# Spaces of norms, determinant of cohomology and Fekete points in non-
Archimedean geometry

Sébastien Boucksom and Dennis Eriksson  CNRS-CMLS
École Polytechnique
F-91128 Palaiseau Cedex
France<EMAIL_ADDRESS>Chalmers University of Technology
and University of Gothenburg
SE-412 96
Gothenburg, Sweden<EMAIL_ADDRESS>
###### Abstract.

We describe the leading asymptotics of the determinant of cohomology of large
powers of a metrized ample line bundle, on a projective Berkovich space over
any complete non-Archimedean field. As consequences, we obtain the existence
of transfinite diameters and equidistribution of Fekete points, as in previous
work of the first author with Berman and Witt Nyström in the complex setting.
Our approach relies on a version of the Knudsen-Mumford expansion for the
determinant of cohomology on models over the (possibly non-Noetherian)
valuation ring, as a replacement for the asymptotic expansion of Bergman
kernels in the complex case, and on the reduced fiber theorem, as a
replacement for the Bernstein-Markov inequalities. Along the way, a systematic
study of spaces of norms and the associated Fubini-Study type metrics is
undertaken.

###### Contents

  1. I Spaces of norms and determinants
    1. 1 Spaces of norms
    2. 2 Determinants and successive minima
    3. 3 Alternative metric structures on spaces of norms
  2. II Models and metrics
    1. 4 Analytification and models
    2. 5 Metrics
    3. 6 Limits of Fubini-Study metrics
  3. III Asymptotics of relative volumes
    1. 7 Monge-Ampère measures and Deligne pairings
    2. 8 Asymptotics of relative volumes
    3. 9 Transfinite diameter and Fekete points
    4. A Determinant of cohomology and Deligne pairings

### Introduction

Fekete points and transfinite diameter are classical notions in logarithmic
potential theory in the plane. For each $m\geq 1$, the _$m$ -diameter_
$\delta_{m}(K)$ of a compact subset $K\subset\mathbb{C}$ is defined as the
supremum of the geometric mean distance between $m+1$ points in $K$,
maximizers being called _Fekete configurations_. The $m$-diameter of $K$
admits a limit $\delta_{\infty}(K)$ as $m\to\infty$, called the _transfinite
diameter_ of $K$, which is proved to coincide with the logarithmic capacity of
$K$. Fekete configurations are also known to become asymptotically unique in
the limit, in the sense that they equidistribute to a certain probability
measure known as the _equilibrium measure_ of $K$.

In the higher dimensional case, a similar understanding was only rather
recently obtained. The first steps were taken by Leja in the 1950’s,
introducing a notion of $m$-diameter $\delta_{m}(K)$ for a compact subset
$K\subset\mathbb{C}^{n}$ in terms of the supremum of certain Vandermonde-type
determinants. The existence of the transfinite diameter
$\delta_{\infty}(K)=\lim_{m\to\infty}\delta_{m}(K)$ was established by
Zaharjuta in the 1970’s, and the next key step came with Rumely’s observation
in [Rum07] that the general results in arithmetic intersection theory
developed in [CLR03] yield in particular an exact formula for
$\delta_{\infty}(K)$ in terms of pluripotential theory
(semipositive/plurisubharmonic envelopes and mixed Monge-Ampère operators in
the sense of Bedford-Taylor). This triggered joint work of the first author
with Berman and Witt Nyström [BB10, BBW11], which built on Bergman kernel
asymptotics to establish a general version of Rumely’s formula in the setting
of polarized projective manifolds, and combined it with a variational argument
to prove the equidistribution of Fekete configurations in this context.

The main purpose of the present paper is to study versions of these results in
non-Archimedean (Berkovich) geometry. While many results hold over an
arbitrary non-Archimedean complete valued field $K$, the full picture relies
on more refined non-Archimedean pluripotiential theory as developed in [BFJ16,
BFJ15, BG+16, BJ18], and hence requires $K$ to be trivially or discretely
valued and of residue characteristic $0$.

#### Asymptotics of relative volumes

The Bouche-Catlin-Tian-Zelditch asymptotic expansion of Bergman kernels is a
fundamental result in complex geometry describing the asymptotic behavior of
the $L^{2}$-norms associated to large tensor powers of a positive Hermitian
line bundle. As noticed in [BB10], it can be reformulated as an asymptotic
expansion for the logarithmic volume ratio of such $L^{2}$-norms. More
specifically, define the _relative volume_ of two norms
$\|\cdot\|,\|\cdot\|^{\prime}$ on an $N$-dimensional complex vector space $V$
as

$\operatorname{vol}(\|\cdot\|,\|\cdot\|^{\prime}):=\log\left(\frac{\det\|\cdot\|^{\prime}}{\det\|\cdot\|}\right),$
(0.1)

where $\det\|\cdot\|$ denotes the induced norm on the determinant line $\det
V$. In terms of the unit balls $B,B^{\prime}\subset V$ of the two norms,
$\operatorname{vol}(\|\cdot\|,\|\cdot\|^{\prime})$ coincides with the
logarithmic volume ratio

$\tfrac{1}{2}\log\left(\frac{\operatorname{vol}B}{\operatorname{vol}B^{\prime}}\right),$

up to an error term $O(N\log N)$ that vanishes when the two norms are
Hermitian. Let $X$ be a smooth, $n$-dimensional complex projective variety
endowed with an ample bundle $L$. Any two smooth positive metrics $\phi,\psi$
on $L$ induce for each $m\in\mathbb{N}$ $L^{2}$-norms
$\|\cdot\|_{L^{2}(m\phi)},\|\cdot\|_{L^{2}(m\psi)}$ on the space of sections
$H^{0}(mL)=H^{0}(X,L^{\otimes m})$, and the asymptotic expansion of Bergman
kernels turns out to be equivalent to the existence of a full asymptotic
expansion

$\frac{1}{mN_{m}}\operatorname{vol}\left(\|\cdot\|_{L^{2}(m\phi)},\|\cdot\|_{L^{2}(m\psi)}\right)=a_{0}+m^{-1}a_{1}+\dots+O(m^{-\infty}),$

with

$N_{m}:=\dim H^{0}(mL)=\frac{m^{n}}{n!}V+O(m^{n-1}).$

Here $V=(L^{n})$ now denotes the degree of $L$. The leading order term $a_{0}$
can further be identified with a fundamental functional in Kähler geometry,
the _relative Monge-Ampère energy_

$E(\phi,\psi):=\frac{1}{n+1}\sum_{j=0}^{n}V^{-1}\int_{X}(\phi-\psi)(dd^{c}\phi)^{j}\wedge(dd^{c}\psi)^{n-j}.$
(0.2)

We use additive notation for metrics on line bundles, so that $\phi-\psi$ is a
function on $X$, and $dd^{c}\phi$, $dd^{c}\psi$ denote the curvature
$(1,1)$-forms of $\phi,\psi$.

If $\phi,\psi$ are now arbitrary continuous metrics on $L$, denote by
$\|\cdot\|_{m\phi}$, $\|\cdot\|_{m\psi}$ the sup-norms they define on
$H^{0}(mL)$. Combining the previous result with a regularization argument and
a growth estimate for the distortion between sup-norms and $L^{2}$-norms
(Bernstein-Markov inequality), it was proved in [BB10] that

$\frac{1}{mN_{m}}\operatorname{vol}\left(\|\cdot\|_{m\phi},\|\cdot\|_{m\psi}\right)\to
E\left(P(\phi),P(\psi)\right).$ (0.3)

Here $P(\phi),P(\psi)$ are the _semipositive envelopes_ of $\phi,\psi$, whose
relative Monge-Ampère energy $E(P(\phi),P(\psi))$ can be defined as in (0.2)
using the Bedford-Taylor theory of mixed Monge-Ampère operators.

