==== TABLE OF CONTENTS ====
Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory//main.tex
Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory//references.tex
Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory//sections/01-introduction.tex
Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory//sections/02-local.tex
Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory//sections/03-residual-curves.tex
Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory//sections/04-support.tex
Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory//sections/05-curve-fields.tex
Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory//sections/06-good-functions.tex
Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory//sections/07-field.tex
Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory//sections/08-compatibility.tex
===========================
==== BEGIN Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory//main.tex ====
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urlcolor=blue!45!black,pdftitle={Reconstruction of Function Fields from Mod-ell Milnor K-Theory},
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\title{Reconstruction of Function Fields\texorpdfstring{\\}{ }
from Mod-\texorpdfstring{$\ell$}{ell} Milnor \texorpdfstring{$K$}{K}-Theory}
\author{OpenAI}
\date{October 5, 2026}
\begin{document}
\maketitle
\begin{abstract}
We prove a mod-$\ell$ Bogomolov--Pop reconstruction theorem for function
fields of transcendence degree at least two over arbitrary algebraically
closed fields of characteristic different from $\ell$. The groups
$K^{\mathrm M}_1/\ell$ and $K^{\mathrm M}_2/\ell$, together with their full
bilinear product, determine the perfect closure and its constant field.
Every compatible isomorphism of these data is induced by a field isomorphism
up to a single scalar in $\F_\ell^\times$, with only Frobenius ambiguity in
the field isomorphism in positive characteristic.
\end{abstract}
\setcounter{tocdepth}{1}
\tableofcontents
\clearpage
\input{sections/01-introduction}
\input{sections/02-local}
\input{sections/03-residual-curves}
\input{sections/04-support}
\input{sections/05-curve-fields}
\input{sections/06-good-functions}
\input{sections/07-field}
\input{sections/08-compatibility}
\clearpage
\input{references}
\end{document}
==== END Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory//main.tex ====
==== BEGIN Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory//references.tex ====
\begin{thebibliography}{99}
\bibitem{Bogomolov1991}
F.~A.~Bogomolov,
\emph{On two conjectures in birational algebraic geometry},
in Algebraic Geometry and Analytic Geometry (Tokyo, 1990),
ICM-90 Satellite Conference Proceedings, Springer, Tokyo, 1991, pp.~26--52.
\href{https://doi.org/10.1007/978-4-431-68172-4_2}{doi:10.1007/978-4-431-68172-4\_2}.
\bibitem{BT2008}
F.~Bogomolov and Y.~Tschinkel,
\emph{Reconstruction of function fields},
Geom. Funct. Anal. \textbf{18} (2008), no.~2, 400--462.
\href{https://arxiv.org/abs/math/0303075}{arXiv:math/0303075}.
\bibitem{BT2009}
F.~Bogomolov and Y.~Tschinkel,
\emph{Milnor $K_2$ and field homomorphisms},
in Surveys in Differential Geometry, Vol.~XIII,
International Press, Somerville, MA, 2009, pp.~223--244.
\href{https://arxiv.org/abs/0902.4875}{arXiv:0902.4875}.
\bibitem{BT2011}
F.~Bogomolov and Y.~Tschinkel,
\emph{Reconstruction of higher-dimensional function fields},
Mosc. Math. J. \textbf{11} (2011), no.~2, 185--204.
\href{https://doi.org/10.17323/1609-4514-2011-11-2-185-204}{doi:10.17323/1609-4514-2011-11-2-185-204}.
\bibitem{CadoretPirutka2021}
A.~Cadoret and A.~Pirutka,
\emph{Reconstructing function fields from Milnor $K$-theory},
Algebra Number Theory \textbf{15} (2021), no.~9, 2261--2288.
\href{https://doi.org/10.2140/ant.2021.15.2261}{doi:10.2140/ant.2021.15.2261}.
\bibitem{InternalBP}
OpenAI,
\emph{The Bogomolov--Pop reconstruction theorem},
OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/The-Bogomolov-Pop-reconstruction-theorem-September-23-2026/paper.pdf}{OAI:The-Bogomolov-Pop-reconstruction-theorem-September-23-2026},
2026, Sections~4--6.
\bibitem{PopBI2012}
F.~Pop,
\emph{On the birational anabelian program initiated by Bogomolov I},
Invent. Math. \textbf{187} (2012), no.~3, 511--533.
\href{https://doi.org/10.1007/s00222-011-0331-x}{doi:10.1007/s00222-011-0331-x}.
\bibitem{Pop2015}
F.~Pop,
\emph{On the minimized decomposition theory of valuations},
Bull. Math. Soc. Sci. Math. Roumanie \textbf{58(106)} (2015), no.~3, 331--357,
with the appendix \emph{On the nature of base fields} by A.~Topaz.
\url{https://ssmr.ro/bulletin/pdf/58-3/articol_10.pdf}.
\bibitem{Pop2018}
F.~Pop,
\emph{Reconstruction of function fields from their pro-$\ell$ abelian divisorial inertia},
preprint, 2018.
\href{https://arxiv.org/abs/1810.04768}{arXiv:1810.04768}.
\bibitem{Tanaka2024}
H.~Tanaka,
\emph{Boundedness of regular del Pezzo surfaces over imperfect fields},
Manuscripta Math. \textbf{174} (2024), no.~1--2, 355--379.
\href{https://doi.org/10.1007/s00229-023-01517-z}{doi:10.1007/s00229-023-01517-z}.
\bibitem{Stacks}
The Stacks Project Authors,
\emph{The Stacks Project}, 2026.
\url{https://stacks.math.columbia.edu}.
Specific tags are cited in the text.
\bibitem{Topaz2016}
A.~Topaz,
\emph{Reconstructing function fields from rational quotients of mod-$\ell$ Galois groups},
Math. Ann. \textbf{366} (2016), no.~1--2, 337--385.
\href{https://doi.org/10.1007/s00208-015-1327-4}{doi:10.1007/s00208-015-1327-4}.
\bibitem{Topaz2020}
A.~Topaz,
\emph{Recovering function fields from their integral $\ell$-adic cohomology with the Galois action},
preprint, 2019, version~1.
\href{https://arxiv.org/abs/1910.03563v1}{arXiv:1910.03563v1}.
\bibitem{Topaz2021Torelli}
A.~Topaz,
\emph{A Torelli theorem for higher-dimensional function fields},
preprint, 2021, version~3.
\href{https://arxiv.org/abs/1705.01084v3}{arXiv:1705.01084v3}.
\bibitem{Topaz2022}
A.~Topaz,
\emph{Algebraic dependence and Milnor $K$-theory},
preprint, 2022.
\href{https://arxiv.org/abs/2211.14665}{arXiv:2211.14665}.
\end{thebibliography}
==== END Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory//references.tex ====
==== BEGIN Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory//sections/01-introduction.tex ====
\section{Introduction}\label{sec:introduction}
Birational anabelian geometry asks how much of a function field can be
recovered from its Galois-theoretic invariants. In the Bogomolov program,
the field has algebraically closed constants, and the relevant invariants
are much smaller than the full absolute Galois group. The present paper
proves the reconstruction assertion for the first two Milnor $K$-groups
modulo a single prime, with their multiplication. The result includes
the isomorphism statement: it identifies every compatible linear isomorphism,
as well as the ambiguity in the field isomorphism that induces it.
\subsection{The reconstruction theorem}
Fix a prime $\ell$ and put $\Lambda=\F_\ell$. For a field $F$ of
characteristic different from $\ell$, set
\[
V_F=F^\times/(F^\times)^\ell,
\qquad
R_F=\Span_\Lambda\{[x]\otimes[1-x]:x\in F\setminus\{0,1\}\}.
\]
We write $V_F$ additively, so $[xy]=[x]+[y]$. Let
\[
W_F=(V_F\otimes_\Lambda V_F)/R_F,
\qquad m_F:V_F\otimes_\Lambda V_F\longrightarrow W_F.
\]
Thus $V_F=K^{\mathrm M}_1(F)/\ell$ and
$W_F=K^{\mathrm M}_2(F)/\ell$, with $m_F$ the Milnor product.
A $\Lambda$-linear isomorphism $\Theta:V_K\to V_L$ is
\emph{compatible} if
\[
(\Theta\otimes\Theta)(R_K)=R_L.
\]
Equivalently, there is a unique linear isomorphism $\Theta_2:W_K\to W_L$
intertwining the products. Write $\Isom_{\mathrm M}(V_K,V_L)$ for
the set of compatible isomorphisms. Multiplication by
$a\in\Lambda^\times$ acts on this set; the induced degree-two map
is then multiplied by $a^2$.
Let $K/k$ and $L/l$ be finitely generated extensions of algebraically
closed fields. Write $F^i$ for the perfect closure of a field $F$:
it equals $F$ in characteristic zero and
$\bigcup_{r\ge0}F^{1/p^r}$ in characteristic $p>0$. Denote by
$\Isom^i(K,L)$ the field isomorphisms $\alpha:K^i\to L^i$
satisfying $\alpha(k)=l$. In positive characteristic, identify two
such isomorphisms if they differ by postcomposition with
$\Frob_L^n$, $n\in\Z$, where $\Frob_L(z)=z^p$. Denote the resulting
set by $\Isom^i_{\mathrm F}(K,L)$; in characteristic zero take no
quotient, and if the characteristics differ this set is empty.
Purely inseparable extension induces canonical isomorphisms on $V$
and $W$. Indeed, for an extension of exponent at most $r$, inclusion
and the $p^r$-power homomorphism compose to the $p^r$-power map on
each field. On Milnor groups in degrees one and two this composition
is multiplication by $p^r$ and $p^{2r}$, respectively, both invertible
modulo $\ell$. Passing to the union proves the assertion for perfect
closures. Thus $\alpha$ induces a compatible map $\alpha_1:V_K\to V_L$,
and postcomposition by $\Frob_L^n$ multiplies it by $p^n$ in
$\Lambda^\times$. There is consequently a canonical map
\begin{equation}\label{eq:main-map}
\Isom^i_{\mathrm F}(K,L)\longrightarrow
\Isom_{\mathrm M}(V_K,V_L)/\Lambda^\times.
\end{equation}
\begin{theorem}\label{thm:main}
Let $\ell$ be any prime. Let $K/k$ and $L/l$ be finitely generated
extensions of arbitrary algebraically closed fields, with
$\charac k,\charac l\ne\ell$ and
$\trdeg(K/k),\trdeg(L/l)\ge2$. Then \eqref{eq:main-map} is bijective.
In particular, a compatible isomorphism $V_K\to V_L$ forces equality
of the characteristics and of the relative transcendence degrees.
\end{theorem}
Only the two vector spaces and their bilinear product enter this
statement. The proof recovers the valuations and rational-subfield
images needed for reconstruction from this datum. The scalar is one
global element of $\F_\ell^\times$; at $\ell=2$ this ambiguity is trivial.
\subsection{Historical context}
Bogomolov proposed reconstructing higher-dimensional function fields over
algebraically closed constants from the pro-$\ell$ quotient in which
commutators are central \cite{Bogomolov1991}. Bogomolov and Tschinkel
proved reconstruction for surfaces over algebraic closures of finite
fields \cite{BT2008}, and subsequently for higher-dimensional function
fields over those constants \cite{BT2011}. Pop independently completed
reconstruction in this constant-field setting \cite{PopBI2012}.
These results established that a small part of the Galois group can
retain the geometry needed to recover a field. Over more general
algebraically closed constants, Pop also obtained reconstruction from
pro-$\ell$ data endowed with divisorial inertia, in relative
transcendence degree greater than two \cite[Theorem~1.1]{Pop2018}.
The finite-coefficient problem requires additional control of the global
reconstruction. Topaz proved a Milnor-theoretic isomorphism theorem in
relative transcendence degree at least five when the degree-one and
degree-two mod-$\ell$ groups and their product are supplied together
with all rational subgroups \cite[Theorem~B]{Topaz2016}. Those subgroups
are the images of the multiplicative groups of relatively algebraically
closed rational one-variable subfields. Theorem~\ref{thm:main} obtains
the required rational subfields from the product itself, and its
point-value argument applies in every relative dimension at least two.
Algebraic dependence has provided another route from multiplicative
invariants to field structure. Bogomolov and Tschinkel used Milnor
$K$-theory modulo infinitely divisible elements in characteristic zero
\cite{BT2009}. Cadoret and Pirutka reconstructed regular function
fields over perfect constants from the multiplicative quotient by
constants together with algebraic dependence, and derived applications
to integral Milnor $K$-theory \cite{CadoretPirutka2021}. Topaz later
proved reconstruction from rational Milnor $K$-theory in absolute
transcendence degree at least five \cite{Topaz2022}. These results
clarify the role of algebraic dependence, while using invariants
different from the single-prime datum considered here.
Our local inputs are the alternating-pair valuation theory recorded in
\cite[Section~2]{Topaz2020} and Pop's density theorem for minimized
inertia \cite{Pop2015}. We state the precise forms used below.
The bounded-support and incidence method is adapted from
\cite[Sections~4--6]{InternalBP}. The finite-coefficient bounded-support
and incidence arguments are proved here. The support estimate also has an
antecedent in the bounded-genus curve sections and Hurwitz argument of
Bogomolov and Tschinkel \cite[Section~6, Proposition~6.1]{BT2011}.
The proof uses no pro-$\ell$ field-reconstruction theorem as a premise.
\subsection{The proof and its main ingredients}
The argument begins with a compatible map $\Theta$ and its dual on
the compact character spaces
$A_F=\Hom(F^\times,\Lambda)$. Two characters $f,g$ form an
\emph{alternating pair} when
$f(x)g(1-x)=f(1-x)g(x)$ for every $x\ne0,1$. This condition is
visible in the Milnor product and is therefore preserved by $\Theta$.
Established local theory turns alternating subspaces into valuations.
Section~\ref{sec:local} uses it to recover the transcendence degree and
the inertia and decomposition spaces of quasi-divisorial valuations,
and the character spaces of their iterated residue fields. These
valuations may act nontrivially on the constants; their residue
constants can therefore have positive characteristic even when the
original field has characteristic zero.
The next task is to recover curve subfields: relatively algebraically
closed intermediate fields $k\subset E\subset K$ of transcendence
degree one over $k$. Individual
mod-$\ell$ classes record only orders modulo $\ell$, so a divisor can
disappear when its multiplicity is divisible by $\ell$. We control
this loss by passing to residual curves. Section~\ref{sec:residual-curves}
uses Pop's density theorem to recognize residual curves whose constants
are algebraic closures of finite fields, together with all their point
order lines. Section~\ref{sec:support} then constructs a residual
curve of bounded genus for any finite independent list of classes
in $V_L$. It preserves independence and realizes each support degree
on a fixed projective model of $L$ as the number of nonzero point
orders on the residual curve.
For each nonconstant $t\in K$, applying Riemann--Hurwitz to five fixed
members of its pencil bounds the total degree of the divisor support
of every class $\Theta([t-a])$, $a\in k$,
on that model.
For a curve subfield $E$, choose $t\in E\setminus k$. The supporting
prime divisors of the classes $\Theta([t-a])$ form an infinite set of
bounded degree, hence lie in a finite-type family. The valuation
correspondence shows that each class in
$\Theta(V_E)$ has an $\ell$-th-power residue along all but finitely many
of these divisors. Section~\ref{sec:curve-fields} studies the incidence
variety of pairs consisting of a point and a divisor containing it.
Restricting its parameter space to a curve produces a finite extension
of $L$ containing the curve's function field. After passage to this
extension, the classes come from that one-variable parameter field.
The argument then descends them to a curve subfield $P\subset L$
and proves $\Theta(V_E)=V_P$.
Two features are useful specifically for finite coefficients. One
parameter curve works for every class. For each class, a finite change
of parameter space allows normality to extend a generic $\ell$-th root
of a representative across the smooth locus where that representative
is a unit. This
avoids any restriction on the cardinality of the constants. Moreover,
the incidence field is the compositum of $L$ and the parameter field.
This identity permits exact descent, retaining information that a
norm could annihilate modulo $\ell$.
Once curve subfields correspond, Section~\ref{sec:good-functions}
recognizes true divisors, which are trivial on the constants, and a
sufficiently large family of rational
subfields on which every point is detected. A quotient of two local
parameters at a smooth point supplies such a subfield. These quotients
generate the function field, and the construction works already in
dimension two. Matching their point-order lines gives bijections
between the corresponding projective lines of points.
Section~\ref{sec:field} recovers field operations from simultaneous
values of these rational functions. A blowup at a smooth point detects
all coordinates of a finite tuple at once. Additive triples yield
rational expressions, allowing $p$-power roots in characteristic $p$,
for the transported addition law.
Their one-variable slices align all point bijections, outside finite
sets, with a single isomorphism of constant fields. An identity of rational functions
removes the apparent freedom caused by finite exceptional sets in
those slices. The resulting correspondence preserves every algebraic
relation among the chosen generators and hence gives an isomorphism
of perfect fields.
Finally, Section~\ref{sec:compatibility} proves that this isomorphism
induces the original map up to one scalar, and that scalar action of
a field automorphism forces it to be a Frobenius power. Throughout,
the pro-$\ell$ density input concerns individual fields. The given
mod-$\ell$ isomorphism is never lifted to a pro-$\ell$ isomorphism.
\subsection{Conventions}
A function field $F/\kappa$ means a finitely generated field extension.
Its constants $\kappa$ will be algebraically closed. A curve is a
geometrically integral variety of dimension one, and its function field
has a unique smooth projective model over algebraically closed constants.
