residue characteristic $p$ and $\trdeg_{\F_p}\kappa(R)\leq d$,
rational Gersten injectivity holds in every degree $0\leq n\leq N$.
\item For every pair of integers $3\leq n\leq N$ and $1\leq e<d$,
every tower
\[
 \Omega_{p,e}(t)^{\perf}\subseteq K\subsetneq L
            \subseteq\Omega_{p,e+1},
 \qquad [K:\Omega_{p,e}(t)^{\perf}]<\infty,
\]
in which $L/K$ is cyclic Galois of prime degree, and every generator
$\sigma\in\operatorname{Gal}(L/K)$, one has
\begin{equation}
 (\sigma-1)K_n(L)_{\Q}
 \subseteq\im\!\left(
 K_{n+1}(\Frac W(L))_{\Q}\xrightarrow{\partial_{W(L)}}K_n(L)_{\Q}
 \right).
 \label{eq:bounded-induction-condition}
\end{equation}
For each $e$, $t$ is transcendental over $\Omega_{p,e}$ and
$\Omega_{p,e+1}$ is an algebraic closure of $\Omega_{p,e}(t)$.
\end{enumerate}
Requiring \eqref{eq:bounded-induction-condition} for all $n\geq3$
and all $e\geq1$ is equivalent to rational Gersten injectivity in
every degree for all mixed-characteristic henselian DVRs of
residue characteristic $p$.
\end{corollary}
\begin{proof}
Assume \textup{(i)}. For a tower in \textup{(ii)}, $L$ is perfect
and has transcendence degree $e+1\leq d$ over $\F_p$.
Thus \textup{(i)}, applied to $W(L)$, gives $M_n(L)=0$.
Localization gives
$\im\partial_{W(L)}=K_n(L)_{\Q}$, proving
\eqref{eq:bounded-induction-condition}.

Assume \textup{(ii)}. By \cref{lem:low-degrees}, the assertion
is true if $N\leq2$. By \cref{prop:low-transcendence}, it is true
if $d\leq1$. Suppose $N\geq3$ and $d\geq2$.
We prove by induction on $e$, for $1\leq e\leq d$, that
\[
 M_n(\Omega_{p,e})=0\qquad(0\leq n\leq N).
\]
The case $e=1$ is \cref{prop:low-transcendence}.
Suppose $1\leq e<d$ and the displayed assertion holds for $e$.
For each $n$ with $3\leq n\leq N$, it gives both
$M_n(\Omega_{p,e})=0$ and $M_{n-1}(\Omega_{p,e})=0$.
Hypothesis \textup{(ii)} and \cref{thm:dimension-step} therefore give
$M_n(\Omega_{p,e+1})=0$. The degrees $0,1,2$ are
\cref{lem:low-degrees}, which completes the induction.
Apply \cref{thm:universal-witt-test} in each degree $n\leq N$
at residue bound $d$ to obtain \textup{(i)}.

For the last assertion, the necessity follows as in the first
paragraph. Conversely, assume all the stated inclusions.
The finite assertion gives $M_n(\Omega_{p,d})=0$ for every
$n,d\geq0$. With the compatible choices in \eqref{eq:omega},
\[
 \Omega_{p,\infty}=\bigcup_{d\geq0}\Omega_{p,d}.
\]
Indeed, the coefficients of a polynomial over
$\F_p(t_1,t_2,\ldots)$ involve only finitely many variables,
so every element algebraic over that field is algebraic over
$\F_p(t_1,\ldots,t_d)$ for some $d$.
By \cref{lem:witt-kernel-union},
$M_n(\Omega_{p,\infty})=0$ for every $n\geq0$.
The unbounded form of \cref{thm:universal-witt-test} proves the claim.
\end{proof}

\begin{remark}[The remaining condition]
\label{rem:induction-scope}
The initial dimensions $0,1$ are covered by
\cref{prop:low-transcendence}; the first additional step in this induction is $1\to2$.
For a cyclic extension $L/K$ of prime degree $\ell$, the identity
\[
 \res_{W(L)/W(K)}\Tr_{W(L)/W(K)}
       =1+\sigma+\cdots+\sigma^{\ell-1}
\]
is multiplication by $\ell$ on the invariant subspace and zero
on $(\sigma-1)M_n(L)$. Hence averaging alone does not establish
the required transfer injectivity. The required assertion is precisely
that each difference $(\sigma-1)a$, with $a\in K_n(L)_{\Q}$,
is a localization boundary as in
\eqref{eq:bounded-induction-condition}.