The main result of the present paper is the following non-Archimedean
analogue, generalizing in particular [BG+16, Theorem A] (which deals with the
discretely valued case).

######  Theorem A.

Let $X$ be a smooth projective variety over a non-Archimedean complete valued
field $K$, and $L$ be an ample line bundle on $X$. For any two continuous
metrics $\phi,\psi$ on (the Berkovich analytification of) $L$, the scaled
relative volumes of the induced sup-norms $\|\cdot\|_{m\phi}$,
$\|\cdot\|_{m\psi}$ on $H^{0}(mL)$ admit a finite limit

$\lim_{m\to\infty}\frac{1}{mN_{m}}\operatorname{vol}(\|\cdot\|_{m\phi},\|\cdot\|_{m\psi})\in\mathbb{R}.$

If the semipositive envelopes $P(\phi),P(\psi)$ are further continuous, then
this limit coincides with the Monge-Ampère energy $E(P(\phi),P(\psi))$.

Continuity of $P(\phi),P(\psi)$ is expected to be always true. Besides the
trivial case where $\phi,\psi$ are already semipositive, it is known to hold
when $K$ is trivially or discretely valued of residue characteristic $0$
[BFJ16, BJ18] (see also [GJKM17] for partial results when $K$ is discretely
valued of equal positive characteristic). The relative volume of two norms on
a $K$-vector space and the Monge-Ampère energy $E$ are still respectively
defined by (0.1) and (0.2), the latter being this time understood in the sense
of Chamber-Loir and Ducros [CLD12].

#### Sketch of the proof

The main tools involved in the proof of Theorem A can be summarized as
follows.

##### Spaces of norms and determinants

The existence of the limit in Theorem A is closely related to [CMac15, WN14].
We deduce it from the filtered linear series technique introduced in [BC],
itself based on Newton-Okounkov bodies. This relies on an appropriate version
of Minkowski’s second theorem, expressing the relative volume of two norms as
a sum of relative successive minima, up to a negligible error term. This
material is included in a general study of spaces of norms and determinants,
which forms the first part of the present paper.

##### Fubini-Study metrics

The semipositive envelope $P(\phi)$ of a continuous metric $\phi$ appearing in
Theorem A is defined as the limit of the Fubini-Study metrics on $L$ induced
by the sup-norms $\|\cdot\|_{m\phi}$. This gives rise to a systematic study of
Fubini-Study metrics and their limits, partly revisiting with a slightly
different perspective [CMor15], and forming the second part of the paper.

##### The reduced fiber theorem

With these preliminary tools in hand, the proof of Theorem A is easily reduced
to the case where each metric is induced by an ample model
$(\mathcal{X},\mathcal{L})$ of $(X,L)$, i.e. an ample line bundle
$\mathcal{L}$ extending $L$ to a projective model $\mathcal{X}$ of $X$ over
the valuation ring $K^{\circ}$. Besides the sup-norm $\|\cdot\|_{m\phi}$
defined by the model metric $\phi=\phi_{\mathcal{L}}$, the space of sections
$H^{0}(mL)$ is then also equipped with the lattice norm
$\|\cdot\|_{H^{0}(m\mathcal{L})}$ induced by the $K^{\circ}$-module
$H^{0}(m\mathcal{L})=H^{0}(\mathcal{X},\mathcal{L}^{\otimes m})$. This lattice
norm, which is to some extent the analogue of the $L^{2}$-norm in the present
non-Archimedean context, coincides with the sup-norm when $\mathcal{X}$ has
reduced special fiber, but not in general. Using the Bosch-Lütkebohmert-
Raynaud reduced fiber theorem [BLR95], we prove however that the distortion
between $\|\cdot\|_{m\phi}$ and $\|\cdot\|_{H^{0}(m\mathcal{L})}$ remains
bounded as $m\to\infty$, which enables us to replace the sup-norms with the
lattice norms in proving Theorem A.

##### Knudsen-Mumford expansion

Our main tool is then the Knudsen-Mumford expansion of the determinant of
cohomology [KM76], which plays the role of the asymptotic expansion of Bergman
kernels in the complex case and provides for $m\gg 1$ a polynomial expansion
(in additive notation for $\mathbb{Q}$-line bundles over
$\operatorname{Spec}K^{\circ}$)

$\det
H^{0}(m\mathcal{L})=\frac{m^{n+1}}{(n+1)!}\langle\mathcal{L}^{n+1}\rangle+\dots$
(0.4)

with leading order term the Deligne pairing $\langle\mathcal{L}^{n+1}\rangle$.
Since models over $K^{\circ}$ are non-Noetherian in the densely valued case,
some care is however required to apply this result, and the relevant
explanations are provided in Appendix A, based on Ducrot’s approach [Duc05].

##### Metrics on Deligne pairings

The Knudsen-Mumford expansion yields an expression of the limit of the scaled
relative volume as a difference of model metrics on the Deligne pairing
$\langle L^{n+1}\rangle$. The final ingredient in the proof of Theorem A
consists in relating model metrics on Deligne pairings with mixed Monge-Ampère
integrals, which is accomplished building on the Poincaré-Lelong formula of
[CLD12] and a careful monotone regularization argument.

#### Transfinite diameter and Fekete points

Following the strategy developed in [BB10, BBW11], we use Theorem A to show
the existence of transfinite diameters, and combine it with a
differentiability result proved in [BFJ15, BG+16, BJ18] under appropriate
assumptions on the ground field $K$ to infer equidistribution of Fekete
points.

For simplicity, we only consider the case of a compact set equal to the whole
of $X^{\mathrm{an}}$. The $m$-diameter of a continuous metric $\phi$ on $L$ is
to be defined as the sup-norm, with respect to the induced metric, of the
Vandermonde determinant

$\det{\bf s}\in H^{0}\left(X^{N_{m}},(mL)^{\boxtimes N_{m}}\right)$

associated to a basis ${\bf s}=(s_{i})$ of $H^{0}(mL)$, defined by

$(\det{\bf s})(x_{1},\dots,x_{N_{m}})=\det\left(s_{i}(x_{j})\right).$

This requires however to normalize the basis in some way, which can be done
using a reference norm $\|\cdot\|_{\mathrm{ref}}$ on $H^{0}(mL)$. We thus
define define the _$m$ -diameter of $\phi$ normalized by
$\|\cdot\|_{\mathrm{ref}}$_ as

$\delta_{m,\|\cdot\|_{\mathrm{ref}}}(\phi):=\left(\frac{\|\det{\bf
s}\|_{(m\phi)^{\boxtimes N_{m}}}}{\det\|s_{1}\wedge\dots\wedge
s_{N_{m}}\|_{\mathrm{ref}}}\right)^{1/mN_{m}}$

for any choice of basis ${\bf s}=(s_{i})$ of $H^{0}(mL)$.