A field extension is regular if it is separable and its base is
relatively algebraically closed. Points of varieties over an
algebraically closed field mean rational points unless a scheme point
is specified. The phrase ``for general points'' means on some dense
Zariski open subset. We continue to use
$V_F=F^\times/(F^\times)^\ell$ for residue fields of characteristic
$\ell$, as a multiplicative quotient, without a Galois-cohomological
interpretation.
==== END Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory//sections/01-introduction.tex ====
==== BEGIN Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory//sections/02-local.tex ====
\section{Alternating characters and quasi-divisorial valuations}
\label{sec:local}
We first recover valuations from the degree-two relations. The local
theory is most naturally expressed on the dual of the multiplicative
group. Its multiplicative formulation also applies to residue fields
of characteristic~\(\ell\), which will occur in the argument.
For any field~\(F\), put
\[
A_F=\Hom(F^\times,\Lambda)
=\Hom_\Lambda(V_F,\Lambda),\qquad \Lambda=\F_\ell.
\]
Here the definition of~\(V_F\) is used in every characteristic. Give
\(A_F\) the topology of pointwise convergence, with~\(\Lambda\) discrete.
Choosing a basis of~\(V_F\) identifies~\(A_F\) with a product of copies
of~\(\Lambda\). In particular it is compact, and evaluation identifies
\(V_F\) with~\(\Hom_{\mathrm{cont}}(A_F,\Lambda)\): a continuous linear
form on that product depends on only finitely many coordinates.
A pair~\(f,g\in A_F\) is \emph{alternating} if
\begin{equation}
f(x)g(1-x)=f(1-x)g(x)\qquad(x\in F\setminus\{0,1\}).
\label{eq:local-alternating}
\end{equation}
A subset is alternating if every pair of its elements is alternating.
The determinant functional
\[
[x]\otimes[y]\longmapsto f(x)g(y)-f(y)g(x)
\]
annihilates~\(R_F\) exactly when~\(f,g\) are alternating. Thus the
isomorphism~\(\Theta\) in Theorem~\ref{thm:main} induces a continuous
linear isomorphism
\[
\Phi=\Theta^*:A_L\xrightarrow{\ \sim\ }A_K
\]
that preserves alternating pairs in both directions.
We consider valuations up to equivalence and write~\(v\leq w\) when
\(v\) is a coarsening of~\(w\). Write~\(U_v\) and~\(U_v^1\) for its
units and principal units, \(Fv\) for its residue field, and~\(vF\) for
its value group. The \emph{minimized inertia and decomposition spaces}
are
\[
I_v=U_v^\perp\subset D_v=(U_v^1)^\perp\subset A_F,
\]
where perpendiculars refer to evaluation. A subset of~\(A_F\) is
\emph{valuative} if it is contained in some~\(I_v\). Restriction to
units gives a continuous map
\[
\rho_v:D_v\longrightarrow A_{Fv}
\]
with kernel~\(I_v\). Surjectivity will be established for the valuations
we use below.
\subsection{The established local input}
The following form of alternating-pair valuation theory collects
\cite[Fact~2.1, Lemma~2.5, Theorem~2.6, and Corollary~2.7]{Topaz2020}.
We state it for finite coefficients; the field in this statement need
not have characteristic different from~\(\ell\).
\begin{theorem}[Alternating-pair valuation theory]
\label{thm:local-input}
Let~\(F\) be a field and set
\(A_F^{\pm}=\{f\in A_F:f(-1)=0\}\).
\begin{enumerate}
\item Every valuative subset~\(\Sigma\subset A_F\) has a unique coarsest
valuation~\(v_\Sigma\) with~\(\Sigma\subset I_{v_\Sigma}\). In particular,
\(v_\Sigma\) coarsens every valuation whose inertia contains~\(\Sigma\).
For such a valuation~\(w\), this coarsening is obtained by quotienting
\(wF\) by its largest convex subgroup contained in
\(w(\bigcap_{f\in\Sigma}\ker f)\).
\item If~\(S\subset A_F^{\pm}\) is an alternating linear subspace, its
valuative elements form a valuative subspace~\(S_0\) of codimension at
most one. Moreover~\(S\subset D_{v_{S_0}}\).
\item If~\(\Sigma\subset A_F^{\pm}\) is valuative and~\(g\in A_F^{\pm}\)
is alternating with every element of~\(\Sigma\), then
\(g\in D_{v_\Sigma}\).
\end{enumerate}
\end{theorem}
The coefficient convention in \cite[Section~2.1]{Topaz2020} explicitly
allows~\(\Z/\ell\); its fraction field is then~\(\F_\ell\), so its
character space is exactly~\(A_F\). The case~\(\ell=2\) is included
in the local theorem. For the fields over algebraically closed
constants considered here, every character kills the constants and
hence~\(-1\), so~\(A_F^{\pm}=A_F\).
\subsection{Value groups and residue characters}
For the remainder of this section, let~\(F/\kappa\) be a finitely
generated extension of an algebraically closed field, and put
\(s=\trdeg(F/\kappa)\). No restriction is imposed on its characteristic.
We first record the valuation facts that turn the local theorem into
an intrinsic description of codimension-one valuations.
\begin{lemma}
\label{lem:local-abhyankar}
For every valuation~\(v\) of~\(F\), the field~\(\kappa v\) is
algebraically closed, the group~\(vF/v\kappa\) is torsion free, and
\begin{equation}
\rrank(vF/v\kappa)+\trdeg(Fv/\kappa v)\leq s.
\label{eq:local-abhyankar}
\end{equation}
Here~\(\rrank(G)=\dim_\Q(G\otimes_\Z\Q)\). If equality holds, then
\(Fv/\kappa v\) is finitely generated and~\(vF/v\kappa\) is a finitely
generated free abelian group. Equality passes to every coarsening
and to the induced residue valuation. Conversely, equality for a
valuation and an induced residue valuation gives equality for their
composition.
\end{lemma}
\begin{proof}
The value group of an algebraically closed field is divisible, and its
residue field is algebraically closed. If~\(n\gamma\in v\kappa\) for
\(\gamma\in vF\), choose~\(\delta\in v\kappa\) with
\(n\delta=n\gamma\). Ordered abelian groups are torsion free, so
\(\gamma=\delta\). This proves torsion-freeness of the relative group.
Choose elements~\(x_1,\ldots,x_a\in F^\times\) whose values are
rationally independent modulo~\(v\kappa\), and units
\(y_1,\ldots,y_b\) whose residues are algebraically independent
over~\(\kappa v\). These elements are algebraically independent
over~\(\kappa\). Indeed, in a putative polynomial relation, distinct
monomials in the~\(x_i\) have distinct values modulo~\(v\kappa\).
Within a coefficient polynomial in the~\(y_j\), its terms of least
value cannot cancel, by algebraic independence of their residues after
rescaling by a constant. The nonzero summands obtained by grouping
according to monomials in the~\(x_i\) have distinct values, so their
sum cannot vanish. This proves
\eqref{eq:local-abhyankar} by taking maximal such families.
Suppose equality holds and choose these families with~\(a+b=s\).
Then~\(F\) is finite over
\(F'=\kappa(x_1,\ldots,x_a,y_1,\ldots,y_b)\). The same least-value
calculation gives
\[
vF'/v\kappa\cong\Z^a,\qquad
F'v=\kappa v(\bar y_1,\ldots,\bar y_b).
\]
The fundamental inequality for the finite extension~\(F/F'\) shows
that~\([vF:vF']\) and~\([Fv:F'v]\) are finite. Thus the residue
extension is finitely generated, while the relative value group is
finitely generated and torsion free, hence free.
For the assertion about composition, write~\(v=w\circ u\), meaning
that~\(u\) is a coarsening and~\(w\) is the induced valuation on~\(Fu\).
The value groups fit into the exact sequence
\[
0\longrightarrow w(Fu)/w(\kappa u)
\longrightarrow vF/v\kappa
\longrightarrow uF/u\kappa\longrightarrow0.
\]
Apply~\eqref{eq:local-abhyankar} first to~\(u\), and then to~\(w\)
on~\(Fu/\kappa u\). The sum of the two inequalities is precisely the
inequality for~\(v\). Consequently equality for~\(v\) forces equality
in both, and equality in both forces equality for~\(v\). This use of
the inequality on~\(Fu/\kappa u\) is legitimate even before finite
generation is known: the algebraic-independence argument proving it
applies to arbitrary field extensions of finite transcendence degree.
\end{proof}
\begin{definition}
A \emph{quasi-divisorial valuation}, or \emph{quasi-prime divisor}, of
\(F/\kappa\) is a valuation minimal under coarsening among those with
\[
vF/v\kappa\cong\Z,\qquad \trdeg(Fv/\kappa v)=s-1.
\]
A quasi-prime divisor trivial on~\(\kappa\) is called a \emph{prime
divisor}, or a \emph{true prime divisor}.
A \emph{quasi-prime \(r\)-divisor} is a composition of~\(r\)
successive quasi-prime divisors on the resulting residue fields.
\end{definition}
Lemma~\ref{lem:local-abhyankar} shows that for a quasi-prime
\(r\)-divisor~\(v\), the residue field is a function field of
transcendence degree~\(s-r\) over~\(\kappa v\), and
\begin{equation}
vF/v\kappa\cong\Z^r.
\label{eq:local-flag-rank}
\end{equation}
The isomorphism concerns abstract abelian groups; no ordering of
\(\Z^r\) is specified.
\begin{lemma}
\label{lem:residue-duality}
Suppose~\(vF/v\kappa\) is free abelian of finite rank. Then restriction
to units induces a canonical topological isomorphism
\begin{equation}
D_v/I_v\xrightarrow{\ \sim\ }A_{Fv}.
\label{eq:local-residue-duality}
\end{equation}
Two elements of~\(D_v\) are alternating if and only if their images
in~\(A_{Fv}\) are alternating. If~\(w\) is a valuation of~\(Fv\), then
\begin{equation}
I_{w\circ v}=\rho_v^{-1}(I_w),\qquad
D_{w\circ v}=\rho_v^{-1}(D_w),
\label{eq:local-composition}
\end{equation}
where the inverse images are taken inside~\(D_v\).
\end{lemma}
\begin{proof}
There is an exact sequence of abelian groups
\[
1\longrightarrow (Fv)^\times/(\kappa v)^\times
\longrightarrow F^\times/(\kappa^\times U_v^1)
\longrightarrow vF/v\kappa\longrightarrow0.
\]
For the kernel assertion, rescale by a constant an element whose value
lies in~\(v\kappa\), and then take its residue. The last group is free,
so the sequence splits as a sequence of abstract groups. Dualizing
into~\(\Lambda\) is consequently exact. Characters kill the
algebraically closed constants on both sides, giving the asserted
surjection~\(\rho_v\) with kernel~\(I_v\). Its induced bijection is a
homeomorphism because its source is compact and its target Hausdorff.
For alternation, if~\(v(x)>0\), then~\(1-x\in U_v^1\), and the
identity~\eqref{eq:local-alternating} holds for any pair in~\(D_v\).
The case~\(v(1-x)>0\) is the same. If~\(v(x)<0\), then
\((1-x)/(-x)\in U_v^1\), so every character in~\(D_v\) takes the same
value on~\(x\) and~\(1-x\). The remaining case is
\(v(x)=v(1-x)=0\), when the identity is exactly its residue-field
counterpart. Conversely, lift any residue element different from
zero and one to test the residue identity.
Finally~\(U_v^1\) is contained in both~\(U_{w\circ v}\) and
\(U_{w\circ v}^1\). Under reduction of~\(v\)-units, these two groups
map respectively onto~\(U_w\) and~\(U_w^1\). Their annihilators
therefore give~\eqref{eq:local-composition}.
\end{proof}
We will also use the following consequence of valuation approximation.
It explains why intersections of decomposition spaces yield inertia.
\begin{lemma}
\label{lem:local-approximation}
Let~\(v_1,v_2\) be incomparable valuations of a field, and let~\(u\)
be their finest common coarsening. Then
\[
U_{v_1}^1U_{v_2}^1=U_u,
\qquad D_{v_1}\cap D_{v_2}=I_u.
\]
\end{lemma}
\begin{proof}
The induced valuations~\(w_1,w_2\) on the residue field~\(Fu\) are
independent. For~\(a\in(Fu)^\times\), the approximation theorem gives
\(b\in(Fu)^\times\) with
\[
w_1(b-1)>0,\qquad w_2(b-a)>w_2(a).
\]
Thus~\(b\in U_{w_1}^1\) and~\(a/b\in U_{w_2}^1\), proving
\(U_{w_1}^1U_{w_2}^1=(Fu)^\times\). Reduction maps~\(U_{v_i}^1\)
onto~\(U_{w_i}^1\), and both groups~\(U_{v_i}^1\) contain~\(U_u^1\).
Lifting the product identity gives the first equality; taking
annihilators gives the second.
\end{proof}
\subsection{Recovering dimension and quasi-divisors}
The next argument follows the alternating-space method of
\cite[Fact~3.2 and Theorem~3.3]{Topaz2020}. We give the details so that
the argument remains available in characteristic~\(\ell\).
\begin{proposition}
\label{prop:local-dimension}
The maximum dimension of an alternating subspace of~\(A_F\) is~\(s\).
If~\(S\subset A_F\) is alternating of dimension~\(s\), and~\(v\) is
the valuation associated to its valuative part~\(S_0\), then
\(S_0=I_v\) and equality holds in~\eqref{eq:local-abhyankar} for~\(v\).
\end{proposition}
\begin{proof}
For any torsion-free abelian group~\(G\) of finite rational rank,
\[
\dim_\Lambda(G/\ell G)\leq\rrank(G).
\]
Indeed, representatives of linearly independent classes modulo~\(\ell\)
are rationally independent: an integral relation can be divided by
the greatest common divisor of its coefficients, using torsion-freeness,
and the resulting relation has a coefficient nonzero modulo~\(\ell\).
Since characters of~\(vF\) kill its divisible subgroup~\(v\kappa\),
this proves
\[
\dim I_v\leq\rrank(vF/v\kappa).
\]
Now let~\(S\) be alternating. By Theorem~\ref{thm:local-input}, its
valuative part~\(S_0\) is a subspace of codimension at most one and,
for its associated valuation~\(v\),
\(S_0\subset I_v\) and~\(S\subset D_v\). Moreover
\(S\cap I_v=S_0\), since every element of~\(I_v\) is valuative.
If~\(S=S_0\), then
\[
\dim S\leq\dim I_v\leq\rrank(vF/v\kappa)\leq s.
\]
If~\(S\ne S_0\), an element of~\(S\setminus S_0\) has nonzero
image under~\(\rho_v\), whose kernel is always~\(I_v\).
Thus~\(A_{Fv}\ne0\). Since~\(\kappa v\) is algebraically closed,
this implies~\(\trdeg(Fv/\kappa v)\geq1\), and hence
\[
\dim S=\dim S_0+1
\leq\rrank(vF/v\kappa)+\trdeg(Fv/\kappa v)\leq s.
\]
If~\(\dim S=s\), equality throughout the applicable chain gives
\(S_0=I_v\) and equality in~\eqref{eq:local-abhyankar}.
To attain the bound, choose a transcendence basis of~\(F/\kappa\),
take a full coordinate flag on its rational function field, and prolong
the associated valuation to~\(F\). It is trivial on~\(\kappa\), and
its value group is a finite-index extension of~\(\Z^s\), hence is
abstractly~\(\Z^s\). Its inertia has dimension~\(s\) and is
alternating by Lemma~\ref{lem:residue-duality}. For~\(s=0\), the
field is~\(\kappa\), its character space is zero, and the assertion
has the same interpretation.
\end{proof}
We can now identify the one-dimensional inertia spaces solely through
maximal alternating subspaces. Once inertia is identified, alternation
with it identifies decomposition as well.
\begin{proposition}
\label{prop:quasi-recovery}
Assume~\(s\geq2\). A line~\(H\subset A_F\) is the inertia of a
quasi-divisorial valuation if and only if
\begin{equation}
H=S_1\cap S_2
\quad\text{for alternating subspaces }S_1,S_2\subset A_F
\text{ of dimension }s.
\label{eq:local-inertia-intersection}
\end{equation}
The quasi-divisorial valuation~\(v\) with~\(I_v=H\) is unique. For any nonzero
\(h\in H\), its decomposition space is
\begin{equation}
D_v=\{g\in A_F:(h,g)\text{ is alternating}\}.
\label{eq:local-decomposition-recognition}
\end{equation}
\end{proposition}
\begin{proof}
Suppose first that~\(v\) is quasi-divisorial. On~\(Fv/\kappa v\)
choose two independent full discrete flag valuations~\(w_1,w_2\),
trivial on~\(\kappa v\). To obtain them, start with coordinate flags
on a rational subfield of transcendence degree~\(s-1\), with distinct
first prime divisors, and prolong to the finite extension~\(Fv\).
Independence survives prolongation. Indeed, a valuation on a finite
field extension that restricts trivially to the smaller field is
trivial, since its value group is then finite and hence zero. Thus a
common nontrivial coarsening of~\(w_1,w_2\) would restrict to a common
nontrivial coarsening of the original independent flag valuations,
which is impossible.
The inertia spaces~\(I_{w_1},I_{w_2}\) have zero intersection by
approximation. Set~\(v_i=w_i\circ v\) and~\(S_i=I_{v_i}\).
Each~\(v_i\) has relative value group~\(\Z^s\), so its inertia is
alternating of dimension~\(s\). By~\eqref{eq:local-composition}, the
intersection~\(S_1\cap S_2\) is~\(I_v\).
Conversely, suppose~\eqref{eq:local-inertia-intersection} holds.
Let~\(S_{i,0}\) be the valuative part of~\(S_i\), with associated
valuation~\(v_i\). Proposition~\ref{prop:local-dimension} gives
\[
S_{i,0}=I_{v_i},\qquad S_i\subset D_{v_i},
\qquad \dim I_{v_i}\geq s-1\geq1,
\]
and~\(v_i\) has equality in~\eqref{eq:local-abhyankar}.