For a fixed $n\geq1$, the vanishing step in
\cref{thm:dimension-step} uses degrees $n-1$ and $n$.
The retraction in \cref{cor:successive-tests} only requires
vanishing in degree $n-1$. Its construction uses the chosen
intermediate field $\Omega_{p,d}(t)^{\perf}$.
The intermediate base fields $K$ range over all finite extensions
of $\Omega_{p,d}(t)^{\perf}$; they are essential in the proof of
\cref{thm:cyclic-descent}.
\end{remark}

\appendix
\section{Existential embeddings and valuation rings}
\label{app:existential}

The argument in this appendix uses the existential embedding theorem
of Anscombe--Dittmann--Jahnke. It gives another proof of the pure
comparisons for DVRs in the main text and applies more generally to
valuation rings which need not be noetherian. The conclusion about
$K$-groups is a formal consequence of that embedding theorem and
\cref{thm:equational-purity}. We recall the logical definitions so
that the hypotheses can be checked without prior model-theoretic
background.
A language specifies sorts, operations, constants, and relations;
a structure interprets these symbols on a set for each sort.
An embedding is injective on each sort, preserves the operations
and constants, and preserves and reflects the relations.
A term is a finite expression in operations, variables, and
constants. An atomic formula is an equality between terms or
a named relation applied to terms. A quantifier-free formula
is a finite Boolean combination of atomic formulas.
Parameters are elements of the structure used as additional
constants.
We use the ring language
$\mathcal L_{\mathrm{ring}}=\{+,-,\cdot,0,1\}$ and the
ordered-group language
$\mathcal L_{\mathrm{oag}}=\{0,+,<\}$.

\begin{definition}
\label{def:existential}
An embedding $A\subseteq B$ of $\mathcal L$-structures is
\emph{existentially closed}, denoted
$A\preceq_{\exists}^{\mathcal L}B$, if, for every finite tuple
$a$ of parameters in $A$, every integer $r\geq0$, and every
quantifier-free $\mathcal L$-formula
$\varphi(X_1,\ldots,X_r,a)$,
\[
 \bigl(B\models\exists X_1\cdots\exists X_r\,
                       \varphi(X_1,\ldots,X_r,a)\bigr)
 \Longrightarrow
 \bigl(A\models\exists X_1\cdots\exists X_r\,
                       \varphi(X_1,\ldots,X_r,a)\bigr).
\]
Each variable and parameter has a specified sort. The notation
$A\models\psi$ means that $\psi$ is true in $A$.
The superscript $\mathcal L$ is omitted when the language is
specified.
\end{definition}

For an inclusion of fields $k\subseteq\ell$, the condition
$k\preceq_{\exists}^{\mathcal L_{\mathrm{ring}}}\ell$
is equivalent to algebraic purity. Indeed, after disjunctive
normal form, a quantifier-free ring formula is a finite
disjunction of systems
\[
 f_i(X)=0\quad(1\leq i\leq s),\qquad
 g_j(X)\ne0\quad(1\leq j\leq t),
\]
where $r,s,t$ are nonnegative integers and
$f_i,g_j\in k[X_1,\ldots,X_r]$.
A solution in $\ell$ satisfies one disjunct. In a field, the
inequations in that disjunct are equivalent to the existence
of $Y$ satisfying $Y\prod_{j=1}^t g_j(X)=1$.
Thus algebraic purity gives a solution in $k$ of that
disjunct. The converse follows by applying existential
closedness to a conjunction of polynomial equations.

For a valued field $(E,v)$, its valuation ring, maximal ideal,
and residue field are
\[
 \mathcal O_v=\{x\in E:v(x)\geq0\},\qquad
 \mathfrak m_v=\{x\in E:v(x)>0\},\qquad
 \kappa(v)=\mathcal O_v/\mathfrak m_v.
\]
The language $\mathcal L_{\mathrm{val}}$ of
\cite[\arxivhtmlref{2305.12145v2}{S2}{\S2}]{ADJ}
has the three sorts
\[
 E,\qquad\kappa(v),\qquad v(E^\times)\cup\{\infty\}.
\]
The first two have the ring language. The last has
$0,+,<,\infty$, where
$\gamma<\infty$ for $\gamma\in v(E^\times)$ and
$\gamma+\infty=\infty+\gamma=\infty$ for
$\gamma\in v(E^\times)\cup\{\infty\}$.
The connecting functions are the valuation
$v:E\to v(E^\times)\cup\{\infty\}$, with $v(0)=\infty$,
and the map $\operatorname{red}:E\to\kappa(v)$ which is
the quotient map on $\mathcal O_v$ and is zero on
$E\setminus\mathcal O_v$.
For $(E,v)\subseteq(F,w)$, one has
\[
 \mathcal O_w=\{x\in F:w(x)\geq0\},\qquad
 \mathcal O_v=E\cap\mathcal O_w\subseteq\mathcal O_w.
\]
It induces the embeddings on the other two sorts.
The notation $(E,v)\preceq_\exists(F,w)$ refers to
$\mathcal L_{\mathrm{val}}$.