######  Theorem B.

Let $\phi,\psi$ be two continuous metrics on $L$, and $(\|\cdot\|_{m})$ be any
sequence of reference norms on the spaces $H^{0}(mL)$, with subexponential
distortion with respect to the sequence of sup-norms $\|\cdot\|_{m\psi}$. Then

$\delta_{\infty,\psi}(\phi):=\lim_{m\to\infty}\delta_{m,\|\cdot\|_{m}}(\phi)$

exists and is independent of the choice of $(\|\cdot\|_{m})$. We further have

$-\log\delta_{\infty,\psi}(\phi)=\lim_{m\to\infty}\frac{1}{mN_{m}}\operatorname{vol}\left(\|\cdot\|_{m\phi},\|\cdot\|_{m\psi}\right).$

This is inferred from Theorem A via an estimate for the operator norms of the
embeddings $\det H^{0}(mL)\hookrightarrow H^{0}\left((mL)^{\boxtimes
N_{m}}\right)$ with respect to the sup-norms induced by $\phi$ on both sides,
which is again ultimately deduced from the reduced fiber theorem.

An _$m$ -Fekete configuration_ for $\phi$ is a point
$P\in(X^{N_{m}})^{\mathrm{an}}$ that achieves the sup-norm of $\det{\bf s}$
with respect to the induced metric, for some (hence any) choice of basis ${\bf
s}$ of $H^{0}(mL)$.

######  Theorem C.

Assume that $K$ is trivially or discretely valued, of residue characteristic
$0$. Let $\phi$ be a continuous metric, and pick for each $m$ an $m$-Fekete
configuration $P_{m}\in(X^{N_{m}})^{\mathrm{an}}$ for $\phi$. Then $P_{m}$
equidistributes to the Monge-Ampère measure
$\operatorname{MA}(P(\phi))=V^{-1}(dd^{c}P(\phi))^{n}$ as $m\to\infty$.

Indeed, the assumption on $K$ guarantees that the semipositive envelope
$P(\phi)$ of any continuous metric $\phi$ is itself continuous [BFJ16, BJ18],
and that $E(P(\phi),P(\psi))$ is differentiable with respect to $\phi$ [BFJ15,
BG+16, BJ18]. By Theorems A and B, $\log\delta_{\infty,\psi}(\phi)$ is thus
differentiable with respect to $\phi$, and we then simply import the
variational argument of [BBW11], itself based on an idea of [SUZ97].

#### Organization of the paper

The paper is organized as follows:

  * •

Section 1 contains background material on norms on finite dimensional vector
spaces over a complete valued field. We present the results in the Archimedean
and non-Archimedean cases as uniformly as possible.

  * •

In Section 2 we discuss determinants of norms and their relationship to
successive minima.

  * •

Section 3, which stands somewhat apart from the rest of paper, applies the
previous result to construct metrics on spaces of norms, following [Ger81].

  * •

Sections 4 and 5 contain background material on Berkovich spaces and metrics
on line bundles. We recall the standard constructions of model metrics, and
compare them to Fubini-Study metrics. We discuss the relation between the
reduced fiber theorem and finiteness of integral closures, and infer our first
key tool, to wit boundedness of the distortion between the sup-norms and
lattice norms induced by a model.

  * •

In Section 6 we study limits of Fubini-Study metrics, compare them to Zhang’s
definition of semipositive metrics, and discuss the related notion of
semipositive envelope.

  * •

In Section 7 we review the Bedford-Taylor/Chambert-Loir-Ducros mixed Monge-
Ampère operators, and relate them to Deligne pairings.

  * •

Section 8 contains the proof of Theorem A.

  * •

Section 9 shows the existence of transfinite diameters and equidistribution of
Fekete points, i.e. Theorem B and Theorem C. We also show how the results can
be applied in the case of toric varieties.

  * •

In the appendix we explain how Ducrot [Duc05] approach to the Knudsen-Mumford
expansion for the determinant of cohomology and the related notion of Deligne
pairings can be extended from the usual noetherian case to schemes of finite
presentation over the valuation ring of a complete non-Archimedean field.

#### Acknowledgement

We are grateful for many helpful discussions and remarks from many colleagues.
In particular, we thank Robert Berman, Dario Cordero-Erausquin, Antoine
Ducros, Gerard Freixas, Gilles Godefroy, Walter Gubler, Mattias Jonsson, Klaus
Künnemann, Marco Maculan, Florent Martin, David Rydh. The first author was
partially supported by the ANR project GRACK. The second author wants to thank
the Max-Planck Institut in Bonn, where part of this work was effectuated, for
their excellent working conditions and hospitality.

##  Part I Spaces of norms and determinants

###  1\. Spaces of norms

The goal of this section is to review some basic material on finite
dimensional normed vector spaces, treating in parallel the Archimedean and
non-Archimedean cases (including the trivially valued case). All the results
are well-known, but our proof of density of diagonalizable norms among
ultrametric norms (Theorem 1.19) appears to be new.

####  1.1. Complete valued fields

Here and throughout the article, $K$ denotes a field, complete with respect to
a (possibly trivial) absolute value $|\cdot|:K\to\mathbb{R}$. The value group
$|K^{*}|$ is a subgroup of $\mathbb{R}_{+}^{*}$, which is thus either discrete
or dense. We sometimes use the _additive value group_
$\Gamma:=\log|K^{*}|\subset\mathbb{R}$.

Recall that $K$ is _Archimedean_ if, for each non-zero $\alpha\in K$, there
exists $n\in\mathbb{Z}$ with $|n\alpha|>1$. This implies that $K$ contains
$\mathbb{Q}$, and that the restriction of $|\cdot|$ to $\mathbb{Q}$ is
equivalent to the standard absolute $|\cdot|_{\infty}$, by Ostrowski’s
theorem. As a result, $K$ is a complete field extension of $\mathbb{R}$, and
hence $K=\mathbb{R}$ or $\mathbb{C}$, by the Gelfand-Mazur theorem.

Otherwise, $K$ is _non-Archimedean_ , which holds if and only if $|\cdot|$
satisfies the ultrametric inequality
$|\alpha+\alpha^{\prime}|\leq\max\\{|\alpha|,|\alpha^{\prime}|\\}$ for all
$\alpha,\alpha^{\prime}\in K$. We then have a corresponding real-valued
valuation $v_{K}:=-\log|\cdot|$ on $K$, whose valuation ring $K^{\circ}$ is
thus the closed unit ball of $K$, with maximal ideal $K^{\circ\circ}$ the open
unit ball and residue field $\widetilde{K}:=K^{\circ}/K^{\circ\circ}$. Since
the value group is a subgroup of $\mathbb{R}$, the valuation ring $K^{\circ}$
is of Krull dimension at most $1$, and it is Noetherian if and only if
$|K^{*}|$ is discrete. In that case, $K^{\circ\circ}$ is a principal ideal; a
generator $\pi_{K}$ is called a uniformizing parameter, and is unique up to
multiplication by a unit. If $K$ is algebraically closed, then $\widetilde{K}$
is algebraically closed as well, and $|K^{*}|$ is divisible. In particular,
$K$ is then either trivially valued or densely valued. The completion of an
algebraic closure of any non-Archimedean field $K$ is denoted by
$\mathbb{C}_{K}$. It is the smallest complete algebraically closed extension
of $K$.