We first show that~\(H\) is valuative. If, for example,
\(v_1\leq v_2\), then
\[
0\ne I_{v_1}\subset S_1\cap S_2=H,
\]
so~\(H=I_{v_1}\). The other comparable case is symmetric. If the
valuations are incomparable, Lemma~\ref{lem:local-approximation}
puts~\(H\) inside the inertia of their finest common coarsening.
In either case every element of~\(H\) is valuative. Because it lies
in each~\(S_i\), it then lies in each~\(S_{i,0}\), giving
\[
H=I_{v_1}\cap I_{v_2}.
\]
Let~\(u=v_H\). By Theorem~\ref{thm:local-input}, it coarsens each
\(v_i\), and therefore
\[
H\subset I_u\subset I_{v_1}\cap I_{v_2}=H.
\]
Equality in~\eqref{eq:local-abhyankar} passes to~\(u\).
Its relative value group is free, so~\(\dim I_u=1\) implies
\(uF/u\kappa\cong\Z\) and~\(\trdeg(Fu/\kappa u)=s-1\).
If a coarsening~\(u'\leq u\) still has relative rank one, its
one-dimensional inertia is contained in~\(I_u\), hence equals~\(H\).
The defining coarseness of~\(v_H=u\) then gives~\(u\leq u'\), so
\(u=u'\). Thus~\(u\) is quasi-divisorial.
For uniqueness, if~\(v\) is any quasi-divisor with~\(I_v=H\), its
associated coarsening~\(v_H\) has the same inertia. Equality in
\eqref{eq:local-abhyankar} passes to that coarsening, so it too has
relative rank one and residue transcendence degree~\(s-1\).
Minimality of~\(v\) forces~\(v=v_H\).
Finally, an element of~\(D_v\) is alternating with~\(h\) by
Lemma~\ref{lem:residue-duality}, since~\(h\) has zero residue image.
Conversely, an element alternating with~\(h\) belongs to
\(D_{v_H}=D_v\) by Theorem~\ref{thm:local-input}.
\end{proof}
\begin{lemma}
\label{lem:local-flag-uniqueness}
If~\(v\) is a quasi-prime~\(r\)-divisor, with~\(r\geq1\), then its
value group has no nonzero~\(\ell\)-divisible convex subgroup.
Moreover~\(v=v_{I_v}\); in particular its inertia determines the
valuation.
\end{lemma}
\begin{proof}
First suppose~\(r=1\). An~\(\ell\)-divisible convex subgroup~\(C\)
of~\(vF\) maps to zero in~\(vF/v\kappa\cong\Z\), and hence lies
in~\(v\kappa\). Coarsening by~\(C\) therefore leaves the relative
value group unchanged. Lemma~\ref{lem:local-abhyankar} shows that the
residue transcendence degree remains~\(s-1\), so minimality
of~\(v\) forces~\(C=0\).
For a composite flag, the value group of its final quasi-prime step
is a nonzero convex subgroup~\(B\subset vF\) with the property just
proved. If~\(C\subset vF\) were a nonzero~\(\ell\)-divisible convex
subgroup, then~\(B\cap C\ne0\), since convex subgroups are linearly
ordered by inclusion. This intersection is~\(\ell\)-divisible:
division by~\(\ell\) in~\(C\) stays in~\(B\) by convexity. It is
also convex in~\(B\), a contradiction.
Put~\(\Gamma=vF\). Since~\(\Gamma/v\kappa\cong\Z^r\) and
\(v\kappa\) is divisible, the common kernel in~\(\Gamma\) of all
characters in~\(I_v\) is
\[
v\kappa+\ell\Gamma=\ell\Gamma.
\]
Every convex subgroup contained in~\(\ell\Gamma\) is itself
\(\ell\)-divisible: an~\(\ell\)-th part exists in~\(\Gamma\), and
convexity places it in the subgroup. The first assertion therefore
shows that the largest such convex subgroup is zero. The
coarsening description in Theorem~\ref{thm:local-input} now gives
\(v=v_{I_v}\).
\end{proof}
\begin{corollary}
\label{cor:flag-matching}
For the fields~\(K/k,L/l\) of Theorem~\ref{thm:main}, a compatible
isomorphism~\(\Theta\) forces
\(\trdeg(K/k)=\trdeg(L/l)=d\), say. Its dual~\(\Phi\) matches the
inertia--decomposition pairs of quasi-prime divisors, and, successively,
the pairs of quasi-prime~\(r\)-divisors for~\(1\leq r\leq d-1\).
For a matched pair of valuations~\(v\) on~\(K\) and~\(w\) on~\(L\),
it induces a topological linear isomorphism
\[
A_{Lw}\xrightarrow{\ \sim\ }A_{Kv}
\]
preserving alternating pairs in both directions.
\end{corollary}
\begin{proof}
The maximum alternating dimension is preserved by~\(\Phi\), giving
equality of transcendence degrees by
Proposition~\ref{prop:local-dimension}. Proposition~\ref{prop:quasi-recovery}
then identifies corresponding quasi-prime inertia and decomposition
spaces. Quotienting by inertia and using
Lemma~\ref{lem:residue-duality} gives the indicated isomorphism on
residue characters, with its alternating relation.
As long as the residual transcendence degree is at least two, apply
Proposition~\ref{prop:quasi-recovery} again there. The inverse-image
identities~\eqref{eq:local-composition} identify the resulting pairs
with those of compositions on the original fields.
Lemma~\ref{lem:local-flag-uniqueness} shows that each such inertia
determines its composite valuation, so this matching is independent
of any choice of a presentation as a flag. At each step
Lemma~\ref{lem:local-abhyankar} supplies finitely generated residue
fields over algebraically closed residue constants. The local
results apply even if one of these fields has characteristic~\(\ell\).
Iteration ends after~\(d-1\) steps, with function fields of curves.
\end{proof}
==== END Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory//sections/02-local.tex ====
==== BEGIN Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory//sections/03-residual-curves.tex ====
\section{Recognizing points on residual curves}\label{sec:residual-curves}
Corollary~\ref{cor:flag-matching} identifies the residue character spaces at
quasi-prime \((d-1)\)-divisors. These residue fields have transcendence
degree one, but their point valuations are not yet distinguished inside
their character spaces. We now recognize those terminal residues whose
constant fields are algebraic closures of finite fields. For these
residues we also recognize every point inertia line. The construction
uses Pop's density theorem for an individual field, followed by reduction
of its characters modulo \(\ell\).
Fix a function field \(F/\kappa\) over an algebraically closed field
of characteristic different from \(\ell\), of transcendence degree
\(d\geq 2\), and let
\(v\) be a quasi-prime \((d-1)\)-divisor. Put
\[
F_0=Fv,\qquad \kappa_0=\kappa v.
\]
By Lemma~\ref{lem:residue-duality}, restriction to units induces a
surjective map \(\rho_v:D_v\to A_{F_0}\) with kernel \(I_v\).
Define
\begin{equation}\label{eq:residual-inertia-set}
S_F^{\mathrm{in}}
=\overline{\bigcup_{w\ {\rm quasi\text{-}prime\ divisor}} I_w}
\subset A_F,
\qquad
J_v=\rho_v(S_F^{\mathrm{in}}\cap D_v)\subset A_{F_0}.
\end{equation}
Here the bar denotes the closure of the union, rather than the subgroup
generated by that union. In particular \(J_v\) is a set of characters,
not in general a subspace. It is stable under multiplication by
\(\Lambda\), so it makes sense to consider the family of lines contained
in \(J_v\). Both sets in \eqref{eq:residual-inertia-set}, and this family
of lines, are preserved by the isomorphisms already constructed in
Section~\ref{sec:local}.
\subsection{The density input and reduction of coefficients}
A valuation \(c\) of a function field \(B/b\), with \(b\) algebraically
closed, is a
\emph{constant reduction} if
\(\trdeg(Bc/bc)=\trdeg(B/b)\).
The composition of a constant reduction with a prime divisor of
\(Bc/bc\) is called a \emph{c.r.\ quasi-prime divisor}.
The trivial constant reduction is allowed, so ordinary prime divisors
belong to this class.
For a constant reduction, Lemma~\ref{lem:local-abhyankar} gives
\(cB=cb\): the relative value group is torsion free of rational rank
zero. After composition with a prime divisor, the relative value
group is therefore \(\Z\). The final discrete step is an innermost
convex subgroup of the composed value group. Every proper coarsening
kills this step and has all its remaining values supplied by
constants. This proves the minimality required of a quasi-prime
divisor and justifies the terminology.
For the density statement, temporarily use \(\ell\)-adic characters:
\[
\widehat A_E=\Hom(E^\times,\Z_\ell),\qquad
\widehat I_a=\Hom(E^\times/U_a,\Z_\ell),\qquad
\widehat D_a=\Hom(E^\times/U_a^1,\Z_\ell).
\]
These groups have the topology of pointwise convergence. Restriction to
units and passage to residue give a map
\(\widehat\rho_v:\widehat D_v\to\widehat A_{F_0}\).
Define \(\widehat S_F^{\mathrm{in}}\) and \(\widehat J_v\) by the same
formulas as in \eqref{eq:residual-inertia-set}, with hats on the character
groups and on \(\rho_v\).
\begin{theorem}[Pop's residual density input]\label{thm:density-input}
Let \(F/\kappa\) be a function field over an algebraically closed field
of characteristic different from \(\ell\). Suppose that a valuation
\(v\) of \(F\), with \(Fv/\kappa v\) a function field, has the following
properties:
\begin{enumerate}
\item \(vF\) has no nonzero \(\ell\)-divisible convex subgroup;
\item there is a subfield \(\kappa\subset\kappa_1\subset F\) such that
\[
\kappa_1v=\kappa v,\qquad
\trdeg(F/\kappa_1)=\trdeg(Fv/\kappa_1v)=1.
\]
\end{enumerate}
Then \(\widehat J_v\) contains the full minimized inertia group
\(\widehat I_u\) of every c.r.\ quasi-prime divisor \(u\) of
\(Fv/\kappa v\).
\end{theorem}
This is the consequence of \cite[Theorem~1.2 and the proof of
Theorem~4.2, p.~351, item~2]{Pop2015} with the notation introduced on
p.~348. The auxiliary field \(\kappa_1\) is not required to be
algebraically closed. The minimized interpretation in residue
characteristic \(\ell\) is explained in
\cite[Remark~4.1 and the appendix, Section~A]{Pop2015}.
We verify the hypotheses for our terminal valuation \(v\).
Lemma~\ref{lem:local-flag-uniqueness} shows that \(vF\) has no nonzero
\(\ell\)-divisible convex subgroup. It remains to construct the
auxiliary field \(\kappa_1\).
Choose elements \(x_1,\ldots,x_{d-1}\in F^\times\) whose values are
rationally independent modulo \(v\kappa\), and set
\(\kappa_1=\kappa(x_1,\ldots,x_{d-1})\).
Distinct monomials in the \(x_i\) have different values modulo
\(v\kappa\). Thus every nonzero polynomial in these elements has a unique
term of least value, proving algebraic independence over \(\kappa\).
If a quotient of two such polynomials has value zero, their least terms
have the same monomial. Its residue is the residue of the quotient of
their coefficients, and consequently belongs to \(\kappa v\).
This proves \(\kappa_1v=\kappa v\); the two transcendence degrees required
by Theorem~\ref{thm:density-input} are now one.
\begin{lemma}\label{lem:residual-density-mod-ell}
For a quasi-prime \((d-1)\)-divisor \(v\) of \(F/\kappa\), the set \(J_v\)
contains the inertia line \(I_u\subset A_{F_0}\) of every c.r.\
quasi-prime divisor \(u\) of \(F_0/\kappa_0\).
\end{lemma}
\begin{proof}
Reduction of character values gives continuous maps
\[
r_E:\widehat A_E\longrightarrow A_E.
\]
They carry \(\widehat I_a\) into \(I_a\), carry \(\widehat D_a\) into
\(D_a\), and commute with restriction to units. In particular,
\[
r_F(\widehat S_F^{\mathrm{in}})\subset S_F^{\mathrm{in}},
\qquad
\rho_v r_F=r_{F_0}\widehat\rho_v
\quad\hbox{on }\widehat D_v.
\]
The first inclusion follows from continuity and the definition as a
closure of a union.
Fix a c.r.\ quasi-prime divisor \(u\) of \(F_0/\kappa_0\). Characters with
either coefficient ring annihilate the divisible group \(u\kappa_0\).
Since \(uF_0/u\kappa_0\cong\Z\), reduction
\(\widehat I_u\to I_u\) is surjective. Given \(\chi\in I_u\), choose a
lift \(\widehat\chi\in\widehat I_u\).
Theorem~\ref{thm:density-input} supplies
\(\widehat\eta\in\widehat S_F^{\mathrm{in}}\cap\widehat D_v\) with
\(\widehat\rho_v(\widehat\eta)=\widehat\chi\).
Then \(r_F(\widehat\eta)\) belongs to \(S_F^{\mathrm{in}}\cap D_v\) and
has residual image \(\chi\). Hence \(\chi\in J_v\).
\end{proof}
The argument needs only the displayed inclusion between the two closed
unions. It does not assert that reduction commutes with their
intersection with decomposition groups, and it makes no use of a lift
of \(\Theta\).
\subsection{Valuative characters and the relation among point orders}
The density input gives a lower bound for \(J_v\). To control its other
elements, we use the fact that taking this closed union introduces no
nonvaluative characters.
\begin{lemma}\label{lem:residual-valuative}
Every element of \(S_F^{\mathrm{in}}\) is valuative. Every element of
\(J_v\) is valuative as a character of \(F_0\).
\end{lemma}
\begin{proof}
Let \(\mathcal V(F)\) be the space of valuation rings of \(F\), with its
compact patch topology. In
\(A_F\times\mathcal V(F)\), the condition that a character \(\chi\)
annihilates the unit group of a valuation ring \(O\) is closed.
Indeed, for each \(x\in F^\times\), failure of this condition is
witnessed by the open conditions
\[
x,x^{-1}\in O,\qquad \chi(x)\ne0.
\]
The projection of the closed incidence set to \(A_F\) is compact,
hence closed. This projection is precisely the set of valuative
characters, which contains every \(I_w\) and therefore contains
\(S_F^{\mathrm{in}}\).
Now let \(\chi\in I_u\cap D_v\) for some valuation \(u\).
If \(u\leq v\), then \(\chi\in I_v\) and its residual image is zero.
If \(v\leq u\), then restriction to \(U_v\) shows that
\(\rho_v(\chi)\) annihilates the units of the induced valuation \(u/v\)
on \(F_0\).
In the remaining case let \(a\) be the finest common coarsening of
\(u\) and \(v\). Approximation for the independent induced valuations
on \(Fa\) gives
\[
U_a=U_u U_v^1.
\]
For example, a prescribed nonzero residue can be written as an
induced \(u\)-unit times an induced principal \(v\)-unit by choosing
an element close to \(1\) at the first valuation and close to that
residue at the second. Lifting gives the displayed identity; the
remaining factor in \(U_a^1\) is already a unit for both refinements.
Thus \(\chi\) annihilates \(U_a\), and
\(\chi\in I_a\subset I_v\). Its residual image is again zero.
\end{proof}
We next record how the points of a complete curve appear in its
character space, in the finite-coefficient form of Topaz's
calculation in \cite[appendix, Lemma~5, pp.~354--355]{Pop2015}.
This description will also be used for curve subfields later.
\begin{lemma}[Point orders]\label{lem:curve-orders}
Let \(B/b\) be a function field of transcendence degree one over an
algebraically closed field, and let \(C/b\) be its smooth projective
curve. For \(x\in C(b)\), let
\(\delta_x=\ord_x\bmod\ell\in A_B\).
The lines \(\Lambda\delta_x\) are distinct and nonzero.
The family \((\delta_x)_{x\in C(b)}\) tends to zero outside finite
subsets, and the continuous summation map
\begin{equation}\label{eq:curve-summation}
\prod_{x\in C(b)}\Lambda\longrightarrow A_B,\qquad
(a_x)_x\longmapsto\sum_x a_x\delta_x
\end{equation}
has kernel consisting exactly of the constant families.
If \(\charac(b)\ne\ell\), the common kernel in \(V_B\) of the
\(\delta_x\) has dimension \(2g(C)\).
\end{lemma}
\begin{proof}
Approximation at distinct point valuations proves that each
\(\delta_x\) is nonzero and that their lines are distinct.
A function on \(C\) has only finitely many zeros and poles. Hence every
fixed element of \(B^\times\) is annihilated by all but finitely many
\(\delta_x\), which proves the asserted convergence and defines
\eqref{eq:curve-summation}, with continuous evaluation at each function.
A family \((a_x)_x\) defines a homomorphism from the divisor group of
\(C\) to \(\Lambda\), sending the point divisor \(x\) to \(a_x\).
It lies in the kernel of \eqref{eq:curve-summation} precisely when
this homomorphism vanishes on principal divisors, or equivalently
factors through \(\Pic(C)\).
The group \(\Pic^0(C)(b)\) is \(\ell\)-divisible: multiplication by
\(\ell\) on the Jacobian is a surjective isogeny, and \(b\) is
algebraically closed. This remains true in characteristic \(\ell\).
Every homomorphism \(\Pic(C)\to\Lambda\) therefore factors through
the degree map \(\Pic(C)\to\Z\). Since all point divisors have degree
one, its coefficient family is constant. Conversely, constant
families annihilate principal divisors.