\begin{corollary}
\label{cor:valued-purity}
If $(E,v)\preceq_\exists(F,w)$, then, for every spectrum $M$
and every integer $n$, both maps
\[
 \res_{F/E}:K_n(E;M)\longrightarrow K_n(F;M),\qquad
 \res_{\mathcal O_w/\mathcal O_v}:
 K_n(\mathcal O_v;M)\longrightarrow K_n(\mathcal O_w;M)
\]
are pure monomorphisms.
\end{corollary}
\begin{proof}
Fix integers $r,s\in\N$ and polynomials
$f_i\in E[X_1,\ldots,X_r]$, $1\leq i\leq s$.
If $b\in F^r$ satisfies $f_i(b)=0$ for $1\leq i\leq s$,
then
\[
 (F,w)\models\exists X_1\cdots\exists X_r\,
                \bigwedge_{i=1}^s f_i(X)=0.
\]
All coefficients are parameters in the valued-field sort $E$.
Existential closedness gives $a\in E^r$ satisfying
$f_i(a)=0$ for $1\leq i\leq s$. Hence $E\subseteq F$ is
algebraically pure.

Now fix polynomials
$f_i\in\mathcal O_v[X_1,\ldots,X_r]$, $1\leq i\leq s$,
and a solution $b\in\mathcal O_w^r$. Then
$f_i(b)=0$ for every $1\leq i\leq s$ and
$w(b_j)\geq0$ for every $1\leq j\leq r$. Consequently
\[
 (F,w)\models\exists X_1\cdots\exists X_r\,
 \left(\bigwedge_{i=1}^s f_i(X)=0
       \wedge\bigwedge_{j=1}^r v(X_j)\geq0\right).
\]
In this formula $v$ denotes the valuation symbol, interpreted
as $w$ in $(F,w)$. Its coefficients belong to
$\mathcal O_v\subseteq E$.
Existential closedness gives $a\in E^r$ with
$f_i(a)=0$ for $1\leq i\leq s$ and
$v(a_j)\geq0$ for $1\leq j\leq r$.
Thus $a\in\mathcal O_v^r$, proving that
$\mathcal O_v\subseteq\mathcal O_w$ is algebraically pure.
Apply \cref{thm:equational-purity} to these two inclusions.
\end{proof}

A valued field $(E,v)$ of mixed characteristic $(0,p)$ is
\emph{finitely ramified} if
\[
 0<\#\{\gamma\in v(E^\times):0<\gamma\leq v(p)\}<\infty.
\]
This cardinality is its \emph{initial ramification}
\cite[\arxivhtmlref{2305.12145v2}{S2.Thmtheorem1}{Definition~2.1}]{ADJ}.
For a normalized discrete valuation it equals $v(p)$.

\begin{theorem}[Anscombe--Dittmann--Jahnke]
\label{thm:ake}
Let $(E,v)\subseteq(F,w)$ be finitely ramified henselian
valued fields of mixed characteristic $(0,p)$ with the same
initial ramification. Then
\[
 \left.
 \begin{aligned}
  \kappa(v)&\preceq_{\exists}^{\mathcal L_{\mathrm{ring}}}\kappa(w),\\
  v(E^\times)&\preceq_{\exists}^{\mathcal L_{\mathrm{oag}}}w(F^\times)
 \end{aligned}
 \right\}
 \quad\Longrightarrow\quad
 (E,v)\preceq_\exists(F,w).
\]
\end{theorem}