A field $K$ is _local_ if its unit ball $K^{\circ}$ is compact. This holds if
and only if $K$ is either Archimedean, or non-Archimedean with finite residue
field $\widetilde{K}$. In the latter case, $K$ is discretely valued, and is in
fact either finite and trivially valued, or isomorphic to a finite extension
of $\mathbb{Q}_{p}$ or $\mathbb{F}_{p}{(\\!(t)\\!)}$ (up to normalization of
the absolute value).

An _immediate extension_ of a non-Archimedean field $K$ is a complete field
extension $L/K$ such that $|L^{*}|=|K^{*}|$ and $\widetilde{L}=\widetilde{K}$,
and $K$ is _maximally complete_ if it admits no nontrivial immediate
extension. By [Kap42], maximally complete is equivalent to spherically
complete. A discretely valued field is maximally complete.

######  Example 1.1.

Let $k$ be an algebraically closed field of characteristic $0$, and endow the
field $K=k{(\\!(t)\\!)}$ of formal Laurent series with the $t$-adic valuation.
An algebraic closure of $K$ is given by the field of Puiseux series
$\bigcup_{n\geq 1}k{(\\!(t^{1/n})\\!)}$, and its completion $\mathbb{C}_{K}$
is realized as the field of formal power series
$f=\sum_{r\in\mathbb{Q}}a_{r}t^{r}$ with $a_{r}\in k$ whose support
$\operatorname{Supp}f=\\{r\in\mathbb{Q}\mid a_{r}\neq 0\\}$ contains only
finitely many elements with a given upper bound. The Malcev-Neumann field
$k{(\\!(t^{\mathbb{Q}})\\!)}$ of power series
$f=\sum_{r\in\mathbb{Q}}a_{r}t^{r}$ with well-ordered support is an immediate
extension of $\mathbb{C}_{K}$, with $f=\sum_{n\geq 1}t^{-1/n}$ in
$k{(\\!(t^{\mathbb{Q}})\\!)}\setminus\mathbb{C}_{K}$. As a consequence,
$\mathbb{C}_{K}$ is not maximally complete.

Similarly, the completion $\mathbb{C}_{p}$ of an algebraic closure of
$\mathbb{Q}_{p}$ is also not maximally complete. Denoting by $A$ the Witt ring
of $\overline{\mathbb{F}}_{p}$ (i.e. the valuation ring of the completion of
the maximal unramified extension of $\mathbb{Q}_{p}$), an immediate maximally
complete extension of $\mathbb{C}_{p}$ is obtained as the quotient of the
Malcev-Neumann ring $A{(\\!(t^{\mathbb{Q}})\\!)}$ by the ideal of formal power
series $f=\sum_{r\in\mathbb{Q}}a_{r}t^{r}$ such that
$\sum_{n\in\mathbb{Z}}a_{r+n}p^{n}=0$ in $\mathbb{Z}_{p}$ for all
$r\in\mathbb{Q}$, cf. [Poo93, §4].

####  1.2. The space of norms

Let $V$ be a fixed finite dimensional $K$-vector space, and set $N:=\dim V$.

######  Definition 1.2.

A _seminorm_ on $V$ is a function $\|\cdot\|:V\to\mathbb{R}_{+}$ such that

  * (i)

$\|\alpha v\|=|\alpha|\|v\|$ for all $\alpha\in K$, $v\in V$;

  * (ii)

$\|v+w\|\leq\|v\|+\|w\|$ for all $v,w\in V$;

It is a _norm_ if $\|v\|=0\Longleftrightarrow v=0$. A seminorm $\|\cdot\|$ is
_ultrametric_ if $\|v+w\|\leq\max\\{\|v\|,\|w\|\\}$. We denote by
$\mathcal{N}(V)$ the set of all norms on $V$, and by

$\mathcal{N}^{\mathrm{ultr}}(V)\subset\mathcal{N}(V)$

the set of ultrametric norms (which is non-empty if and only if $K$ is non-
Archimedean).

The group $\operatorname{GL}(V)$ acts on $\mathcal{N}(V)$ by composition, and
this action preserves $\mathcal{N}^{\mathrm{ultr}}(V)$.

######  Remark 1.3.

When $K$ is non-Archimedean, ultrametric norms will of course be the relevant
ones, but allowing at first arbitrary norms yields a more uniform treatment of
the Archimedean and non-Archimedean cases.

######  Example 1.4.

If $K$ is Archimedean, mapping a norm $\|\cdot\|$ to its closed unit ball $B$
(centered at $0$) sets up a one-to-one correspondence between $\mathcal{N}(V)$
and the set of all convex bodies of $V$ that are centrally symmetric when
$K=\mathbb{R}$, and $S^{1}$-invariant when $K=\mathbb{C}$. The inverse map is
obtained by setting

$\|v\|=\inf\left\\{r\geq 0\mid v\in rB\right\\}.$

######  Example 1.5.

If $K$ is trivially valued, the closed balls $B_{r}$, of radius $r$, of an
ultrametric norm form an increasing filtration of $V$ by linear subspaces,
which is exhaustive ($B_{r}=V$ for $r\gg 1$), separating ($B_{r}=\\{0\\}$ for
$r\ll 1$), and right-continuous
($B_{r}=\bigcap_{r^{\prime}>r}B_{r^{\prime}}$). Conversely, any such
filtration defines an ultrametric norm by setting

$\|v\|=\inf\\{r\geq 0\mid v\in B_{r}\\}.$

In other words, the data of an ultrametric norm with respect to the trivial
absolute value is equivalent to that of an increasing flag
$\\{0\\}=V_{0}\subset V_{1}\subset\dots\subset V_{n}=V$ of linear subspaces,
together with a increasing sequence $0=r_{0}<r_{1}<\dots<r_{n}$.

Equivalence of norms over $\mathbb{R}$ and $\mathbb{C}$ is usually established
as a consequence of the compactness of the unit cube. Crucially, equivalence
of norms still holds over any complete valued field.

######  Proposition 1.6.

Any two norms $\|\cdot\|$, $\|\cdot\|^{\prime}$ on $V$ are equivalent, i.e.
there exists $C>0$ such that $C^{-1}\|\cdot\|\leq\|\cdot\|^{\prime}\leq
C\|\cdot\|$.

###### Proof.

For the convenience of the reader, we repeat the simple standard argument, in
order to show that it applies to the trivially valued case as well. Note first
that the result implies that $V$ is complete with respect to any norm
$\|\cdot\|$. Indeed, after choosing a basis $(e_{i})$ of $V$, $\|\cdot\|$ will
be equivalent to the $\ell^{\infty}$-norm $\|\cdot\|_{\infty}$ associated to
$(e_{i})$, which is complete since it is isometrically isomorphic to $K^{N}$.
We argue by induction on $N=\dim V$, the desired result being trivial for
$N=1$. We are going to show that any given norm $\|\cdot\|$ on $V$ is
equivalent to $\|\cdot\|_{\infty}$. For each subspace $W\neq V$, the
restriction of $\|\cdot\|$ to $W$ is complete, by induction. As a result, $W$
is closed with respect to $\|\cdot\|$, and hence $\inf_{w\in W}\|v+w\|>0$ for
each $v\in V\setminus W$. In particular,

$c_{i}:=\inf_{\alpha\in K^{N}}\left\|e_{i}+\sum_{j\neq
i}\alpha_{j}e_{j}\right\|>0$

for all $i$. For each $\alpha\in K^{N}$ and each $i$ with $\alpha_{i}\neq 0$,
we get