Finally, if all point orders of \(f\in B^\times\) are divisible by
\(\ell\), write \(\operatorname{div}(f)=\ell D\) and associate to
\([f]\) the class of \(D\) in \(\Pic(C)[\ell]\). This gives an
isomorphism from the common kernel to \(\Pic(C)[\ell]\):
surjectivity follows from the definition of torsion in \(\Pic(C)\);
injectivity follows because \(b^\times\) is \(\ell\)-divisible.
When \(\charac(b)\ne\ell\), the \(\ell\)-torsion of the Jacobian is
isomorphic to \((\Z/\ell)^{2g(C)}\).
\end{proof}
\subsection{An intrinsic test for finite-field constants}
We now adapt the curve-like relation criterion of
\cite[Theorem~4.2 and appendix, Main Theorem~I, p.~353]{Pop2015}
to the family of lines in \(J_v\).
Let \(\mathcal L_v\) be the family of one-dimensional subspaces of
\(A_{F_0}\) contained in \(J_v\). Choose a nonzero generator
\(e_H\in H\) for each \(H\in\mathcal L_v\). Consider the following
conditions:
\begin{enumerate}
\item the family \((e_H)_{H\in\mathcal L_v}\) tends to zero outside
finite subsets;
\item the resulting summation map
\[
\prod_{H\in\mathcal L_v}\Lambda\longrightarrow A_{F_0},
\qquad (a_H)_H\longmapsto\sum_H a_H e_H
\]
has a one-dimensional kernel generated by a family whose every
coordinate is nonzero.
\end{enumerate}
These conditions are independent of the choices of generators.
Changing generators rescales the coordinates of the product, and,
because \(\Lambda\) is finite, preserves the convergence in the first
condition. They are also preserved by topological linear
isomorphisms of the residual character spaces.
\begin{proposition}\label{prop:point-recognition}
For a quasi-prime \((d-1)\)-divisor \(v\) of \(F/\kappa\), the family
\(\mathcal L_v\) satisfies the preceding two conditions if and only if
\(\kappa v\) is algebraic over a finite field.
When this holds, \(\mathcal L_v\) consists exactly of the point inertia
lines of the smooth projective curve of \(Fv/\kappa v\).
Consequently the matching in Corollary~\ref{cor:flag-matching}
recognizes these terminal valuations and matches their full point
families.
\end{proposition}
\begin{proof}
Suppose first that \(\kappa_0\) is algebraic over a finite field.
Every valuation on \(\kappa_0\) is trivial. A nontrivial valuation
of the one-variable field \(F_0\) trivial on \(\kappa_0\) is a point
valuation on its smooth projective curve: properness gives a closed
center, and the discrete valuation ring at that smooth point is
dominated by the valuation ring. Writing each function as a power
of a uniformizer times a local unit identifies the two valuations.
Thus
Lemma~\ref{lem:residual-valuative} shows that every line in
\(\mathcal L_v\) is a point line. The converse inclusion follows
from Lemma~\ref{lem:residual-density-mod-ell}, using the trivial
constant reduction. Lemma~\ref{lem:curve-orders} now proves the two
conditions.
Suppose instead that \(\kappa_0\) is not algebraic over a finite field.
There is a nontrivial valuation \(b\) on \(\kappa_0\): in
characteristic zero extend a nontrivial valuation of the prime
field; in positive characteristic use a transcendental element,
extend its variable valuation to a rational transcendence basis,
and prolong to \(\kappa_0\).
Choose a transcendental \(t\in F_0\) with \(F_0/\kappa_0(t)\) finite.
The Gauss extension of \(b\) to \(\kappa_0(t)\), followed by a
prolongation to \(F_0\), gives a constant reduction \(c\).
Indeed the rational residue field has transcendence degree one,
and finite prolongation gives a finite residue extension.
The value group of \(c\) equals its constant-value subgroup: the
Gauss extension has this property, and a finite prolongation has
finite value-group index, whereas the constant-value group is
divisible.
Choose a point valuation \(q\) on \(F_0c/\kappa_0c\), and form
\(u=q\circ c\). This is a c.r.\ quasi-prime divisor, whose value
group contains the final discrete step as an innermost convex
subgroup. In particular \(uF_0/u\kappa_0\cong\Z\), so \(I_u\) is a
nonzero line, and it belongs to \(\mathcal L_v\).
It is different from every true point line. To see this, let \(p\)
be a point valuation of \(F_0/\kappa_0\). The valuation \(u\) is
nontrivial on constants, so cannot coarsen \(p\). Conversely, a
proper coarsening of \(u\) kills its innermost discrete subgroup
and retains only values supplied by constants; it cannot equal
the nontrivial valuation \(p\) that is trivial on constants.
Thus \(p\) and \(u\) are incomparable. Since \(p\) has rank one,
their finest common coarsening is trivial. Approximation gives
\(I_p\cap I_u=0\), proving the assertion even with
\(\Lambda\)-coefficients.
All point lines already belong to \(\mathcal L_v\), and their
generators have the nonzero relation of
Lemma~\ref{lem:curve-orders}, after rescaling to the chosen
generators. If the first condition holds for the full family,
extend this relation by zero on the additional lines. It is a
nonzero kernel vector with a zero coordinate, contradicting the
second condition. If the first condition fails, the family also
fails the stated test. This proves the equivalence.
\end{proof}
==== END Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory//sections/03-residual-curves.tex ====
==== BEGIN Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory//sections/04-support.tex ====
\section{A uniform bound for the support of a pencil}
\label{sec:support}
Fix a normal integral projective model $X\subset\Pj_l^n$ of $L/l$.
For $h\in V_L$, define
\[
\supp_X(h)=\{D\subset X:\ D\text{ is a prime divisor and }
\ord_D(h)\ne0\text{ in }\Lambda\}.
\]
Its degree is
\[
s_X(h)=\sum_{D\in\supp_X(h)}\deg(D).
\]
The orders modulo $\ell$, and hence this finite set and its degree, depend
only on the class $h$. We will prove that, for each $t\in K\setminus k$,
the integers $s_X(\Theta([t-a]))$ are bounded independently of $a\in k$.
The method is to realize any finite selection of these classes on a
residual curve of bounded genus. The point correspondence from
Section~\ref{sec:residual-curves} then turns the desired bound into a
Riemann--Hurwitz estimate. The bounded-genus and Hurwitz method appears
in \cite[Section~6, proof of Proposition~6.1]{BT2011}; we adapt the
bounded-support construction of \cite[Sections~4--6]{InternalBP} to
finite coefficients.
\subsection{Finite Kummer tests}
We first record why only finitely many classes can disappear in a
finitely generated extension.
\begin{lemma}\label{lem:kummer-kernel}
Let $E/F$ be a finitely generated field extension, where
$\charac F\ne\ell$ and $\mu_\ell\subset F$. The kernel of
$V_F\longrightarrow V_E$ is finite dimensional over $\Lambda$.
\end{lemma}
\begin{proof}
Let $F'$ be the relative algebraic closure of $F$ in $E$.
Choose a transcendence basis ${\bf x}$ for $E/F$ such that
$[E:F({\bf x})]=N<\infty$. For every finite extension $F_1/F$
contained in $F'$, the tuple ${\bf x}$ remains algebraically independent
over $F_1$, and
\[
[F_1:F]=[F_1({\bf x}):F({\bf x})]\le N.
\]
The finite subextensions of $F'/F$ form a directed system with bounded
degrees. One of them has maximal degree and contains all the others;
thus $F'/F$ is finite.
If $r$ independent classes in $V_F$ vanish in $V_E$, choose
representatives $f_1,\ldots,f_r\in F^\times$ and their $\ell$-th roots
in $E$. Those roots belong to $F'$, and Kummer theory gives
\[
[F(f_1^{1/\ell},\ldots,f_r^{1/\ell}):F]=\ell^r.
\]
Hence $\ell^r\le [F':F]$, which bounds the kernel dimension.
\end{proof}
The following geometric observation explains why a single smooth curve
can preserve any specified finite Kummer extension. Its irreducibility
assertion is useful because the cover can have arbitrarily large degree.
\begin{lemma}\label{lem:support-bertini}
Let $U$ be a smooth integral locally closed subvariety of projective space
over an algebraically closed field, of dimension at least two. Let
$Y\to U$ be a finite \'{e}tale morphism with $Y$ integral.
For general hyperplanes $H$, both $U\cap H$ and
$Y\times_U(U\cap H)$ are nonempty, smooth, and integral.
\end{lemma}
\begin{proof}
Smoothness of $U\cap H$ is Bertini's smoothness theorem for an embedded
smooth variety, valid in every characteristic
\cite[Lemma~33.47.3, Tag~0FD6]{Stacks}. It implies smoothness
upstairs because the cover is \'{e}tale. We give the irreducibility
argument, applied to either $Y\to\Pj^n$ or $U\to\Pj^n$.
Write $f:Y\to\Pj^n$ for the resulting quasi-finite morphism and
$a=\dim Y\ge2$. Let $\mathcal H=(\Pj^n)^\vee$ be the space of
hyperplanes. In the incidence of triples $(y_1,y_2,H)$ with
$f(y_1),f(y_2)\in H$, the locus $f(y_1)\ne f(y_2)$ is a projective
space bundle with fiber $\Pj^{n-2}$ over a nonempty open of $Y\times Y$.
It is therefore irreducible of dimension $2a+n-2$.
The locus of pairs with equal images has dimension at most $a$, by
quasi-finiteness; its hyperplane incidence has dimension at most
$a+n-1<2a+n-2$.
For a general hyperplane, $f^{-1}(H)$ is nonempty and has pure dimension
$a-1$. Nonemptiness follows because the image of $f$ contains an open
of its $a$-dimensional closure; purity follows from the principal ideal
theorem on the integral variety $Y$. Thus every component of the square
of the generic hyperplane section has dimension $2a-2$ over the function
field of $\mathcal H$. No such component can be contained in the
equal-image incidence, whose total dimension is too small. The square
of the generic section is consequently irreducible.
This implies geometric irreducibility of the generic section. Indeed,
if it had more than one geometric component, ordered pairs of points
lying on the same component and on different components would give two
distinct Galois-invariant unions of components of its square.
Its already established generic smoothness excludes nonreducedness.
Geometric integrality spreads to a nonempty open of $\mathcal H$,
which proves the assertion for general hyperplanes.
\end{proof}
\begin{proposition}[Finite curve test]\label{prop:finite-curve-test}
There is an integer $G_X\ge0$, depending only on the embedded model $X$,
with the following property. Given $g_1,\ldots,g_m\in L^\times$ whose
classes in $V_L$ are independent, there is a quasi-prime
$(d-1)$-divisor $w$ of $L/l$ such that:
\begin{enumerate}
\item $lw$ is algebraically closed and algebraic over a finite field of
characteristic different from $\ell$, and $Lw$ is the function field of a
smooth projective curve $C_w/lw$ of genus at most $G_X$;
\item the $g_j$ are $w$-units and their residue classes are independent
in $V_{Lw}$;
\item for every $j$,
\[
\#\{x\in C_w(lw):\ord_x(\overline{g_j})\not\equiv0\pmod\ell\}
=s_X([g_j]).
\]
\end{enumerate}
\end{proposition}
\begin{proof}
We first construct a curve over $l$, then specialize its finite defining
data, and finally realize that specialization by a valuation of $L$.
\emph{A curve preserving orders and independence.}
Let $D_1,\ldots,D_q$ be the prime divisors occurring in the divisors of
the $g_j$. There is a closed subset $B\subset X$ of codimension at least
two such that $X\setminus B$ is smooth, the $D_i\setminus B$ are smooth
pairwise disjoint Cartier divisors, and near each $D_i\setminus B$ every
$g_j$ is a power of a local equation of $D_i$ times a unit.
Outside their union the functions are units. To obtain $B$, remove the
singular locus of the normal variety $X$, the singular and non-Cartier
loci of the $D_i$, their pairwise intersections, and the exceptional
loci of these local expressions. Each has codimension at least two.
On the smooth open
$U=X\setminus(B\cup D_1\cup\cdots\cup D_q)$, the simultaneous root
cover defined by the equations $z_j^\ell=g_j$ is finite \'{e}tale.
Kummer theory and independence say that it is integral of degree
$\ell^m$. Successively choose $d-1$ general hyperplanes. Apply
Lemma~\ref{lem:support-bertini} to the cover on each successive open
section, and Bertini smoothness to the base and its specified divisors.
We obtain a smooth integral projective curve $C\subset X$ that avoids
$B$, meets every $D_i$ transversely in exactly $\deg D_i$ distinct
points, and has an integral root cover over $C\cap U$.
The latter cover still has degree $\ell^m$, so the restricted classes
of the $g_j$ are independent. At a point of $C\cap D_i$ their orders
are exactly their orders along $D_i$. These are all their zeros and
poles on $C$, proving the required equality of support counts over $l$.
The genus of this curve is independent of the chosen functions.
At each stage we may also require the hyperplane to avoid the associated
points of the preceding scheme section. Multiplication by its equation
is then injective, so the Hilbert polynomial of a section is the first
difference of the preceding Hilbert polynomial. The final scheme
section avoids $B$, and on $X\setminus B$ the successive sections are
smooth by Bertini. Hence the final scheme section is reduced and equals
$C$, even if intermediate sections had embedded components supported
in $B$.
Its Hilbert polynomial, and thus its genus, depend only on the Hilbert
polynomial of the embedded $X$. Denote this genus by $G_X$.
This argument does not require $X$ to be Cohen--Macaulay.
\emph{Specialization of the finite data.}
Choose a finitely generated subring $R\subset l$ in which $\ell$ is
invertible and over which the preceding data are defined. Enlarge $R$
by finitely many elements and localize it as follows. The smooth open
of $X$ containing $C$, the flag sections in that open, and their
inclusions descend with their dimensions, smoothness, and geometric
integrality preserved in every geometric fiber. The final curve
descends as a smooth projective curve of genus $G_X$.
Include the coordinates of its finitely many zeros and poles and the
local parameter--unit expressions for the functions. After shrinking
$\Spec R$, they give disjoint point sections with the same orders and
no other zeros or poles. The root cover over the complement of those
sections remains finite \'{e}tale of degree $\ell^m$, with geometrically
integral fibers. Thus independence and the support counts persist in
every geometric fiber under consideration.
Here the spreading statements have their usual finite-presentation
meaning: finitely many schemes, morphisms, functions, and identities
descend after adjoining finitely many coefficients
\cite[Tags~01ZM, 0C0C, 081F]{Stacks}; smoothness and
flatness hold after shrinking; and geometric integrality at the generic
point persists on an open \cite[Tags~0578, 0559]{Stacks}.
The Hilbert polynomial in the projective
flat family of curves is constant; its constant term is the fiberwise
Euler characteristic, locally constant in this proper flat family
\cite[Lemma~36.32.2, Tag~0B9T]{Stacks}. For the orders, the identities
$g_j=u\pi^e$ on finitely many neighborhoods, with $u$ invertible and
$\pi$ a parameter for the relevant section, preserve the integer $e$.
On the complement of these neighborhoods the functions and their
inverses are regular. Properness of the curve ensures that these
finitely many open conditions still cover every fiber after shrinking.
We also include a presentation of the function field, so that the
specialized variety will be the residue field of a valuation.
Choose a separating transcendence basis $x_1,\ldots,x_d$ for $L/l$ and
a primitive element $y$ for the finite separable extension
$L/l(x_1,\ldots,x_d)$. Write its monic minimal polynomial as
$Q(Y)\in l({\bf x})[Y]$. After inverting a polynomial in ${\bf x}$,
this gives a finite integral model over an open of affine $d$-space.
Descend that model, a common dense open with the model already chosen,
and the rational expressions for all the $g_j$.
By geometric integrality and further localization of $R$, the reduced
polynomial in every geometric fiber is irreducible of the same degree
as $Q$ and gives that fiber's function field. All denominators in the
coefficients and in the expressions being used remain nonzero, and the
$g_j$ agree with the specified nonzero functions there.
\emph{Realization by a valuation.}
Choose a closed point $s\in\Spec R$. Its residue field is finite, of
characteristic different from $\ell$. A valuation of $\operatorname{Frac}R$
dominating $R_s$ extends to $l$; equivalently, one may directly choose a
valuation ring of $l$ dominating $R_s$
\cite[Lemma~10.50.2, Tag~00IA]{Stacks}.
Write its residue field as $\lambda_0$.
Since $l$ is algebraically closed, so is $\lambda_0$.
We may arrange that the ultimate residue field is algebraic over the
finite field $\kappa(s)$. Indeed let $b$ be the algebraic closure of
$\kappa(s)$ inside $\lambda_0$, and choose a transcendence basis
$\mathcal T$ for $\lambda_0/b$. On $b(\mathcal T)$ give the basis
elements rationally independent values in an ordered free abelian group
with basis $\mathcal T$ and give $b^\times$ value zero.
Every polynomial has a unique term of least value, so this defines a
valuation with residue $b$. Extend it to the algebraic extension
$\lambda_0/b(\mathcal T)$. Its residue extension is algebraic and thus
still equals $b$. Composing valuations gives a valuation $u$ of $l$
with residue $\lambda=b$. Its center on $R$ remains $s$, since the
second valuation is trivial on the finite field $\kappa(s)$.
An ordered free abelian group exists for any cardinality of
$\mathcal T$, so this construction imposes no cardinality condition on $l$.
Give $l({\bf x})$ the Gauss extension of $u$: the values of the $x_i$
are zero and their residues are algebraically independent over
$\lambda$. Prolong it to $L$, obtaining $w_X$.