This is \cite[\arxivhtmlref{2305.12145v2}{S5.Thmtheorem14}{Theorem~5.14}]{ADJ}.
Adjoining $\infty$ to the value-group sort gives the same
existential-closedness condition: each variable can be
separated into the cases where it is finite and where it is
$\infty$, reducing a formula to finitely many formulas in
$\mathcal L_{\mathrm{oag}}$.
We use the corrected arXiv proof. As the authors explain in
\cite[\arxivhtmlref{2305.12145v2}{Thmspecialremark1}{Remark~A}]{ADJ},
the published proof of Lemma~4.2 incorrectly treated an arbitrary
$p$-basis as a separating transcendence basis. The revised proof
repairs that step; the statement and numbering of Theorem~5.14
are unchanged.


\begin{corollary}[Purity from residue fields and value groups]
\label{cor:adj-k-theory}
Under the hypotheses of \cref{thm:ake}, assume
\[
 \kappa(v)\preceq_{\exists}^{\mathcal L_{\mathrm{ring}}}\kappa(w),
 \qquad
 v(E^\times)\preceq_{\exists}^{\mathcal L_{\mathrm{oag}}}w(F^\times).
\]
For every spectrum $M$ and every $n\in\Z$, restriction induces
pure monomorphisms
\[
 K_n(\mathcal O_v;M)\longrightarrow K_n(\mathcal O_w;M),
 \qquad K_n(E;M)\longrightarrow K_n(F;M).
\]
Consequently injectivity of
$K_n(\mathcal O_w;M)\to K_n(F;M)$ implies injectivity of
$K_n(\mathcal O_v;M)\to K_n(E;M)$.
\end{corollary}
\begin{proof}
\Cref{thm:ake} gives $(E,v)\preceq_\exists(F,w)$, and
\cref{cor:valued-purity} gives the two pure monomorphisms.
For the last assertion, the commutative square
\[
\begin{tikzcd}[column sep=large,row sep=large]
 K_n(\mathcal O_v;M)\arrow[r]\arrow[d,hook]
   &K_n(E;M)\arrow[d,hook]\\
 K_n(\mathcal O_w;M)\arrow[r]&K_n(F;M)
\end{tikzcd}
\]
sends the upper kernel to the lower kernel; injectivity of the
left vertical map proves the implication.
\end{proof}

For mixed-characteristic DVRs, the completion map and the maps in
\cref{thm:perfect-extension}\textup{(iii)} preserve the uniformizer
and have identical value groups $\Z$. Their residue embeddings
are existentially closed, by the field case of
\cref{def:existential}. \Cref{cor:adj-k-theory} therefore recovers
the purity assertions proved in the main text by regular approximation.

\begin{example}[A valuation of rank two]
\label{ex:rank-two}
Let $k\subseteq\ell$ be perfect fields of characteristic $p$ with
$k\preceq_\exists^{\mathcal L_{\mathrm{ring}}}\ell$.
For $h=k$ or $h=\ell$, consider the subring
\[
 W(h)+t\,\Frac W(h)[[t]]\ \subseteq\ \Frac W(h)((t)).
\]
For a nonzero Laurent series $\sum_{i\geq m}a_it^i$ with
$a_m\ne0$, define
\[
 v_h\!\left(\sum_{i\geq m}a_it^i\right)
       =(m,v_{W(h)}(a_m))\in\Z\times\Z
\]
with the lexicographic order, and define $v_h(0)=\infty$.
Products of leading coefficients and the valuation inequality for
sums show that this is a valuation. Its value group is all of
$\Z\times\Z$, since
\[
 v_h(t^ap^b)=(a,b)\qquad((a,b)\in\Z\times\Z).
\]
Its valuation ring is the displayed subring, and its maximal ideal is
\[
 \mathfrak m_{v_h}=pW(h)+t\Frac W(h)[[t]]=p\mathcal O_{v_h}.
\]
Its residue field is therefore $h$. Since $v_h(p)=(0,1)$,
its initial ramification is one.

These valuation rings are henselian. Indeed, let $f(X)$ be monic
over the displayed ring with a simple residue root in $h$.
Taking constant terms gives a polynomial $f_0(X)\in W(h)[X]$.
Henselianity of $W(h)$ gives $a\in W(h)$ with
$f_0(a)=0$ and $f_0'(a)\in W(h)^\times$, lifting that root.
In $\Frac W(h)[[t]]$ one has
\[
 f(a)\in t\Frac W(h)[[t]],\qquad
 f'(a)\in\Frac W(h)[[t]]^\times.
\]
Hensel's lemma in this complete DVR gives a root
$b\in a+t\Frac W(h)[[t]]$. It belongs to the displayed
valuation ring and has the required residue, proving henselianity.