$\left\|\sum_{j}\alpha_{j}e_{j}\right\|=|\alpha_{i}|\left\|e_{i}+\sum_{j\neq
i}\frac{\alpha_{j}}{\alpha_{i}}e_{j}\right\|\geq c_{i}|\alpha_{i}|,$

and hence
$\left\|\sum_{j}\alpha_{j}e_{j}\right\|\geq(\min_{j}c_{j})\max_{j}|\alpha_{j}|$.
By the triangle inequality, we also have
$\left\|\sum_{j}\alpha_{j}e_{i}\right\|\leq N\max_{j}|\alpha_{j}|\|e_{i}\|$,
which proves that $\|\cdot\|$ and $\|\cdot\|_{\infty}$ are indeed equivalent.
∎

As noticed during the proof, each linear subspace $W\subset V$ is closed with
respect to any norm $\|\cdot\|$, and $\|\cdot\|$ thus induces a quotient norm
$\|\cdot\|_{V/W}$ on $V/W$, defined as usual by

$\|\bar{v}\|_{V/W}:=\inf_{w\in W}\|v+w\|$

for each $v\in V$ with image $\bar{v}\in V/W$.

By Proposition 1.6, we can endow $\mathcal{N}(V)$ with the _Goldman-Iwahori_
metric $\operatorname{\operatorname{d}_{\infty}}$ (named after [GI]), defined
by

$\operatorname{\operatorname{d}_{\infty}}\left(\|\cdot\|,\|\cdot\|^{\prime}\right):=\sup_{v\in
V\setminus\\{0\\}}\left|\log\|v\|-\log\|v\|^{\prime}\right|.$ (1.1)

The exponential of
$\operatorname{\operatorname{d}_{\infty}}\left(\|\cdot\|,\|\cdot\|^{\prime}\right)$
is thus the distortion betwen the two norms, i.e. the smallest constant $C\geq
1$ such that

$C^{-1}\|\cdot\|\leq\|\cdot\|^{\prime}\leq C\|\cdot\|.$

The action of $\operatorname{GL}(V)$ on $\mathcal{N}(V)$ preserves
$\operatorname{\operatorname{d}_{\infty}}$.

######  Example 1.7.

Assume $\dim V=1$, and pick a nonzero $v\in V$. Then
$\|\cdot\|\mapsto\log\|v\|$ defines an isometry
$(\mathcal{N}(V),\operatorname{\operatorname{d}_{\infty}})\simeq\mathbb{R}$,
and the action of $\operatorname{GL}(V)$ on $\mathcal{N}(V)$ is equivalent to
the action of the additive value group $\Gamma=\log|K^{*}|$ on $\mathbb{R}$ by
translation.

The basic topological properties of $\mathcal{N}(V)$ are as follows.

######  Proposition 1.8.

The metric space $(\mathcal{N}(V),\operatorname{\operatorname{d}_{\infty}})$
is complete, and the subset $\mathcal{N}^{\mathrm{ultr}}(V)$ of ultrametric
norms is closed. If $K$ is local, i.e. $K^{\circ}$ is compact, any closed
bounded subset of $\mathcal{N}(V)$ is compact.

###### Proof.

If $(\|\cdot\|_{n})$ is a Cauchy sequence in $\mathcal{N}(V)$, then
$\log\|v\|_{n}$ is a Cauchy sequence for each nonzero $v\in V$. We easily
conclude that $\|\cdot\|_{n}$ converges in $\mathcal{N}(V)$, which proves the
first assertion. Assume now that $K$ is local. After choosing a basis, we may
assume that $V=K^{N}$, which we equip with the $\ell^{\infty}$-norm
$\|\cdot\|_{\infty}$. By the triangle inequality, each norm $\|\cdot\|$ with
$\operatorname{\operatorname{d}_{\infty}}(\|\cdot\|,\|\cdot\|_{\infty})\leq C$
restricts to a $C$-Lipschitz continuous function on the compact set
$(K^{\circ})^{N}$. By the Arzelà-Ascoli theorem, the closed balls of
$\mathcal{N}(V)$ (and hence any bounded closed subset) are thus compact. ∎

######  Remark 1.9.

Conversely, if $\dim V>1$, one can show that $\mathcal{N}(V)$ is locally
compact only if $K^{\circ}$ is compact.

####  1.3. Diagonalizable norms

In order to treat in parallel the Archimedean and non-Archimedean cases, we
will use the following terminology.

######  Definition 1.10.

A norm $\|\cdot\|$ on $V$ is _diagonalizable_ if there exists a basis
$(e_{i})$ such that we have for all $\alpha\in K^{N}$:

  * (i)

$\|\sum_{i}\alpha_{i}e_{i}\|^{2}=\sum_{i}\|\alpha_{i}e_{i}\|^{2}$ (Archimedean
case);

  * (ii)

$\|\sum_{i}\alpha_{i}e_{i}\|=\max_{i}\|\alpha_{i}e_{i}\|$ (non-Archimedean
case).

The basis $(e_{i})$ is then said to be _orthogonal_ for $\|\cdot\|$, and it is
_orthonormal_ if it further satisfies $\|e_{i}\|=1$. We say that a
diagonalizable norm $\|\cdot\|$ is _pure_ if it admits an orthonormal basis.
We denote by

$\mathcal{N}^{\mathrm{diag}}(V)\subset\mathcal{N}(V)$

the set of diagonalizable norms.

A diagonalizable norm is pure iff it takes values in $|K|$. In the Archimedean
case, a norm is diagonalizable if and only if it derives from a
Euclidian/Hermitian scalar product, and every such norm is pure. In the non-
Archimedean case, every diagonalizable norm is ultrametric. The converse
depends on the specific field $K$ we are dealing with, but diagonalizable
norms are always dense in the set of ultrametric norms, see §1.4 below.

######  Example 1.11.

If $K$ is trivially valued, any ultrametric norm $\|\cdot\|$ is
diagonalizable. Indeed, $\|\cdot\|$ determines a flag of subspaces (cf.
Example 1.5), and a basis $(e_{i})$ of $V$ is orthogonal for $\|\cdot\|$ if
and only if $(e_{\sigma(i)})$ is compatible with the flag, for some
permutation $\sigma\in\mathfrak{S}_{N}$.

######  Example 1.12.

By [BGR, 2.4.2/3], any ultrametric norm is diagonalizable as soon as $K$ is
discretely valued (see also [CMor15, Proposition 1.3]). By [BGR, 2.4.4], this
is more generally true if $K$ is maximally complete; conversely, $K^{2}$
admits a non-diagonalizable ultrametric norm as soon as $K$ is not maximally
complete (e.g. $K=\mathbb{C}_{p}$). Indeed, let $L/K$ be a nontrivial
immediate extension, pick $\pi\in L-K$, and denote by $\|\cdot\|$ the
restriction of $|\cdot|_{L}$ to $K+\pi K\simeq K^{2}$. If $\|\cdot\|$ were
diagonalizable, it would admit an orthonormal basis $(e_{1},e_{2})$, since
$\|\cdot\|$ takes values in $|K|=|L|$. Using $\widetilde{L}=\widetilde{K}$, we
then find a unit $\alpha\in K^{\circ}$ such that $e_{1}-\alpha e_{2}\in
L^{\circ\circ}$, i.e. $\|e_{1}-\alpha e_{2}\|<1$, contradicting the
orthonormality of $(e_{1},e_{2})$.