The coefficients of $Q$ are integral at the Gauss valuation and its
reduction is the specified irreducible polynomial over
$\lambda(\overline{\bf x})$. Since $Q$ is monic, $y$ is integral;
its residue therefore has degree $\deg Q$. The fundamental inequality
for a finite extension of valued fields now forces
\[
w_XL=ul,\qquad
Lw_X=\lambda(\overline{\bf x},\overline y).
\]
Thus $w_X$ is a constant reduction with precisely the specialized
function field as residue, and each $g_j$ has the prescribed residue.
On $Lw_X$, take the successive true divisor valuations of the
specialized smooth flag, ending at its curve. Compose them with $w_X$.
Write $w_1$ for the first composition. It is quasi-divisorial: its value group $\Gamma$
has a convex kernel $\Z$ over $w_XL$, while the constant values map
isomorphically onto $w_XL=ul$. Consequently
$\Gamma/w_1l\cong\Z$. Every nonzero convex subgroup of $\Gamma$
contains this innermost $\Z$, so any proper coarsening has value group
generated by constant values and loses the relative rank-one
contribution. This proves the required minimality.
The remaining steps are true divisor steps. Their composition $w$ is
therefore a quasi-prime $(d-1)$-divisor with residue the specialized
curve field. All $g_j$ are units at the successive generic points.
The genus, independence, and support counts already preserved in the
specialization give the three conclusions.
\end{proof}
\subsection{Five fibers control the degree}
We now transfer the finite tests across $\Theta$.
The curve used for a test is allowed to depend on the tested classes;
its genus bound depends only on $X$.
\begin{proposition}\label{prop:uniform-support}
For every $t\in K\setminus k$,
\[
\sup_{a\in k}s_X\bigl(\Theta([t-a])\bigr)<\infty.
\]
\end{proposition}
\begin{proof}
Put $h_a=\Theta([t-a])$. The classes $[t-a]$, $a\in k$, are
independent in $V_{k(t)}$, as is seen from their orders at the finite
points of the projective line.
Lemma~\ref{lem:kummer-kernel} shows that the kernel of
$V_{k(t)}\to V_K$ is finite dimensional. A basis for its intersection
with the span of these classes involves only finitely many indices.
After removing those indices, we have an infinite set $A\subset k$
such that the classes $[t-a]$, $a\in A$, are independent in $V_K$.
Choose five distinct anchors $a_1,\ldots,a_5\in A$ and put
$N_i=s_X(h_{a_i})$. If $\charac k=0$, choose the anchors to include
$b_0,b_0+1,b_0+\ell$; this is possible because $k\setminus A$ is finite.
Fix one further $a\in A$ distinct from the anchors and put $a_6=a$.
Choose representatives $g_1,\ldots,g_6\in L^\times$ of
$h_{a_1},\ldots,h_{a_6}$, and apply
Proposition~\ref{prop:finite-curve-test} to these representatives.
Let $w$ be the resulting valuation and $C_w$ its residual curve.
Their residue classes are independent, so their evaluations on $D_w$
are independent by Lemma~\ref{lem:residue-duality}.
Let $v$ be the quasi-prime $(d-1)$-divisor of $K$ corresponding to $w$
under Corollary~\ref{cor:flag-matching}.
The evaluations of the six functions $u_j=t-a_j$, $1\le j\le6$,
on $D_v$ are independent.
We first normalize these functions so that their residues are a pencil
on $Kv$. We claim that $v(k(t))=vk$.
Otherwise the nonzero relative value group of $k(t)/k$ has rational
rank one, and the valuation transcendence-degree inequality forces
$k(t)v=kv$. Equality in that inequality makes the relative value
group cyclic. Any element with value in $vk$ can then be multiplied
by a constant to become a unit, and its residue can be removed by a
further constant. Its class consequently vanishes on $D_v$.
Thus evaluation of $k(t)^\times$ on $D_v$ factors through the cyclic
relative value group and has dimension at most one, a contradiction.
The values of all six $u_j$ are equal. If $v(u_i)>v(u_j)$, then
$u_j/(a_i-a_j)$ is a principal unit, contradicting its nonzero
evaluation on $D_v$. Write $\gamma$ for the common value.
For distinct indices, $v(a_i-a_j)\ge\gamma$; a strict inequality
would make $u_i/u_j$ a principal unit and give equal evaluations.
Hence every difference $a_i-a_j$ has value exactly $\gamma$.
In positive characteristic the residue characteristic equals
$\charac k\ne\ell$. In characteristic zero our three specified
anchors give $v(1)=v(\ell)=\gamma=0$, so again
$\charac(kv)\ne\ell$.
Choose $b\in k^\times$ with $v(b)=\gamma$ and set
\[
\tau_0=\overline{(t-a_1)/b},\qquad
\beta_j=\overline{(a_j-a_1)/b}.
\]
The $\beta_j\in kv$ are distinct and
$\overline{u_j/b}=\tau_0-\beta_j$.
These residue classes remain independent: the functions $u_j/b$ are
units, their evaluations factor through $D_v/I_v$, and multiplication
by $b$ changes no class because constants are $\ell$-divisible.
In particular $\tau_0\notin kv$.
More precisely, the dual of the induced isomorphism
$D_w/I_w\longrightarrow D_v/I_v$ carries
$[\tau_0-\beta_j]\in V_{Kv}$ to $[\overline{g_j}]\in V_{Lw}$.
Indeed their evaluations on corresponding residual characters are the
evaluations of $[u_j]$ and $\Theta([u_j])=[g_j]$ on the original
decomposition spaces. This identifies the individual tested classes,
as well as preserving their independence.
Proposition~\ref{prop:point-recognition} says that $kv$, like $lw$,
is algebraic over a finite field, and that the complete point families
on the two residual curves correspond. Let $C_v$ be the smooth
projective curve with function field $Kv$.
Both residue characteristics are different from $\ell$; by
Lemma~\ref{lem:curve-orders}, the dimensions of the common point-order
kernels are twice the genera. The residual isomorphism preserves
these kernels, so $g(C_v)=g(C_w)\le G_X$.
It also preserves the number of nonzero point orders of each tested
class. For the five anchor classes these numbers are $N_1,\ldots,N_5$.
Write $p=\charac(kv)>0$ and $\tau_0=\tau^{p^r}$ with $r$ maximal.
Such an $r$ exists because a nonconstant function has a nonzero point
order, and $p^r$ must divide that fixed nonzero integer.
The function $\tau$ defines a separable morphism
$C_v\to\Pj_{kv}^1$; write $n'$ for its degree.
Multiplication of orders by $p^r$ does not change their vanishing
modulo $\ell$.
Consider the fiber of this morphism over $\beta_i^{1/p^r}$.
At most $N_i$ points in the fiber have ramification index not divisible
by $\ell$. The other points have index at least $\ell$, so there are
at most $n'/\ell$ of them. If $r_i$ is the total number of points
in the fiber, then
$r_i\le N_i+n'/\ell$.
Since the sum of the ramification indices in a fiber is $n'$, and
the different exponent at a point is at least its index minus one,
the fiber contributes at least
$n'-r_i\ge(1-1/\ell)n'-N_i$ to the different.
The five fibers are disjoint. Riemann--Hurwitz, with its different term
in arbitrary characteristic \cite[Lemma~53.12.4, Tag~0C1F]{Stacks},
therefore yields
\[
2g(C_v)-2+2n'
\ge 5(1-1/\ell)n'-\sum_{i=1}^5N_i,
\]
or
\begin{equation}\label{eq:support-five-fibers}
\bigl(5(1-1/\ell)-2\bigr)n'
\le 2G_X-2+\sum_{i=1}^5N_i.
\end{equation}
The coefficient on the left is positive for every prime $\ell$,
including $\ell=2$. Hence $n'$ has a bound depending only on the five
anchors and $X$.
For the additional parameter, the zeros and poles of
$\tau_0-\beta_6$ lie in two fibers of this same morphism.
It has at most $2n'$ nonzero point orders modulo $\ell$.
The point correspondence and the finite curve test identify this
number with $s_X(h_a)$. This proves a uniform bound for all
non-anchor members of $A$. Including the anchors and the finitely
many parameters outside $A$ completes the proof.
\end{proof}
==== END Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory//sections/04-support.tex ====
==== BEGIN Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory//sections/05-curve-fields.tex ====
\section{Recovering curve subfields}
\label{sec:curve-fields}
The support bound of Proposition~\ref{prop:uniform-support} allows us to
pass from individual multiplicative classes to subfields. We prove that
$\Theta$ carries the mod-$\ell$ multiplicative group of every relatively
algebraically closed one-variable subfield of $K$ onto that of such a
subfield of $L$. The geometric step is an incidence construction: infinitely
many divisors of bounded degree form a family, and a curve in its parameter
space produces the one-variable field. This follows the bounded-support
and incidence method of \cite[Sections~4--6]{InternalBP}. We give the
construction and the descent argument in the finite-coefficient setting.
A \emph{curve subfield} of a function field $F/\kappa$ is a subfield
$E\subset F$ containing $\kappa$, relatively algebraically closed in $F$,
and of transcendence degree one over $\kappa$. It is finitely generated:
for any $t\in E\setminus\kappa$, it is the relative algebraic closure of
$\kappa(t)$ in $F$, which is finite over $\kappa(t)$. The natural map
$V_E\to V_F$ is injective, since an $\ell$-th root in $F$ of an element
of $E$ is algebraic over $E$. We henceforth regard $V_E$ as a subspace
of $V_F$.
\subsection{Two descent facts}
We first record the field-theoretic facts needed to descend the field
produced by incidence. Recall that a finitely generated extension $F/B$
is \emph{regular} if it is separable and $B$ is relatively algebraically
closed in $F$; equivalently, $F$ and an algebraic closure $\overline B$ are
linearly disjoint over $B$.
\begin{lemma}[Descent from algebraically closed constants]
\label{lem:constant-descent}
Let $F/B$ be a finitely generated regular extension, where
$\charac(B)\ne\ell$ and $\mu_\ell\subset B$. If $g\in F^\times$
has an $\ell$-th root in $F\overline B$, then its class in $V_F$ lies
in the image of $V_B$.
\end{lemma}
\begin{proof}
Choose $r\in F\overline B$ with $r^\ell=g$. The element $r$ already
belongs to $FB'$ for a finite Galois extension $B'/B$. Indeed it is
separable over $F$, so purely inseparable constant extensions are
unnecessary, and we may take a finite Galois closure of the remaining
constant extension. By regularity,
$\Gal(FB'/F)=\Gal(B'/B)$. For $\gamma$ in this group,
\[
c_\gamma=\frac{\gamma(r)}r\in\mu_\ell.
\]
These multipliers form a multiplicative cocycle with values in $B'^\times$.
Hilbert's Theorem~90 gives $b'\in B'^\times$ with
$\gamma(b')/b'=c_\gamma$ for every $\gamma$. Thus $r/b'\in F$ and
$b'^\ell\in B$. The equality
$g=(r/b')^\ell b'^\ell$ proves the assertion.
\end{proof}
\begin{lemma}[Intersections of curve subfields]
\label{lem:curve-intersection}
Let $F/\kappa$ be a finitely generated extension of an algebraically
closed field of characteristic different from $\ell$. If $P,Q\subset F$
are distinct curve subfields, then
\[
\dim_\Lambda(V_P\cap V_Q)<\infty.
\]
Each $V_P$ is infinite dimensional. In particular, an inclusion
$V_P\subset V_Q$ forces $P=Q$.
\end{lemma}
\begin{proof}
The compositum $PQ$ has transcendence degree two over $\kappa$.
Otherwise every element of $Q$ would be algebraic over $P$, so relative
algebraic closedness of $P$ in $F$ would give $Q\subset P$; reversing
their roles would give equality.
Let $C_P,C_Q$ be their smooth projective curves. The natural map to
$C_P\times C_Q$ is dominant, since its image has dimension two;
consequently $PQ$ is the function field of this product.
Inside $V_{PQ}$, any class coming from both fields has zero order along
every divisor $\{x\}\times C_Q$. Its representative from $P$ therefore
lies in the common kernel of all point orders of $C_P$. This kernel has
dimension $2g(C_P)$ by Lemma~\ref{lem:curve-orders}, so the intersection
inside $V_{PQ}$ is finite dimensional. The maps from $V_P$ and $V_Q$
to $V_{PQ}$ are injective: each factor field is relatively algebraically
closed in the function field of the product.
The kernel of $V_{PQ}\to V_F$ is finite dimensional by
Lemma~\ref{lem:kummer-kernel}. To see that passing to $F$ preserves the
finite-intersection conclusion, consider pairs $(a,b)\in V_P\oplus V_Q$
whose images in $V_F$ agree. The difference $a-b$ lies in this finite
kernel; the kernel of the difference map on such pairs is the intersection
already computed in $V_{PQ}$. The space of these pairs, and hence
$V_P\cap V_Q$ in $V_F$, is finite dimensional.
Finally, choose $t\in P\setminus\kappa$. The classes $[t-a]$,
$a\in\kappa$, are linearly independent in $V_{\kappa(t)}$, as their
orders at the distinct finite points show. Lemma~\ref{lem:kummer-kernel}
gives only a finite-dimensional kernel on passing to $V_P$, so $V_P$
is infinite dimensional.
\end{proof}
\subsection{A family of divisors and a field of constants}
The next lemma isolates the geometric construction. Its hypothesis says
that every class under consideration becomes an $\ell$-th power on almost
every divisor in one fixed infinite family. The conclusion realizes all
these classes as classes from a single curve field after a finite extension
of the ambient field.
\begin{lemma}[Incidence descent]
\label{lem:curve-incidence}
Let $X\subset\Pj_l^n$ be a normal integral projective variety of dimension
$d\ge2$ over an algebraically closed field $l$ of characteristic different
from $\ell$, and put $L=l(X)$. Let $\mathcal S$ be an infinite set of
prime divisors on $X$ whose degrees are bounded. Suppose that
$H\subset V_L$ is a subspace with the following property: for every
$g\in L^\times$ with $[g]\in H$,
\begin{equation}
\label{eq:curve-residue-trivial}
\ord_D(g)=0\quad\hbox{and}\quad
g|_D\in l(D)^{\times\ell}
\qquad\hbox{for all but finitely many }D\in\mathcal S.
\end{equation}
Then there are a finite extension $M/L$ and a one-variable field
$P_0/l$ contained in $M$ such that $M/P_0$ is regular,
\begin{equation}
\label{eq:curve-compositum}
M=LP_0,
\end{equation}
and the image of $H$ in $V_M$ is contained in the image of $V_{P_0}$.
In particular, every $g$ as above has an $\ell$-th root in
$M\overline{P_0}$.
\end{lemma}
\begin{proof}
\emph{A bounded parameter space.}
We recall why a degree bound gives a parameter space of finite type.
Integral subvarieties of fixed dimension
and bounded degree in a fixed projective space have only finitely many
Hilbert polynomials; see also \cite[Proposition~5.3]{Tanaka2024}.
One way to establish this boundedness is to cut them
out set-theoretically by forms of bounded degree. If an integral
subvariety $D\subset\Pj^n$ has dimension $r$ and degree at most $b$, and
$x\notin D$, choose a linear projection to $\Pj^{r+1}$ whose center
avoids the join of $x$ and $D$. When $D$ is already a hypersurface no
projection is necessary. The projection is defined on $D$ and is finite
there: a positive-dimensional fiber would contradict ampleness of the
pulled-back hyperplane bundle. Its image is a hypersurface of degree at
most $b$, and its equation pulls back to a form vanishing on $D$ but not
at $x$. Multiplication by a form nonzero at $x$ makes the degree exactly
$b$ if necessary. It follows that the degree-$b$ forms vanishing on $D$
cut it out as a reduced set.
A fixed number of such forms, namely the dimension of the space of
degree-$b$ forms, therefore places all these reduced subvarieties among
the geometric reductions of fibers of one projective family of finite
type. Such geometric reductions have finitely many Hilbert polynomials.
For completeness, over an integral base descend the reduction of the
geometric generic fiber to a finite extension of the base function field,
and normalize the base in that extension. On a dense open of this finite
base change, spread the reduction as a closed subscheme, with nilpotent
defining ideal, flat and with geometrically reduced fibers. These
properties follow after shrinking from generic flatness and spreading
geometric reducedness in the resulting projective flat family
\cite[Tag~\texttt{0578}]{Stacks}. Thus
it gives exactly the geometric reductions there, with constant Hilbert
polynomial. The complement upstairs has image in a proper closed subset
of the original base because the normalization is finite. Noetherian
induction on that closed subset proves the assertion. Applying this to
the divisors in $\mathcal S$ gives the required finite union of Hilbert
schemes on $X$.
\smallskip\noindent
\emph{The incidence fields.}
In that union take a positive-dimensional irreducible component of the
closure of the points corresponding to $\mathcal S$. After replacing it
by an integral locally closed open subset $T$, the tested points are dense
in $T$, and its universal family
\[
Z\subset X\times T\longrightarrow T
\]
is flat with geometrically integral fibers of dimension $d-1$. The latter
condition is available by constructibility of geometric integrality
\cite[Tags~\texttt{0579} and~\texttt{055B}]{Stacks}: the
tested fibers are integral over the algebraically closed field $l$, and
their parameter points are dense. Choose an integral locally closed curve
$T'\subset T$, once and for all, and put $Z'=Z\times_T T'$.
Both $Z$ and $Z'$ are integral, by flatness and geometric integrality of
their generic fibers. Their maps to $X$ are dominant. Indeed a proper
closed subset of $X$ can contain only finitely many distinct prime
divisors, whereas the fibers over the infinitely many distinct points of
$T$, or of $T'$, are distinct divisors.
The two projections now give
\[
\begin{array}{ccccc}
&& Z' &&\\[-2pt]
&\swarrow&&\searrow&\\[-2pt]
X&&&&T'
\end{array}
\qquad\qquad
\begin{array}{ccccc}
&& M=l(Z') &&\\[-2pt]
&\nearrow&&\nwarrow&\\[-2pt]
L=l(X)&&&&P_0=l(T').