The ideals generated by $t/p^m$, for $m\in\N$, form a strictly
increasing chain: equality of two successive ideals would imply
$1/p$ belongs to the valuation ring. Thus these rings are not
noetherian. The coefficient embedding induces the identity on
the value groups $\Z\times\Z$ and the prescribed embedding on
residue fields. \Cref{cor:adj-k-theory} gives, for every $n\in\Z$
and every spectrum $M$, pure monomorphisms
\begin{gather*}
 K_n\bigl(W(k)+t\Frac W(k)[[t]];M\bigr)
 \longrightarrow
 K_n\bigl(W(\ell)+t\Frac W(\ell)[[t]];M\bigr),\\
 K_n\bigl(\Frac W(k)((t));M\bigr)
 \longrightarrow K_n\bigl(\Frac W(\ell)((t));M\bigr).
\end{gather*}
The example illustrates \cref{cor:adj-k-theory} for nonnoetherian
valuation rings. Its conclusion also follows from regular
approximation, as the next remark shows.
\end{example}

\begin{remark}[An alternative proof of \cref{ex:rank-two}]
The proof of \cref{prop:regular-comparison} makes
$W(k)\subseteq W(\ell)$ algebraically pure. It also reflects
finite equations over $\Frac W(k)$ with specified coordinates
required to lie in $W(k)$. To verify this, take $r,s\in\N$,
polynomials $f_i\in\Frac W(k)[X_1,\ldots,X_r]$ for
$1\leq i\leq s$, and $b\in\Frac W(\ell)^r$ with
$f_i(b)=0$ for $1\leq i\leq s$. Take a subset
$J\subseteq\{1,\ldots,r\}$ such that $b_j\in W(\ell)$ for
$j\in J$. Choose $N\in\N$ with $p^Nb_j\in W(\ell)$ for
$j\notin J$. Substitute $X_j=p^{-N}Y_j$ for $j\notin J$,
leave $X_j=Y_j$ for $j\in J$, and clear powers of $p$ in the
finitely many coefficients. Algebraic purity solves the resulting
equations in $W(k)$. Reversing the substitutions gives a solution
over $\Frac W(k)$ whose $J$-coordinates belong to $W(k)$.

Now take $r,s\in\N$, polynomials
\[
 f_i\in\bigl(W(k)+t\Frac W(k)[[t]]\bigr)[X_1,\ldots,X_r]
 \quad(1\leq i\leq s),
\]
and a solution $b\in\bigl(W(\ell)+t\Frac W(\ell)[[t]]\bigr)^r$.
Smooth approximation for
$\Frac W(k)[[t]]\subseteq\Frac W(\ell)[[t]]$, as in
\eqref{eq:smooth-approximation}, factors the corresponding map
\[
 \Frac W(k)[[t]][X_1,\ldots,X_r]/(f_1,\ldots,f_s)
 \longrightarrow\Frac W(\ell)[[t]]
\]
through a smooth $\Frac W(k)[[t]]$-algebra $B$.
Denote the images of $X_j$ in $B$ by $x_j$.
Choose $u,v\in\N$ and a finite presentation
\[
 B/tB=\Frac W(k)[Y_1,\ldots,Y_u]/(g_1,\ldots,g_v)
\]
and representatives $q_j(Y)$ of $x_j\bmod tB$, for $1\leq j\leq r$.
The induced point with values in $\Frac W(\ell)$ solves
\[
 g_i(Y)=0\ (1\leq i\leq v),\qquad
 Z_j=q_j(Y)\ (1\leq j\leq r),\qquad Z_j\in W(\ell),
\]
because the constant coefficient of each $b_j$ belongs to $W(\ell)$.
The preceding argument gives a $\Frac W(k)$-point of $B/tB$
at which every $x_j\bmod tB$ belongs to $W(k)$.
Smooth henselian lifting \cite[\stackstag{0H74}]{Stacks} gives
$B\to\Frac W(k)[[t]]$ lifting that point. The images of the
$x_j$ have constant coefficients in $W(k)$ and satisfy every
$f_i=0$; they lie in $W(k)+t\Frac W(k)[[t]]$.
This proves algebraic purity of the valuation-ring embedding.
For the fraction-field embedding, apply
\cref{prop:regular-comparison} to
$\Frac W(k)[[t]]\subseteq\Frac W(\ell)[[t]]$; its residue
embedding is algebraically pure by the first paragraph.
Finally apply \cref{thm:equational-purity}.
\end{remark}

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\end{document}