In the Archimedean case, diagonalizable norms are of course preserved by
restriction and quotient. This is also true in the non-Archimedean case (cf.
[BGR, 2.4.1/5]):

######  Lemma 1.13.

Let $0\to V^{\prime}\to V\to V^{\prime\prime}\to 0$ be an exact sequence of
vector spaces. If $\|\cdot\|$ is a diagonalizable norm on $V$, then the
induced norms on $V^{\prime}$ and $V^{\prime\prime}$ are also diagonalizable.

The following codiagonalization result is crucial for what follows. It will be
proved in §1.5, after the basic facts on duality have been discussed.

######  Proposition 1.14.

For any two diagonalizable norms
$\|\cdot\|,\|\cdot\|^{\prime}\in\mathcal{N}^{\mathrm{diag}}(V)$, there exists
a basis $(e_{i})$ of $V$ that is orthogonal for both $\|\cdot\|$ and
$\|\cdot\|^{\prime}$.

This can be used to give a simple description of the restriction of
$\operatorname{\operatorname{d}_{\infty}}$ to
$\mathcal{N}^{\mathrm{diag}}(V)$.

######  Lemma 1.15.

If $\|\cdot\|,\|\cdot\|^{\prime}\in\mathcal{N}^{\mathrm{diag}}(V)$ are
codiagonalized in a basis $(e_{i})$, then

$\operatorname{\operatorname{d}_{\infty}}(\|\cdot\|,\|\cdot\|^{\prime})=\max_{i}\left|\log\frac{\|e_{i}\|}{\|e_{i}\|^{\prime}}\right|.$

If $K$ is non-Archimedean, we have more generally

$\operatorname{\operatorname{d}_{\infty}}(\|\cdot\|,\|\cdot\|^{\prime})=\log\max\left\\{\max_{i}\frac{\|e_{i}\|^{\prime}}{\|e_{i}\|},\max_{i}\frac{\|e^{\prime}_{i}\|}{\|e^{\prime}_{i}\|^{\prime}}\right\\}$

whenever $(e_{i})$ and $(e^{\prime}_{i})$ are orthogonal bases for $\|\cdot\|$
and $\|\cdot\|^{\prime}$, respectively.

###### Proof.

Assume first that $K$ is Archimedean. We trivially have

$\operatorname{\operatorname{d}_{\infty}}(\|\cdot\|,\|\cdot\|^{\prime})=\sup_{v\in
V\setminus\\{0\\}}\left|\log\frac{\|v\|}{\|v\|^{\prime}}\right|\geq
m:=\max_{i}\left|\log\frac{\|e_{i}\|}{\|e_{i}\|^{\prime}}\right|$

Consider conversely $v=\sum_{i}\alpha_{i}e_{i}$ with $\alpha\in K^{N}$. Then

$\|v\|^{\prime 2}=\sum_{i}|\alpha_{i}|^{2}\|e_{i}\|^{\prime 2}\leq
e^{2m}\sum_{i}|\alpha_{i}|\|e_{i}\|^{2}=e^{2m}\|v\|^{2}.$

By symmetry, this shows that $e^{-m}\|\cdot\|\leq\|\cdot\|^{\prime}\leq
e^{m}\|\cdot\|$, i.e.
$\operatorname{\operatorname{d}_{\infty}}(\|\cdot\|,\|\cdot\|^{\prime})\leq
m$. In the non-Archimedean case the result follows from Lemma 1.16 below. ∎

######  Lemma 1.16.

Assume that $K$ is non-Archimedean. Let $\|\cdot\|$ be a diagonalizable norm
with orthogonal basis $(e_{i})$, and $\|\cdot\|^{\prime}$ be any ultrametric
seminorm on $V$. Then

$\sup_{v\in
V\setminus\\{0\\}}\frac{\|v\|^{\prime}}{\|v\|}=\max_{i}\frac{\|e_{i}\|^{\prime}}{\|e_{i}\|}.$

###### Proof.

As above, we trivially have $\sup_{v\in
V\setminus\\{0\\}}\|v\|^{\prime}/\|v\|\geq
m:=\max_{i}\|e_{i}\|^{\prime}/\|e_{i}\|$, and $v=\sum_{i}\alpha_{i}e_{i}$
satisfies $\|v\|^{\prime}\leq\max_{i}|\alpha_{i}|\|e_{i}\|^{\prime}\leq
m\max_{i}|\alpha_{i}|\|e_{i}\|=m\|v\|$. ∎

We now discuss in more detail the structure of the set
$\mathcal{N}^{\mathrm{diag}}(V)$ of diagonalizable norms. To each basis ${\bf
e}=(e_{i})$ of $V$ is associated an injective map

$\iota_{\bf e}:\mathbb{R}^{N}\hookrightarrow\mathcal{N}^{\mathrm{diag}}(V),$

which takes $\lambda\in\mathbb{R}^{N}$ to the unique norm $\|\cdot\|_{{\bf
e},\lambda}$ that is diagonalized in $(e_{i})$ and such that $\|e_{i}\|_{\bf
e,\lambda}=e^{-\lambda_{i}}$. The image

$\mathbb{A}_{\bf e}:=\iota_{\bf
e}(\mathbb{R}^{N})\subset\mathcal{N}^{\mathrm{diag}}(V)$

is thus the set of norms that are diagonalized in the given basis ${\bf e}$,
and is called an _apartment_ (or _flat_) of $\mathcal{N}^{\mathrm{diag}}(V)$.
By definition, $\mathcal{N}^{\mathrm{diag}}(V)=\bigcup_{\bf e}\mathbb{A}_{\bf
e}$, and the Goldman-Iwahori metric $\operatorname{\operatorname{d}_{\infty}}$
can then be characterized as follows.

######  Proposition 1.17.

The restriction of $\operatorname{\operatorname{d}_{\infty}}$ to
$\mathcal{N}^{\mathrm{diag}}(V)$ is the unique metric such that each
$\iota_{\bf e}:\mathbb{R}^{N}\hookrightarrow\mathcal{N}^{\mathrm{diag}}(V)$ is
an isometric embedding with respect to the $\ell^{\infty}$-norm on
$\mathbb{R}^{N}$.

###### Proof.

Each $\iota_{\bf e}$ is an isometric embedding by Lemma 1.15, and uniqueness
follows from the fact that any two points of $\mathcal{N}^{\mathrm{diag}}(V)$
belong to the image of some $\iota_{\bf e}$, by codiagonalization. ∎

This picture will be generalized to any symmetric norm on $\mathbb{R}^{N}$
(and in particular to the $\ell^{2}$-norm) in Section 3, leading to the
description of $\mathcal{N}^{\mathrm{diag}}(V)$ as a Riemannian symmetric
space/Euclidian building. In the present setting, the general construction of
retractions onto an apartment in building theory specializes as follows
(compare [Ger81]).

######  Definition 1.18.

Let ${\bf e}=(e_{i})$ be a basis of $V$, with apartment $\mathbb{A}_{\bf
e}=\iota_{\bf e}(\mathbb{R}^{N})\subset\mathcal{N}^{\mathrm{diag}}(V)$. The
_Gram-Schmidt projection_ $\rho_{\bf e}:\mathcal{N}(V)\to\mathbb{A}_{\bf e}$
is defined by sending a norm $\|\cdot\|$ to the unique norm $\|\cdot\|_{\bf
e}$ that is diagonalized in ${\bf e}$ and such that

$\|e_{i}\|_{\bf e}=\inf_{\alpha\in
K^{N}}\left\|e_{i}+\sum_{j<i}\alpha_{j}e_{j}\right\|$

for $i=1,\dots,N$.