\end{array}
\]
Since $\dim Z'=d$, the extension $M/L$ is finite. Geometric integrality
of the generic fiber of $Z'\to T'$ says precisely that $M/P_0$ is regular.
The inclusion $Z'\subset X\times T'$ also shows that its function field
is generated by the coordinates from the two factors, proving
\eqref{eq:curve-compositum}.
\smallskip\noindent
\emph{Descent along the fixed curve.}
We next show that the chosen fields work for every class in $H$. Fix
$g\in L^\times$ with $[g]\in H$. Let $Y$ be the intersection of the
relative smooth locus of $Z\to T$ with the inverse image of the open
subset of $X$ on which $g$ is a unit. Every nonempty geometric fiber of
$Y\to T$ is integral. Its fibers over the generic points of $T$ and
$T'$ are nonempty: the corresponding incidence families dominate $X$,
and the smooth locus is dense in their geometrically integral fibers.
On $Y$ consider the finite \'etale cover defined by adjoining an
$\ell$-th root of $g$.
This cover splits on a dense set of the tested fibers. Indeed
\eqref{eq:curve-residue-trivial} gives a rational $\ell$-th root on each
such divisor, and on its smooth unit locus the root and its inverse are
regular, by normality. The cover therefore splits on the geometric
generic fiber over $T$ as well. Otherwise, since $\ell$ is prime and
the constants contain $\mu_\ell$, its generic Kummer polynomial would
be irreducible over that geometric function field. The geometric generic
cover would then be integral. Geometric integrality spreads to a
nonempty open of $T$
\cite[Tags~\texttt{0578} and~\texttt{0559}]{Stacks}, contradicting the
dense set of split fibers.
It remains to pass this splitting to our fixed curve $T'$. Shrinking $T$
separately for each $g$ would not justify this passage: the resulting open
could miss $T'$. Instead, descend the geometric generic splitting to a
finite extension of $l(T)$ and let $\widetilde T\to T$ be the finite
surjective normalization in that extension. The pullback
$\widetilde Y=Y\times_T\widetilde T$ is normal, since it is smooth over
the normal scheme $\widetilde T$ \cite[Tag~\texttt{034F}]{Stacks}.
It is also integral. Its generic
fiber is geometrically integral, and every irreducible component dominates
$\widetilde T$: on a normal scheme the components are open, and a smooth
map has open image. Thus there is only one component.
The pulled-back finite \'etale cover of $\widetilde Y$ is generically
split, and hence split everywhere. To justify the last implication,
each component is finite and birational over the normal integral base;
it is therefore isomorphic to that base. Take a geometric point of
$\widetilde T$ lying over the generic point of $T'$. The corresponding
fiber of $\widetilde Y$ is a base change of the nonempty fiber of $Y$
over that generic point. Its split cover shows
\begin{equation}
\label{eq:curve-geometric-root}
g\in(M\overline{P_0})^{\times\ell}.
\end{equation}
This reasoning applies to each $g$ separately, while $T'$, $M$, and
$P_0$ stay fixed. Lemma~\ref{lem:constant-descent}, applied to the
regular extension $M/P_0$, now gives the asserted containment in $V_M$.
\end{proof}
\subsection{Descent to the original function field}
We apply incidence to the image of a curve subfield under $\Theta$.
The last step below uses the equality $M=LP_0$ to descend all the way to
$L$, rather than merely finding a curve field after a finite extension.
\begin{theorem}[Correspondence of curve subfields]
\label{thm:curve-matching}
For every curve subfield $E\subset K$ there is a unique curve subfield
$P\subset L$ such that $\Theta(V_E)=V_P$. This assignment is a bijection
between the curve subfields of $K/k$ and those of $L/l$.
\end{theorem}
\begin{proof}
Fix a normal integral projective model $X\subset\Pj_l^n$ of $L/l$.
Let $E\subset K$ be a curve subfield, choose $t\in E\setminus k$, and
put $H=\Theta(V_E)$. For $a\in k$, let $h_a=\Theta([t-a])$, and let
$\mathcal S$ be the union of their mod-$\ell$ supports on $X$.
Proposition~\ref{prop:uniform-support} bounds the degrees of these
supports, and hence the degree of every prime divisor in $\mathcal S$.
The set $\mathcal S$ is infinite. To see this, first observe that the
kernel of all divisor-order maps on $X$ is finite dimensional. If it
contained more than $2G_X$ independent classes, apply
Proposition~\ref{prop:finite-curve-test} to representatives of a finite
independent subset of that size. Their independent residues on the
testing curve would all have zero point orders, contradicting
Lemma~\ref{lem:curve-orders} and its genus bound. On the other hand,
the $[t-a]$ span an infinite-dimensional subspace of $V_E$, and therefore
the $h_a$ span an infinite-dimensional subspace of $V_L$. If
$\mathcal S$ were finite, their order vectors would lie in a finite
dimensional space; the finite-dimensional kernel just proved would give
a contradiction.
We verify \eqref{eq:curve-residue-trivial}. For each $D\in\mathcal S$,
some $h_a$ has nonzero order at $D$. Match the valuation of $D$ to the
quasi-divisorial valuation $v$ of $K$ using
Proposition~\ref{prop:quasi-recovery}. Its inertia $I_v$ has nonzero
restriction to $E$. Consequently $vE/vk\ne0$; this group is a subgroup
of the cyclic group $vK/vk$, and the valuation transcendence-degree
inequality gives $Ev=kv$. Restriction of $D_v$ to $E$ therefore factors
through $vE/vk$ and has dimension at most one. Explicitly, an element
whose value is in $vk$ can be rescaled by a constant to a unit; its
residue is in $kv$, so a further constant rescaling makes it a principal
unit. Since inertia already has nonzero restriction, the images of
$I_v$ and $D_v$ on $E$ are the same line.
Now let $g\in L^\times$ with $[g]\in H$. Whenever $\ord_D(g)=0$,
inertia at $D$ annihilates this class. The preceding equality of
restriction images, transported through $\Phi$, implies that decomposition
at $D$ annihilates it too. By Lemma~\ref{lem:residue-duality}, its
residue is an $\ell$-th power in $l(D)$. Since a fixed rational function
has nonzero integral order at only finitely many prime divisors, this
proves \eqref{eq:curve-residue-trivial}. Lemma~\ref{lem:curve-incidence}
now supplies $M/L$ and $P_0\subset M$ with $M=LP_0$, $M/P_0$ regular,
and the image of $H$ contained in the image of $V_{P_0}$.
Choose a finite normal extension $N/L$ containing $M$, allowing
inseparability, and let $P_N$ be the relative algebraic closure of $P_0$
in $N$. It is a curve subfield of $N/l$. The image $H_N$ of $H$ in
$V_N$ is infinite dimensional, by Lemma~\ref{lem:kummer-kernel}, and
lies in $V_{P_N}$. Every $L$-automorphism $\gamma$ of $N$ fixes $H_N$
pointwise. Thus $H_N\subset V_{P_N}\cap V_{\gamma(P_N)}$, and
Lemma~\ref{lem:curve-intersection} forces $\gamma(P_N)=P_N$.
Put $P_L=P_N\cap L$. This is relatively algebraically closed in $L$.
It also has transcendence degree one over $l$. Indeed the invariant
field $P_N^{\Aut(N/L)}$ has transcendence degree one, and it is contained
in the fixed field $N^{\Aut(N/L)}$. For a finite normal extension the
latter is $L$ in characteristic zero and is a finite purely inseparable
extension of $L$ in positive characteristic. In the latter case, a
uniform $p$-power of the invariant field lies in $P_N\cap L$, preserving
transcendence degree. Thus $P_L$ is a curve subfield of $L$.
We now use the compositum identity to obtain containment already in
$V_L$. Since $P_L\subset L\subset M$ and every element of
$P_L\subset P_N$ is algebraic over $P_0$, relative algebraic closedness
of $P_0$ in $M$ gives
$P_L\subset P_0$. The extension $P_0/P_L$ is therefore finite algebraic.
Choose the algebraic closures inside a common algebraically closed
overfield. Then
\begin{equation}
\label{eq:curve-exact-descent}
M\overline{P_0}=L\overline{P_L}.
\end{equation}
Here we used both $M=LP_0$ and the fact that $P_0$ and $P_L$ have the
same algebraic closure in that overfield.
The extension $L/P_L$ is regular. Relative algebraic closedness is
already known, and separability is automatic in characteristic zero.
In characteristic $p>0$, choose a separating variable $u$ for $P_L/l$.
It cannot become a $p$-th power in $L$: a $p$-th root would be algebraic
over $P_L$ and hence belong to $P_L$, contrary to the choice of $u$.
Over the perfect field $l$, such an element extends to a separating
transcendence basis of $L/l$, by the usual $p$-basis criterion. Thus
$L/l(u)$, and therefore $L/P_L$, is separable.
For each $[g]\in H$, Lemma~\ref{lem:curve-incidence} and
\eqref{eq:curve-exact-descent} give an $\ell$-th root of $g$ in
$L\overline{P_L}$. Applying Lemma~\ref{lem:constant-descent} once more,
now to $L/P_L$, yields the exact inclusion $\Theta(V_E)\subset V_{P_L}$.
Apply the same argument to $\Theta^{-1}$ and $P_L$. For a curve
subfield $E_1$ of $K$, it gives
\[
\Theta(V_E)\subset V_{P_L}\subset\Theta(V_{E_1}).
\]
Lemma~\ref{lem:curve-intersection} and the infinite dimension of $V_E$
force $E=E_1$. Both inclusions are consequently equalities. The same
argument with $K$ and $L$ reversed proves surjectivity of the matching,
and Lemma~\ref{lem:curve-intersection} proves its uniqueness.
\end{proof}
==== END Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory//sections/05-curve-fields.tex ====
==== BEGIN Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory//sections/06-good-functions.tex ====
\section{True divisors and rational subfields}
\label{sec:good-functions}
The curve-subfield correspondence lets us distinguish valuations trivial
on the constants from the quasi-divisorial valuations recovered in
Section~\ref{sec:local}. Their restrictions to curve subfields will then
identify a family of rational functions with enough point information to
recover field operations.
A \emph{true prime divisor}, or \emph{divisorial valuation}, of $F/\kappa$
is a quasi-prime divisor trivial on $\kappa$. Its value group is $\Z$;
we use the discrete normalization when writing its order character.
The true-divisor criterion below has as a methodological antecedent
Topaz's rational-character criterion \cite[Lemma~A.7]{Topaz2021Torelli},
via the companion reconstruction manuscript \cite[Sections~4--6]{InternalBP}.
Here finite coefficients require the finite-dimensional-kernel argument
given below; the rational-coefficient statement is not being invoked as
a proof of this proposition.
\begin{proposition}[Recognition of true divisors]
\label{prop:true-divisors}
Let $F/\kappa$ be one of $K/k$ or $L/l$. A quasi-divisorial valuation
$v$ of $F/\kappa$ is a true prime divisor if and only if there is a curve
subfield $E\subset F$ for which
\begin{equation}
\label{eq:good-true-test}
\dim_\Lambda(V_E\cap D_v^\perp)<\infty.
\end{equation}
Consequently $\Phi$ matches the inertia and decomposition groups of true
prime divisors of $L/l$ with those of $K/k$.
\end{proposition}
\begin{proof}
Suppose first that $v$ is trivial on $\kappa$. Since the residue field
has transcendence degree $d-1\ge1$, choose a unit $t$ with transcendental
residue, and let $E$ be the relative algebraic closure of $\kappa(t)$
in $F$. The valuation is trivial on $\kappa(t)$ and therefore on its
algebraic extension $E$. Reduction embeds $E$ into $Fv$. Its induced
map $V_E\to V_{Fv}$ has finite-dimensional kernel by
Lemma~\ref{lem:kummer-kernel}. Lemma~\ref{lem:residue-duality} identifies
that kernel with $V_E\cap D_v^\perp$, proving
\eqref{eq:good-true-test}.
Conversely, suppose that the restriction of $v$ to $\kappa$ is nontrivial,
and let $E$ be any curve subfield. Choose $x\in E\setminus\kappa$ with
$v(x)\ge0$, replacing a nonconstant element by its inverse if necessary.
There are infinitely many $c\in\kappa^\times$ with $v(c)>0$, for
example the positive powers of one such element. Each $1+cx$ is a
principal unit for $v$. Their classes are independent in
$V_{\kappa(x)}$: up to constant factors they are $[x+c^{-1}]$, with
distinct zeros on the rational curve. The map to $V_E$ has only a
finite-dimensional kernel by Lemma~\ref{lem:kummer-kernel}. These
principal units therefore span an infinite-dimensional subspace of
$V_E\cap D_v^\perp$. This disproves \eqref{eq:good-true-test} for every
$E$. The final assertion now follows from
Theorem~\ref{thm:curve-matching} and the already established matching of
quasi-divisorial pairs.
\end{proof}
For a curve subfield $E\subset F$, restriction of characters gives a
continuous surjection $A_F\to A_E$, dual to the inclusion $V_E\subset V_F$.
If a true divisor restricts nontrivially as a valuation on $E$, that
restriction is a positive integer multiple of a point valuation of the
smooth projective curve of $E$. Its inertia image in $A_E$ is either
the corresponding point line or zero; the latter occurs when that
integer is divisible by $\ell$. Thus we retain the nonzero inertia
images when discussing which point lines are detected.
\begin{definition}
\label{def:good-subfield}
A curve subfield $E\subset F$ is \emph{good} if it is rational over
the constants and every point line in $A_E$ is the nonzero image of the
inertia of some true divisor of $F$. A \emph{good function} is an element
$t\in F$ generating a good subfield $\kappa(t)$.
\end{definition}
\begin{proposition}[Recognition of good subfields]
\label{prop:good-matching}
The curve-subfield correspondence of Theorem~\ref{thm:curve-matching}
restricts to a bijection on good subfields. For matched good subfields
$E\subset K$ and $P\subset L$, the induced character isomorphism gives
a bijection between the points of their smooth projective rational curves.
\end{proposition}
\begin{proof}
For each curve subfield $E$, let $\mathcal Q_E$ be the collection of
nonzero images of true divisorial inertia in $A_E$. By
Proposition~\ref{prop:true-divisors}, these collections correspond under
the induced character isomorphisms. They are subfamilies of the point
lines of $E$. Choose a nonzero generator of each line. The generators
tend to zero outside finite sets, so summation defines a continuous map
from a product of copies of $\Lambda$ into $A_E$.
By Lemma~\ref{lem:curve-orders}, when every point occurs, this summation
map has one-dimensional kernel, generated by a coefficient family with
every coordinate nonzero. A proper subfamily of the point lines has no
relation: extending any proposed relation by zero to the missing points
would contradict the description of the full relation kernel. Hence
the stated kernel property recognizes exactly when $\mathcal Q_E$
contains every point line. Once this holds, the common kernel of their
orders on $V_E$ has dimension $2g(E)$. Its vanishing recognizes genus
zero, equivalently rationality over the algebraically closed constants.
Both conditions are preserved by the character isomorphisms. Finally,
distinct points have distinct inertia lines, so their matching gives the
asserted point bijection.
\end{proof}
We now construct enough good functions for the later reconstruction.
The construction is local on a smooth model: a ratio of two transverse
parameters has a multiplicity-one divisor over every value, and the
exceptional divisor of the blowup ensures that the resulting rational
subfield is relatively algebraically closed.
\begin{lemma}[A supply of good functions]
\label{lem:good-functions}
Let $F/\kappa$ be a finitely generated extension of an algebraically
closed field with $\trdeg(F/\kappa)\ge2$ and
$\charac(\kappa)\ne\ell$. On an integral model of $F/\kappa$, let
$q$ be a smooth closed point and let $a,b$ be rational functions regular
at $q$, vanishing there, with independent differentials in the cotangent
space at $q$. Then $t=a/b$ is a good function. Moreover, for every
$f\in F^\times$ there is a good function $t$ such that $ft$ is good.
Consequently good functions generate $F$ over $\kappa$, and their classes
span $V_F$.
\end{lemma}
\begin{proof}
Blow up the smooth point $q$ and let $v$ be the valuation of the
exceptional divisor. Write $e=\trdeg(F/\kappa)$. Its residue field is
the rational field of $\Pj_\kappa^{e-1}$. The leading linear forms of
$a$ and $b$ give independent homogeneous coordinates on the exceptional
divisor. Their ratio is the residue of $t$ and is a rational coordinate
in $Fv$. In particular $v$ is trivial on $\kappa(t)$, and
$\kappa(\overline t)$ is relatively algebraically closed in $Fv$.
The relative algebraic closure of $\kappa(t)$ in $F$ is a finite
algebraic extension on which $v$ is still trivial. It therefore embeds
by reduction into $Fv$, over $\kappa(t)\cong\kappa(\overline t)$.
Relative algebraic closedness in the residue field forces this extension
to be $\kappa(t)$ itself.
For each $c\in\kappa$, the function $a-cb$ has nonzero linear term at
$q$ and cuts out a prime divisor locally there, with order one.
The function $b$ is a unit at its generic point because the linear terms
of $a$ and $b$ are independent. The resulting true divisor has
$\ord(t-c)=1$ and detects the point $c$ of $\Pj^1_\kappa$.
The divisor locally cut out by $b$ similarly detects infinity: $a$
is a unit at its generic point and $t$ has a simple pole. Every point
line is thus detected, proving that $t$ is good.