Setting $W_{i}:=\operatorname{Vect}(e_{1},\dots,e_{i})$ defines a complete
flag $W_{\bullet}$, and $\|\cdot\|$ induces a subquotient norm on each graded
piece $W_{i}/W_{i-1}$, and hence a diagonalizable norm on the graded object
$\operatorname{Gr}V=\bigoplus_{1\leq i\leq N}W_{i}/W_{i-1}$. The norm
$\|\cdot\|_{\bf e}$ can then be described as the corresponding norm on $V$
under the isomorphism $V\simeq\operatorname{Gr}V$ defined by $(e_{i})$. It is
straightforward to see that $\rho_{\bf e}:\mathcal{N}(V)\to\mathbb{A}_{\bf e}$
is a retraction, i.e. restricts to the identity on $\mathbb{A}_{\bf e}$.

The chosen terminology comes from the Archimedean case, where the Gram-Schmidt
orthogonalization process associates to a Euclidian/Hermitian norm $\|\cdot\|$
and a basis $(e_{i})$ the orthogonal basis $(e^{\prime}_{i})$ obtained by
projection of each $e_{i}$ orthogonal to $W_{i-1}$, which satisfies
$\|e_{i}\|_{\bf e}=\|e^{\prime}_{i}\|$.

####  1.4. Approximation by diagonalizable norms

The goal of this section is to study the closure in $\mathcal{N}(V)$ of the
set $\mathcal{N}^{\mathrm{diag}}(V)$ of diagonalizable norms.

######  Theorem 1.19.

The space of diagonalizable norms $\mathcal{N}^{\mathrm{diag}}(V)$ satisfies
the following properties.

  * (i)

Each norm $\|\cdot\|\in\mathcal{N}(V)$ is at distance at most $\log N$ of
$\mathcal{N}^{\mathrm{diag}}(V)$.

  * (ii)

If $K$ is Archimedean, then $\mathcal{N}^{\mathrm{diag}}(V)$ is closed in
$\mathcal{N}(V)$.

  * (iii)

If $K$ is non-Archimedean, then the closure of
$\mathcal{N}^{\mathrm{diag}}(V)$ in $\mathcal{N}(V)$ coincides with the set
$\mathcal{N}^{\mathrm{ultr}}(V)$ of ultrametric norms.

In the Archimedean case, (i) in Theorem 1.19 can be deduced from the John
ellipsoid theorem; one can also use the simpler Auerbach lemma, whose proof
will be basically repeated below. (ii) follows from the fact that
Euclidian/Hermitian norms are characterized by the paparallelogram law

$\|u+v\|^{2}+\|u-v\|^{2}=2\|u\|^{2}+2\|v\|^{2}.$

Finally, the first part of (iii) is equivalent to the existence of
$\alpha$-cartesian bases in the sense of [BGR, 2.6.1/3], which will be
recovered below by imitating the Auerbach argument.

Denote by $V^{\vee}$ the dual of $V$, and by $\det
V^{\vee}=\bigwedge^{N}V^{\vee}$ its determinant line. Viewing an element
$\omega\in\det V^{\vee}$ as a multilinear form on $V$, we define its _operator
norm_ as

$\|\omega\|_{\mathrm{op}}:=\sup_{(v_{1},\dots,v_{N})\in
V^{N}}\frac{|\omega(v_{1},\dots,v_{N})|}{\|v_{1}\|\cdots\|v_{N}\|}.$

This supremum is indeed finite by equivalence of norms.

######  Lemma 1.20.

Let $\|\cdot\|$ be a norm on $V$, and pick a non-zero $\omega\in\det
V^{\vee}$. For each basis $(e_{i})$ of $V$ and all $\alpha\in K^{N}$, we then
have

$\max_{i}\|\alpha_{i}e_{i}\|\leq\left(\frac{\|\omega\|_{\mathrm{op}}\|e_{1}\|\cdots\|e_{N}\|}{|\omega(e_{1},\dots,e_{N})|}\right)\|\sum_{i}\alpha_{i}e_{i}\|.$

###### Proof.

The dual basis $(e_{i}^{\vee})$ satisfies

$\langle
e_{i}^{\vee},v\rangle=\frac{\omega(e_{1},\dots,e_{i-1},v,e_{i+1},\dots,e_{N})}{\omega(e_{1},\dots,e_{N})},$

and hence

$\max_{i}|\langle
e_{i}^{\vee},v\rangle|\|e_{i}\|\leq\left(\frac{\|\omega\|_{\mathrm{op}}\|e_{1}\|\dots\|e_{N}\|}{|\omega(e_{1},\dots,e_{N})|}\right)\|v\|,$

which is equivalent to the desired result. ∎

###### Proof of Theorem 1.19.

Assume first that $K$ is Archimedean. As noted above, (ii) follows from the
characterization of diagonalizable norms in terms of the parallelogram law.
Let $\|\cdot\|$ be any norm on $V$, and fix a nonzero determinant
$\omega\in\det V^{\vee}$. By compactness, we may choose a basis $(e_{i})$ of
$V$ with $\|e_{i}\|=1$ and
$\|\omega\|_{\mathrm{op}}=|\omega(e_{1},\dots,e_{N})|$. For each
$p\in[1,\infty]$, denote by $\|\cdot\|_{p}$ the $\ell^{p}$-norm in the basis
$(e_{i})$. Lemma 1.20 and the triangle inequality yield
$\|\cdot\|_{\infty}\leq\|\cdot\|\leq\|\cdot\|_{1}$. Since
$N^{-1/2}\|\cdot\|_{1}\leq\|\cdot\|_{2}\leq N^{1/2}\|\cdot\|_{\infty}$, it
follows that $N^{-1/2}\|\cdot\|_{2}\leq\|\cdot\|\leq N^{1/2}\|\cdot\|_{2}$,
and hence
$\operatorname{\operatorname{d}_{\infty}}(\|\cdot\|,\|\cdot\|_{2})\leq\log N$,
which proves (i) in the Archimedean case since
$\|\cdot\|_{2}\in\mathcal{N}^{\mathrm{diag}}(V)$.