Given $f\in F^\times$, choose the smooth point $q$ in an open set where
$f$ is a unit, and choose $a,b$ as above. The functions $fa,b$ also
vanish at $q$ with independent differentials, since
$d(fa)_q=f(q)\,da_q$. Hence both $ft=(fa)/b$ and $t=a/b$ are good.
The equality $f=(ft)/t$ proves field generation, and
$[f]=[ft]-[t]$ proves the spanning assertion for $V_F$.
\end{proof}
\begin{lemma}[A common partner]
\label{lem:common-partner}
For any finite list of good functions $t_1,\ldots,t_m$ of $F/\kappa$
there is a good function $s$ such that $s$ is algebraically independent
of each $t_i$ over $\kappa$ and
\[
t_i+cs\ \hbox{is good for every }i
\hbox{ and every }c\in\kappa^\times.
\]
\end{lemma}
\begin{proof}
Choose a smooth closed point $q$ on a model where all the $t_i$ are
regular, and choose $a,b$ vanishing at $q$ with independent differentials.
Put $s=a/b$. It is good by Lemma~\ref{lem:good-functions}. For
$c\ne0$, write
\[
t_i+cs=\frac{t_i b+ca}{b}.
\]
At $q$ the numerator has differential $t_i(q)\,db_q+c\,da_q$, which
is independent of $db_q$. The same lemma therefore makes every indicated
sum good.
Let $v$ be the exceptional-divisor valuation at $q$ used in the preceding
lemma. The residue of $s$ is transcendental over $\kappa$. By contrast,
the restriction of $v$ to $\kappa(t_i)$ is centered at the finite point
$t_i(q)$: the nonzero function $t_i-t_i(q)$ has positive order at $q$.
Its residue field is consequently $\kappa$. If $s$ were algebraic over
$\kappa(t_i)$, relative algebraic closedness of that good subfield in
$F$ would imply $s\in\kappa(t_i)$, contradicting its transcendental
residue. This proves the required independence.
\end{proof}
==== END Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory//sections/06-good-functions.tex ====
==== BEGIN Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory//sections/07-field.tex ====
\section{Recovering the field from simultaneous values}\label{sec:field}
The preceding section recovers the good rational subfields and the points of
their projective lines. We now recover the field operations from this
information. A single true divisor can detect the values of several good
functions at once. The resulting correspondence of algebraic relations
will first identify the constant fields and then give an isomorphism of
the perfect closures.
For every good function \(t\in K\), choose a generator \(t'\in L\) of the good
rational subfield matched with \(k(t)\) by
Proposition~\ref{prop:good-matching}. Its point correspondence, expressed in
these coordinates, is a bijection
\[
B_t:\Pj^1(k)\longrightarrow\Pj^1(l).
\]
Changing \(t'\) by a projective linear transformation, we arrange
\(B_t(\infty)=\infty\). These choices are made independently for the
different good functions. At this stage the maps \(B_t\) are only
bijections of sets.
\subsection{Simultaneous values}
For a finite tuple \(\mathbf t=(t_1,\ldots,t_m)\) of good functions, let
\(X_{\mathbf t}\subset(\Pj^1_k)^m\) be the reduced closure of the image of
the rational map defined by the \(t_i\) on a model of \(K/k\). Thus
\(X_{\mathbf t}\) is the irreducible variety of algebraic relations among
the \(t_i\). Define \(Y_{\mathbf t}\subset(\Pj^1_l)^m\) in the same way
from the chosen \(t_i'\).
\begin{lemma}[Simultaneous values]\label{lem:simultaneous-values}
For every finite tuple \(\mathbf t\) of good functions there are dense open
subsets \(U\subset X_{\mathbf t}\) and \(U'\subset Y_{\mathbf t}\) such that
\begin{equation}\label{eq:field-simultaneous}
\left(\prod_i B_{t_i}\right)(U(k))\subset Y_{\mathbf t}(l),
\qquad
\left(\prod_i B_{t_i}^{-1}\right)(U'(l))\subset X_{\mathbf t}(k).
\end{equation}
\end{lemma}
\begin{proof}
Choose a smooth model open on which all the \(t_i\) are regular and all the
differentials \(dt_i\) are nowhere zero. Such an open exists. In positive
characteristic, \(t_i\) cannot be a \(p\)-th power in \(K\), since its
\(p\)-th root would be algebraic over the relatively algebraically closed
subfield \(k(t_i)\). As \(k\) is perfect, this gives \(dt_i\ne0\); in
characteristic zero the same conclusion is immediate.
At a closed point \(P\) of this open, write \(a_i=t_i(P)\). The exceptional
divisor of the blowup at \(P\) defines a true divisorial valuation \(v\) with
\[
\ord_v(t_i-a_i)=1\qquad\text{for every }i.
\]
Indeed \(dt_i(P)\ne0\) says that \(t_i-a_i\) has order one in the maximal
ideal of the regular local ring at \(P\). Thus restriction of \(I_v\)
to each \(k(t_i)\) is the nonzero point line at \(a_i\).
By Proposition~\ref{prop:true-divisors}, the matched valuation on \(L\) is a
true divisor. Restriction of characters commutes with the correspondence
of good subfields, so its center in the \(t_i'\)-line is \(B_{t_i}(a_i)\)
for every \(i\). Specializing the target tuple along this valuation shows
that \((B_{t_i}(a_i))_i\) belongs to \(Y_{\mathbf t}(l)\): the generic tuple
lies in that closed subvariety of the product of projective lines, and so
does its specialization.
The image of the chosen model open in \(X_{\mathbf t}\) is constructible
and dense, hence contains a dense open subset \(U\). Every \(k\)-point of
\(U\) has a closed preimage, since its nonempty fiber is of finite type
over the algebraically closed field \(k\). This proves the first inclusion
in \eqref{eq:field-simultaneous}. The same argument for \(\Theta^{-1}\)
proves the second.
\end{proof}
These inclusions provide information in the Zariski topology without
asserting that any \(B_t\) preserves that topology. We next apply them to
the relation \(z=t+cs\). This will force the coordinate bijections to have
a common field-theoretic form.
\subsection{Aligning the constant fields}
An element of \(l(X_1,\ldots,X_r)^i\) will be called a \emph{perfect rational
function}. It has a well-defined value at a general tuple of \(l\)-points:
in positive characteristic one first raises it to a sufficiently large
\(p\)-power and then takes the unique corresponding root in \(l\).
Here and below, \emph{general} means belonging to a suitable dense open.
Let \(\mathcal H_l\) be the group of permutations of \(\Pj^1(l)\) generated
by \(\PGL_2(l)\), together with \(x\mapsto x^p\) when \(\charac l=p>0\).
\begin{lemma}\label{lem:cofinite-rational}
A perfect rational function in one variable that is injective on a cofinite
subset of \(\Pj^1(l)\) agrees there with a unique member of \(\mathcal H_l\).
In characteristic zero,
\(\mathcal H_l=\PGL_2(l)\); in characteristic \(p>0\), every member has a
unique expression
\[
M\circ\Frob^n,\qquad M\in\PGL_2(l),\quad n\in\Z.
\]
In particular, \(\mathcal H_l\) embeds in the group of permutations of
\(\Pj^1(l)\) modulo agreement outside finite sets.
\end{lemma}
\begin{proof}
In characteristic zero, injectivity on a cofinite set implies that a
rational map has degree one. In characteristic \(p\), choose \(N\ge0\)
such that the given function \(h\) satisfies \(h^{p^N}=r\in l(X)\).
Write \(r=q^{p^a}\), with \(a\ge0\) maximal and \(q\in l(X)\) separating.
The map \(q\) is also injective on a cofinite set, so its degree is one.
Indeed a separating map of degree greater than one has more than one
distinct point over a general value, and deleting finitely many points
cannot change this. Thus \(q\) is projective linear, and
\(h=\Frob^{a-N}\circ q\) on points.
Conjugating a projective linear transformation by Frobenius raises its
coefficients to \(p\)-th powers. This proves the asserted normal form.
No nonzero power of Frobenius is projective linear: for a positive exponent
the corresponding map has inseparable degree greater than one, and a
negative exponent reduces to this case by inversion. The normal form is
therefore unique. Finally, two perfect rational functions agreeing
cofinitely are equal, as is seen after clearing Frobenius powers and
comparing ordinary rational functions.
\end{proof}
Write \(T_a(x)=x+a\) and \(m_c(x)=cx\), fixing \(\infty\); these
transformations generate \(\Aff(k)\). We compare bijections of projective
point sets modulo finite disagreement. Composition is well defined with
this convention, because bijections carry finite sets to finite sets.
Membership in \(\mathcal H_l\) in the next lemma means membership after
this identification. Lemma~\ref{lem:cofinite-rational} ensures that the
representing member of \(\mathcal H_l\) is unique.
\begin{lemma}[Transported addition]\label{lem:field-affine}
Let \(t,s\) be algebraically independent good functions such that
\(z_c=t+cs\) is good for every \(c\in k^\times\).
For every \(c\in k^\times\) there is a perfect rational function
\(R_c\in l(X,Y)^i\) such that
\begin{equation}\label{eq:field-sum}
R_c(x,y)=B_{z_c}\bigl(B_t^{-1}(x)+cB_s^{-1}(y)\bigr)
\qquad\text{for general }(x,y)\in l^2.
\end{equation}
The one-variable slices of these functions give
\begin{equation}\label{eq:field-affine}
B_sB_t^{-1}\in\mathcal H_l,
\qquad
B_t\Aff(k)B_t^{-1}\subset\mathcal H_l.
\end{equation}
The second inclusion is an embedding of groups.
\end{lemma}
\begin{proof}
Consider the relation variety \(Y\) for \((t,s,z_c)\). Its projection to
the first two factors is dominant. Indeed \(k(t)\) and \(k(s)\) are
distinct curve subfields, so their matched subfields \(l(t')\) and
\(l(s')\) are distinct. Any two distinct curve subfields have compositum
of transcendence degree two: if their compositum had transcendence degree
one, relative algebraic closedness would make them equal.
On the dense open of \(Y\) given by the inverse inclusion in
Lemma~\ref{lem:simultaneous-values}, the third coordinate is forced by the
first two to be
\[
B_{z_c}\bigl(B_t^{-1}(x)+cB_s^{-1}(y)\bigr).
\]
We restrict to finite coordinates, which is permitted because all the
coordinate functions are nonconstant and the \(B\)'s fix \(\infty\).
If \(\dim Y=3\), then \(Y=(\Pj^1_l)^3\), and a dense open has infinitely
many third coordinates above a general pair \((x,y)\), a contradiction.
Consequently \(\dim Y=2\).
The projection \(Y\to(\Pj^1_l)^2\) is proper, dominant and generically
finite. The complement of the open just used has dimension at most one,
so its image is a proper closed subset of the base. After removing that
image and restricting to the locus of finite fibers, every closed fiber
has precisely one point. Thus the finite extension \(l(Y)/l(x,y)\)
has separable degree one. Indeed, in positive characteristic write the
minimal polynomial of the third coordinate as \(g(Z^{p^e})\), with \(g\)
separable; in characteristic zero take the minimal polynomial itself.
After shrinking the base to preserve the degree and nonzero discriminant
of the separable polynomial,
the separable degree counts the distinct points of a general fiber,
since taking \(p^e\)-th roots is a bijection on algebraically closed
points. The extension is therefore purely
inseparable, and the third coordinate gives a perfect rational function
\(R_c\) satisfying \eqref{eq:field-sum}.
Fixing a general \(y\) in \eqref{eq:field-sum} gives a perfect rational
function of \(x\), agreeing cofinitely with a bijection. A dense open in
\((\Pj^1_l)^2\) has cofinite fibers outside finitely many values of \(y\).
Since \(B_s\) is a bijection, Lemma~\ref{lem:cofinite-rational} therefore gives
\[
D_b:=B_{z_c}T_{cb}B_t^{-1}\in\mathcal H_l
\qquad\text{for cofinitely many }b\in k.
\]
For two such parameters,
\[
D_b^{-1}D_{b'}=B_tT_{c(b'-b)}B_t^{-1}.
\]
Every element of \(k\) is a difference of two members of a cofinite subset:
for any prescribed difference, the two required cofinite sets intersect.
Hence all translations conjugated by \(B_t\) belong to \(\mathcal H_l\).
The products \(D_bD_{b'}^{-1}\) give the same assertion for \(B_{z_c}\),
and then \(D_b\) gives \(B_{z_c}B_t^{-1}\in\mathcal H_l\).
Fixing a general \(x\) instead yields
\[
B_{z_c}T_a m_cB_s^{-1}\in\mathcal H_l
\qquad\text{for cofinitely many }a\in k.
\]
Composing on the left successively with
\[
B_{z_c}T_{-a}B_{z_c}^{-1}
\quad\text{and}\quad
B_tB_{z_c}^{-1},
\]
both already in \(\mathcal H_l\), gives
\[
B_t m_cB_s^{-1}\in\mathcal H_l\qquad(c\ne0).
\]
At \(c=1\) this gives \(B_sB_t^{-1}\in\mathcal H_l\). Composing again
gives all conjugated multiplications \(B_t m_cB_t^{-1}\). This proves
\eqref{eq:field-affine}. Finally, two distinct affine transformations
cannot agree outside a finite set; conjugating by a bijection preserves
this property. Thus the resulting homomorphism is injective.
\end{proof}
We have obtained an affine group action by perfect rational transformations.
Its translation subgroup will identify the addition on the constants.
The two-variable function \(R_1\) is then needed to show that the resulting
embedding of constant fields is onto.
\begin{proposition}[Alignment of the constants]\label{prop:constant-alignment}
The fields \(k\) and \(l\) have the same characteristic. There is a field
isomorphism \(\sigma:k\longrightarrow l\) such that, for every good function
\(t\), there is \(H_t\in\mathcal H_l\) with
\begin{equation}\label{eq:field-alignment}
B_t=H_t\circ\sigma
\qquad\text{outside a finite subset of }\Pj^1(k),
\end{equation}
where \(\sigma(\infty)=\infty\).
\end{proposition}
\begin{proof}
Choose a good function \(t\), and use Lemma~\ref{lem:common-partner} to
choose a good \(s\) satisfying the hypotheses of
Lemma~\ref{lem:field-affine}. First consider the embedding
\(B_t\Aff(k)B_t^{-1}\subset\mathcal H_l\), with the finite-agreement
convention of that lemma.
\emph{The affine image is projective linear.}
In positive target characteristic the Frobenius exponent defines a
homomorphism \(\mathcal H_l\to\Z\). Its restriction to the affine image is
zero: \(k^\times\) is divisible, while the additive group of \(k\) is
divisible in characteristic zero and torsion in positive characteristic.
Both groups therefore have zero image in \(\Z\), and they generate
\(\Aff(k)\). In characteristic zero for \(l\), there is no exponent to
consider. We have in either case an embedding into \(\PGL_2(l)\).
\emph{Translations yield a field embedding.}
Let \(U\) be the image of the translation subgroup. This is an infinite
abelian subgroup of \(\PGL_2(l)\), normalized by the whole affine image.
It cannot contain a nonidentity semisimple element. To see this, such an
element has two fixed points, and every element of \(U\) preserves their
pair. The subgroup fixing both points has index at most two in \(U\);
it is infinite, hence contains an element of order greater than two.
The centralizer of that element is the torus fixing the two points, so
all of \(U\) lies in that torus. Any transformation normalizing \(U\)
preserves the same pair and acts on \(U\) by either identity or inversion.
This contradicts the faithful conjugation action of \(k^\times\) on its
additive group in \(\Aff(k)\).
Every nonidentity element of \(U\) is consequently unipotent. Conjugate
the image in \(\PGL_2(l)\) so that one such element is a translation with
unique fixed point \(\infty\). Its centralizer consists of translations,
and hence so does \(U\). Write this conjugation as postcomposition
\(B=MB_t\), with \(M\in\PGL_2(l)\). We obtain an injective additive map
\(\chi:k\to l\) for which
\begin{equation}\label{eq:field-translation}
BT_bB^{-1}=T_{\chi(b)}
\qquad\text{modulo finite disagreement}.
\end{equation}
After rescaling the target coordinate, assume \(\chi(1)=1\).
The image of \(m_c\) normalizes \(U\), hence fixes \(\infty\), and is an
affine transformation of some slope \(\lambda_c\).
Conjugating translations gives
\[
\chi(cb)=\lambda_c\chi(b).
\]
Setting \(b=1\) gives \(\lambda_c=\chi(c)\). Thus \(\chi\) is
multiplicative as well as additive and is a field embedding. In
particular, the characteristics of \(k\) and \(l\) agree.
\emph{The field embedding is onto.}
Set \(z=t+s\). Lemma~\ref{lem:field-affine} gives members
\(M_s,M_z\in\mathcal H_l\) such that
\[
B_s=M_sB,\qquad B_z=M_zB
\qquad\text{modulo finite disagreement}.
\]
Using \(R_1\) from \eqref{eq:field-sum}, define the perfect rational function
\[
F(X,Y)=M_z^{-1}\bigl(R_1(M^{-1}X,M_sY)\bigr).
\]
For cofinitely many \(b\in k\), and for each such \(b\) for cofinitely many
\(X\in l\), equations \eqref{eq:field-sum} and
\eqref{eq:field-translation} give
\begin{equation}\label{eq:field-slice}
F(X,B(b))=B\bigl(B^{-1}(X)+b\bigr)=X+\chi(b).
\end{equation}
Here is the order of the exclusions. First exclude the finitely many
values of \(b\) at which \(B(b)\) is infinite, the equality \(B_s=M_sB\)
fails, or the corresponding vertical fiber misses the open of
\eqref{eq:field-sum}. For each remaining \(b\), that open excludes only
finitely many \(X\); the equality \(B_z=M_zB\) excludes finitely many more,
since its argument is a bijective function of \(X\). Finally
\eqref{eq:field-translation} excludes a finite set of \(X\), which may
depend on \(b\).
For each fixed remaining \(b\), \eqref{eq:field-slice} is therefore an
identity of perfect rational functions in \(X\).
Choose \(X_0\in l\) such that \(F(X_0,Y)\) is defined and finite for general
\(Y\). This choice is possible by expressing a power of \(F\) as a
quotient of polynomials and choosing \(X_0\) for which its denominator
does not vanish identically as a polynomial in \(Y\).
Define
\[
h(Y)=F(X_0,Y)-X_0.
\]
For all but finitely many of the remaining \(b\), the denominator is
nonzero at \(Y=B(b)\); evaluating the rational identity then gives
\[
\chi(b)=h(B(b)).
\]
In particular, this evaluation does not require \(X_0\) to avoid the
original exceptional sets for all \(b\) at once.
The injectivity of \(\chi\) implies that \(h\) is injective on a cofinite
subset of \(\Pj^1(l)\). By Lemma~\ref{lem:cofinite-rational}, it
represents a member of \(\mathcal H_l\), so \(\chi(k)\) contains a
cofinite subset of \(l\). An infinite proper subfield cannot be
cofinite: if \(a\notin\chi(k)\), the infinite coset \(a+\chi(k)\) is
disjoint from it. Thus \(\chi(k)=l\). Taking \(\sigma=\chi\), the
equality \(\sigma=hB=hMB_t\) outside a finite set proves
\eqref{eq:field-alignment} for our initial \(t\).
\emph{The same constant isomorphism works for every good function.}
For any other good function \(r\), Lemma~\ref{lem:common-partner}
supplies one good \(s_0\) that is algebraically independent of both
\(t\) and \(r\), with \(t+cs_0\) and \(r+cs_0\) good for every \(c\ne0\).
Lemma~\ref{lem:field-affine} gives
\(B_{s_0}B_t^{-1},B_{s_0}B_r^{-1}\in\mathcal H_l\), hence
\(B_rB_t^{-1}\in\mathcal H_l\). The alignment already obtained for \(t\)
therefore gives \eqref{eq:field-alignment} for \(r\), with the same
\(\sigma\).
\end{proof}
\subsection{The isomorphism of perfect closures}
Fix \(\sigma\) and the \(H_t\) given by
Proposition~\ref{prop:constant-alignment}, and define
\[
u_t=H_t^{-1}(t')\in L^i
\qquad\text{for every good function }t.
\]
These are field elements, obtained by projective linear operations and
Frobenius powers. They satisfy
\begin{equation}\label{eq:field-good-perfect}
l(u_t)^i=l(t')^i.
\end{equation}
We show that the assignments \(t\mapsto u_t\) preserve every algebraic
relation. This is the point where the finite exceptions in the individual
coordinate alignments cease to matter.
\begin{theorem}\label{thm:field-isomorphism}
There is a field isomorphism
\[
\alpha:K^i\xrightarrow{\ \sim\ }L^i,\qquad \alpha|_k=\sigma,
\]
such that \(\alpha(t)=u_t\) for every good function \(t\).
In particular, if \(k(t)\) and \(l(t')\) are matched good rational
subfields, then \(\alpha(k(t)^i)=l(t')^i\).
\end{theorem}
\begin{proof}
Let \(\mathbf t=(t_1,\ldots,t_m)\) be a finite tuple of good functions,
and let \(Z_{\mathbf t}\subset(\Pj^1_l)^m\) be the relation variety of
\((u_{t_1},\ldots,u_{t_m})\). We first prove
\begin{equation}\label{eq:field-relations}
\sigma\bigl(X_{\mathbf t}(k)\bigr)=Z_{\mathbf t}(l),
\end{equation}
where \(\sigma\) acts coordinatewise.
The point map
\[
\mathcal H=\prod_i H_{t_i}^{-1}:
(\Pj^1(l))^m\longrightarrow(\Pj^1(l))^m
\]
is a Zariski homeomorphism. Projective linear maps are isomorphisms; in
positive characteristic the coordinate map
\([X:Y]\mapsto[X^p:Y^p]\) is finite, radicial and surjective, hence a
homeomorphism on point spaces, as is its inverse.
Products with different powers in different coordinates have the same
property. It follows that
\[
\mathcal H\bigl(Y_{\mathbf t}(l)\bigr)=Z_{\mathbf t}(l).
\]
To justify this last equality at the level of relation varieties, place
the finite tuple of \(u_{t_i}\) in one finite purely inseparable extension
of \(L\). On a common model, the tuples \((t_i')_i\) and
\((u_{t_i})_i\) are related by \(\mathcal H\) wherever they are defined.
Their images are dense in their respective relation varieties.
Taking closures under the homeomorphism gives the equality.
Apply now the first inclusion of
Lemma~\ref{lem:simultaneous-values}. Delete from its source open the
finitely many exceptional values in each coordinate where
\eqref{eq:field-alignment} fails. Each deleted coordinate fiber is a
proper closed subset, since every \(t_i\) is nonconstant. The resulting
open is still dense, and on it
\(\mathcal H\circ\prod_iB_{t_i}=\sigma\).
Consequently a dense subset of \(\sigma(X_{\mathbf t}(k))\) lies in
\(Z_{\mathbf t}(l)\). The former is the point set of the variety obtained
from \(X_{\mathbf t}\) by applying \(\sigma\) to coefficients, so taking
closures gives
\[
\sigma(X_{\mathbf t}(k))\subset Z_{\mathbf t}(l).
\]
For the reverse inclusion, use the second inclusion of
Lemma~\ref{lem:simultaneous-values}. Delete the finitely many target
coordinate values at which
\(B_{t_i}^{-1}=\sigma^{-1}H_{t_i}^{-1}\) fails. Again this leaves a
dense open, now in \(Y_{\mathbf t}\), because each target coordinate is
nonconstant. Its image under \(\mathcal H\) is dense in
\(Z_{\mathbf t}\) and lies in \(\sigma(X_{\mathbf t}(k))\).
Taking closures proves \eqref{eq:field-relations}.
Thus, for every polynomial \(P\in k[T_1,\ldots,T_m]\),
\[
P(t_1,\ldots,t_m)=0
\quad\Longleftrightarrow\quad
\sigma(P)(u_{t_1},\ldots,u_{t_m})=0,
\]
where \(\sigma(P)\) is obtained by applying \(\sigma\) to the coefficients.
Good functions generate \(K\) by Lemma~\ref{lem:good-functions}.
The displayed equivalence for every finite tuple makes the assignments
\(a\mapsto\sigma(a)\) for \(a\in k\) and \(t\mapsto u_t\) for good \(t\)
well defined on all polynomial expressions, and preserves their nonzero
values. Passing to quotients therefore gives an embedding
\(\alpha_0:K\hookrightarrow L^i\).
Because \(L^i\) is perfect, this embedding extends uniquely to an embedding
\(\alpha:K^i\hookrightarrow L^i\). Its image contains \(l=\sigma(k)\)
and every \(u_t\), and is itself perfect; by
\eqref{eq:field-good-perfect} it contains the perfection of every matched
good rational subfield of \(L\).
All good rational subfields are matched, and their generators generate
\(L\). The image of \(\alpha\) therefore contains \(L^i\), proving
surjectivity and the final assertion.
\end{proof}
==== END Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory//sections/07-field.tex ====
==== BEGIN Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory//sections/08-compatibility.tex ====
\section{Compatibility and uniqueness}\label{sec:compatibility}
Theorem~\ref{thm:field-isomorphism} gives an isomorphism of perfect fields
that carries the perfection of every good rational subfield to the
perfection of its match under $\Theta$. We must prove that its action on all multiplicative classes is
$\Theta$ up to a single scalar. We then determine which field
automorphisms act by scalars.
\subsection{Distinguishing divisors by rational subfields}
For a true prime divisor $v$ and a good rational subfield $E\subset K$,
consider the restriction $D_v\to A_E$. Its rank here means the dimension
of its image as a $\Lambda$-vector space.
\begin{lemma}\label{lem:restriction-rank}
Let $v$ be a true prime divisor of $K/k$ and let $E\subset K$ be a curve
subfield. The restriction $v|_E$ is trivial if and only if the image of
$D_v\to A_E$ is infinite-dimensional. If $v|_E$ is nontrivial, this image
has dimension at most one.
\end{lemma}
\begin{proof}
If $v|_E$ is trivial, reduction embeds $E$ in $Kv$. The kernel of
$V_E\to V_{Kv}$ is finite-dimensional by Lemma~\ref{lem:kummer-kernel},
whereas $V_E$ is infinite-dimensional. Dualizing and using
Lemma~\ref{lem:residue-duality} shows that the restriction of $D_v$ to
$A_E$ has infinite-dimensional image.
Otherwise $v|_E$ is a positive multiple of the order at a point $x$ of
the smooth projective curve of $E/k$. If $f$ is a unit there, its residue
is some $c\in k^\times$, and $f/c$ is a principal $v$-unit in $K$.
Every character in $D_v$ therefore kills $f$. Its restriction to $E$
factors through the value group of $v|_E$, which is cyclic. The image
has dimension at most one. This argument uses decomposition characters;
it remains valid if the ramification multiplicity is divisible by $\ell$.
\end{proof}
\begin{lemma}\label{lem:separate-divisors}
For distinct true prime divisors $v_1,v_2$ of $K/k$, there is a good
rational subfield $E\subset K$ such that $v_1|_E$ is trivial and
$v_2|_E$ is nontrivial.
\end{lemma}
\begin{proof}
Choose a normal projective model $X$ on which both valuations have
codimension-one centers $D_1,D_2$. We recall why this is possible.
For each valuation, lifts of a residue transcendence basis, together
with a uniformizer, define a rational subfield over which $K$ is finite.
Normalization of a suitable projective model of that subfield realizes
the valuation as a prime divisor. A common normal projective model
dominating the two models retains codimension-one centers: the residue
field of each new center contains that of the old center, of
transcendence degree $d-1$ over $k$.
Choose two distinct closed points $P,Q$ of $D_1\setminus D_2$, and a
smooth closed point $R$ of $X\setminus(D_1\cup D_2)$. The first choice
is possible because $d\ge2$. There is an affine open containing these
points and meeting $D_2$; write its coordinate ring as $A$, and let
$\mathfrak p_2$ be the prime ideal of $D_2$ there. Choose two independent
cotangent vectors at $R$. The Chinese remainder theorem, applied to
$\mathfrak p_2,\mathfrak m_P,\mathfrak m_Q,\mathfrak m_R^2$, produces
$a,b\in A$ satisfying
\[
\begin{array}{c|ccc}
& D_2 & P & Q\\ \hline
a &0&0&1\\
b &1&1&1
\end{array}
\]
and having value zero and the prescribed independent first-order terms
at $R$. These ideals are pairwise comaximal, so the prescriptions are
independent. By Lemma~\ref{lem:good-functions}, $t=a/b$ is good.
Its residue on $D_1$ is nonconstant, since it takes different values
at $P,Q$, and its value along $D_2$ is positive. Thus $v_1$ is trivial
on $k(t)$ and $v_2$ is nontrivial there.
\end{proof}
\subsection{The scalar is global}
\begin{proposition}\label{prop:global-scalar}
Let $\Psi:V_K\to V_K$ be a compatible $\Lambda$-linear automorphism.
If $\Psi(V_E)=V_E$ for every good rational subfield $E\subset K$,
then $\Psi=a\,\mathrm{id}$ for one $a\in\Lambda^\times$.
\end{proposition}
\begin{proof}
By Proposition~\ref{prop:true-divisors}, $\Psi^*$ permutes the pairs
$I_v\subset D_v$ for true prime divisors. Since $\Psi$ fixes each
$V_E$, it preserves the rank of restriction of $D_v$ to $A_E$.
Lemmas~\ref{lem:restriction-rank} and~\ref{lem:separate-divisors}
therefore force this permutation of true divisors to be the identity.
Fix a good rational subfield $E$. Every point line of $A_E$ is the
nonzero restriction of some $I_v$. The induced dual automorphism
$\Psi_E^*=(\Psi|_{V_E})^*$ of $A_E$ therefore fixes every point line.
Write
\[
\Psi_E^*(\ord_x)=a_x\ord_x,\qquad a_x\in\Lambda^\times,
\quad x\in C_E(k),
\]
where $C_E$ denotes the smooth projective model of $E$. The sum of
these order characters is zero, and its coefficient families have
exactly the diagonal kernel by Lemma~\ref{lem:curve-orders}.
Continuity permits applying $\Psi_E^*$ to this sum, so all $a_x$ equal
one scalar $a_E$. Since $E$ is rational, its point orders separate
$V_E$, and hence $\Psi|_{V_E}=a_E\,\mathrm{id}$.
Let $E=k(t)$ and $F=k(s)$ be any two good rational subfields. On a
common smooth model open, $t,s$ are regular and both differentials
are nonzero. Blow up a closed point of that open. Its exceptional
divisor $v$ has
\[
\ord_v(t-t(P))=\ord_v(s-s(P))=1.
\]
Thus the inertia line $I_v$ restricts nontrivially to a point line
of both $A_E$ and $A_F$. The scalar by which $\Psi^*$ acts on $I_v$
must be both $a_E$ and $a_F$. All these scalars agree. Good-function
classes span $V_K$ by Lemma~\ref{lem:good-functions}, proving the claim.
\end{proof}
Apply the proposition to $\Psi=\alpha_1^{-1}\Theta$, where
$\alpha:K^i\to L^i$ is supplied by Theorem~\ref{thm:field-isomorphism}.
The canonical identifications under perfection are compatible with
Milnor multiplication. Also $\alpha(E^i)$ is the perfection of the
good field matched to $E$, so $\Psi$ fixes $V_E$ for every such $E$.
Consequently $\Theta=a\alpha_1$ for a single $a\in\Lambda^\times$.
\subsection{Uniqueness up to Frobenius}
\begin{proposition}\label{prop:uniqueness}
Suppose $\beta:K^i\to K^i$ is a field automorphism preserving $k$
whose action on $V_K$ is a scalar. Then $\beta$ is the identity in
characteristic zero and an integral power of Frobenius in positive
characteristic.
\end{proposition}
\begin{proof}
Let $E=k(t)$ be good. Its perfection is relatively algebraically
closed in $K^i$: if an element is algebraic over $E^i$, clearing
Frobenius powers puts an algebraic element over $E$ in $K$, and
relative algebraic closedness then puts it in $E$. The same applies
to $\beta(E^i)$. Choose a nonconstant element of $\beta(E^i)\cap K$
by clearing powers, and let $P$ be the relative algebraic closure
of the rational field it generates in $K$. Relative algebraic closedness
of $\beta(E^i)$ gives $P^i\subset\beta(E^i)$. Conversely, every element
of $\beta(E^i)$ is algebraic over this rational field; clearing powers
puts it in $P$. Hence $\beta(E^i)=P^i$. Since $\beta_1$ is scalar, $V_P=V_E$ inside $V_K$.
Lemma~\ref{lem:curve-intersection} forces $P=E$.
It follows that $\beta(t)=h(t)$ for a perfect rational function $h$
generating $k(t)^i$ over $k$ up to perfection. This implies that $h$
belongs to the group $\mathcal H_k$ of projective transformations and
Frobenius powers from Section~\ref{sec:field}. Indeed, after clearing
powers, its separable degree must be one; a larger separable degree
could not disappear on passing to perfections. In characteristic
zero this says simply that $h$ is projective linear.
Write $\rho=\beta|_k$. For any two distinct points $a,b\in\Pj^1(k)$,
choose a rational function $f_{a,b}(t)$ with divisor $[a]-[b]$.
The point-order vector of $\beta_1([f_{a,b}])$ is supported exactly at
\[
h^{-1}(\rho(a)),\qquad h^{-1}(\rho(b)).
\]
Here $\rho$ fixes infinity. For negative Frobenius powers, the order
vector is interpreted by clearing powers; multiplication by a power
of $p$ is invertible in $\Lambda$. Both displayed coordinates therefore
have nonzero orders. Scalar action on $V_E$ implies that the
permutation $h^{-1}\rho$ fixes every unordered pair of distinct
points. On a set with at least three points this forces every point
to be fixed: intersect the fixed pairs $\{a,b\}$ and $\{a,c\}$.
Thus $h$ and $\rho$ agree on all points.
In particular $h$ fixes $0,1,\infty$. An element of $\mathcal H_k$
with these three fixed points is a pure Frobenius power, or the
identity in characteristic zero. In positive characteristic its
exponent is determined by $\rho$ and therefore is the same for every
good $t$. Distinct Frobenius powers act differently on the infinite
algebraically closed field $k$: equality of two powers would force
every element to satisfy $X^{p^n}-X=0$ for some $n>0$.
Good functions generate $K$, so $\beta$ is this same Frobenius power
on $K$ and then on its perfection.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:main}]
A compatible $\Theta$ gives equal transcendence degrees by
Proposition~\ref{prop:local-dimension}. Theorem~\ref{thm:field-isomorphism}
produces a field isomorphism $\alpha:K^i\to L^i$ respecting constants;
in particular, the characteristics agree. Proposition~\ref{prop:global-scalar}
shows that $\Theta$ and $\alpha_1$ differ by a single scalar, proving
surjectivity of the stated map. If two field isomorphisms have the
same image modulo scalars, their quotient satisfies
Proposition~\ref{prop:uniqueness}, proving injectivity modulo Frobenius.
Conversely, every integral Frobenius power acts on $V_L$ by the
corresponding nonzero scalar. The map is therefore well-defined and
bijective with exactly the stated equivalence relations.
\end{proof}
==== END Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory//sections/08-compatibility.tex ====