Assume now that $K$ is non-Archimedean, and pick any norm
$\|\cdot\|\in\mathcal{N}(V)$, not necessarily ultrametric at this point. For
any $\varepsilon>0$, there exists a basis $(e_{i})$ such that

$\|\omega\|_{\mathrm{op}}\leq(1+\varepsilon)\frac{|\omega(e_{1},\dots,e_{N})|}{\|e_{1}\|\dots\|e_{N}\|},$

and Lemma 1.20 yields

$\max_{i}\|\alpha_{i}e_{i}\|\leq(1+\varepsilon)\|\sum_{i}\alpha_{i}e_{i}\|$
(1.2)

for all $\alpha\in K^{N}$. Since

$\left\|\sum_{i}\alpha_{i}e_{i}\right\|\leq\sum_{i}\left\|\alpha_{i}e_{i}\right\|\leq
N\max_{i}\left\|\alpha_{i}e_{i}\right\|,$

it follows that the diagonalizable norm
$\|\cdot\|^{\prime}\in\mathcal{N}^{\mathrm{diag}}(V)$ defined by
$\left\|\sum_{i}\alpha_{i}e_{i}\right\|^{\prime}:=\max_{i}|\alpha_{i}|\|e_{i}\|$
satisfies

$\operatorname{\operatorname{d}_{\infty}}\left(\|\cdot\|,\|\cdot\|^{\prime}\right)\leq\max\\{\log
N,\log(1+\varepsilon)\\},$

which proves the non-Archimedean case of (i) since $\epsilon>0$ was arbitrary.
Finally, if $\|\cdot\|$ is assumed to be ultrametric, then (1.2) implies
$(1-\varepsilon)\|\cdot\|^{\prime}\leq\|\cdot\|\leq\|\cdot\|^{\prime}$, hence
$\operatorname{\operatorname{d}_{\infty}}\left(\|\cdot\|,\|\cdot\|^{\prime}\right)\leq\log(1+\varepsilon)$,
and $\|\cdot\|$ is thus in the closure of $\mathcal{N}^{\mathrm{diag}}(V)$. ∎

####  1.5. Duality

To each norm $\|\cdot\|$ on $V$ is associated a dual norm $\|\cdot\|^{\vee}$
on the dual vector space $V^{\vee}$, defined by the usual formula

$\|\mu\|^{\vee}=\sup_{v\in
V\setminus\\{0\\}}\frac{|\langle\mu,v\rangle|}{\|v\|}.$

Again, the supremum is finite by equivalence of norms, and it is
straightforward to see that the duality map
$\mathcal{N}(V)\to\mathcal{N}(V^{\vee})$ so defined is $1$-Lipschitz
continuous with respect to the $\operatorname{\operatorname{d}_{\infty}}$
metric. Diagonalizable norms are preserved by duality:

######  Lemma 1.21.

If $\|\cdot\|\in\mathcal{N}^{\mathrm{diag}}(V)$ is diagonalizable, then so is
$\|\cdot\|^{\vee}$. Further, if $(e_{i})$ is an orthogonal basis for
$\|\cdot\|$, then the dual basis $(e_{i}^{\vee})$ is orthogonal for
$\|\cdot\|^{\vee}$, and $\|e_{i}^{\vee}\|^{\vee}=\|e_{i}\|^{-1}$.

###### Proof.

In the Archimedean case, $\|\cdot\|$ is Euclidian/Hermitian, and the result is
well-known. In the non-Archimedean case, the result is a simple consequence of
Lemma 1.16. ∎

######  Lemma 1.22.

If $W\subset V$ is a linear subspace, the canonical embedding
$(V/W)^{\vee}\hookrightarrow V^{\vee}$ identifies the dual of the quotient
norm $\|\cdot\|_{V/W}$ with the restriction of $\|\cdot\|^{\vee}$.

###### Proof.

The image of $(V/W)^{\vee}$ in $V^{\vee}$ is the space $W^{\perp}$ of linear
forms $\mu\in V^{\vee}$ that vanish on $W$. Denoting by $\bar{v}\in V/W$ the
image of $v\in V$ and by $\tilde{\mu}\in V^{\vee}$ the image of
$\mu\in(V/W)^{\vee}$, we have by definition

$\|\tilde{\mu}\|^{\vee}=\sup_{v\in
V\setminus\\{0\\}}\frac{|\langle\tilde{\mu},v\rangle|}{\|v\|}$

and

$\|\mu\|_{V/W}^{\vee}=\sup_{v\in
V-W}\frac{|\langle\tilde{\mu},v\rangle|}{\|\bar{v}\|_{V/W}}.$

Since $\|\bar{v}\|_{V/W}=\inf_{w\in W}\|v+w\|\leq\|v\|$ and
$\langle\tilde{\mu},v\rangle=0$ for $v\in W$, we trivially have
$\|\tilde{\mu}\|^{\vee}\leq\|\mu\|_{V/W}^{\vee}$. Conversely, we have for each
$v\in V-W$ and $w\in W$

$\frac{|\langle\tilde{\mu},v\rangle|}{\|v+w\|}=\frac{|\langle\tilde{\mu},v+w\rangle|}{\|v+w\|}\leq\|\tilde{\mu}\|^{\vee},$

hence

$\frac{|\langle\tilde{\mu},v\rangle|}{\|\bar{v}\|_{V/W}}=\sup_{w\in
W}\frac{|\langle\tilde{\mu},v\rangle|}{\|v+w\|}\leq\|\tilde{\mu}\|^{\vee},$

and taking the supremum over $v$ yields the inequality in the other direction
$\|\mu\|_{V/W}^{\vee}\leq\|\tilde{\mu}\|^{\vee}$ and we conclude. ∎

Biduality holds in the Archimedean case, as a consequence of the Hahn-Banach
theorem:

######  Proposition 1.23.

If $K$ is Archimedean, the duality map
$\mathcal{N}(V)\to\mathcal{N}(V^{\vee})$ is an involutive isometry.

In the non-Archimedean case, biduality fails, for the simple reason that the
dual of any norm is automatically ultrametric. This is however the only
obstruction (compare for instance [CMor15, Corollary 1.7]):

######  Proposition 1.24.

Assume that $K$ is non-Archimedean. The bidual $\|\cdot\|^{\vee\vee}$ of a
norm $\|\cdot\|\in\mathcal{N}(V)$ is the largest ultrametric norm such that
$\|\cdot\|^{\vee\vee}\leq\|\cdot\|$. In particular,
$\|\cdot\|^{\vee\vee}=\|\cdot\|$ if and only if $\|\cdot\|$ is ultrametric,
and the duality map $\|\cdot\|\mapsto\|\cdot\|^{\vee}$ restricts to an
involutive isometry
$\mathcal{N}^{\mathrm{ultr}}(V)\simeq\mathcal{N}^{\mathrm{ultr}}(V^{\vee})$.

###### Proof.

By definition of the dual norm, we have
$|\langle\mu,v\rangle|\leq\|\mu\|^{\vee}\|v\|$ for all $v\in V$, $\mu\in
V^{\vee}$, and hence $(\|\cdot\|^{\vee})^{\vee}\leq\|\cdot\|$. Since
$\|\cdot\|\mapsto\|\cdot\|^{\vee\vee}$ is non-decreasing, it is thus enough to
show that any ultrametric norm $\|\cdot\|$ satisfies
$\|\cdot\|^{\vee\vee}=\|\cdot\|$. By density of diagonalizable norms in
$\mathcal{N}^{\mathrm{ultr}}(V)$ (Theorem 1.19), we may assume that
$\|\cdot\|$ is diagonalizable, in which case the result follows from Lemma
1.21. ∎

We are now in a position to prove the codiagonalization result promised in
Proposition 1.14.

###### Proof of Proposition 1.14.

That any two diagonalizable norms
$\|\cdot\|,\|\cdot\|^{\prime}\in\mathcal{N}^{\mathrm{diag}}(V)$ are
codiagonalizable is a standard fact in the Archimedean case, and we henceforth
assume that $K$ is non-Archimedean. Our argument extends the classical one of
[GI], which treats the case of a local field, following a suggestion of Marco
Maculan, whom we warmly thank. Recall that a direct sum decomposition
$V=V_{1}\oplus\dots\oplus V_{r}$ is _orthogonal for $\|\cdot\|$_ if

$\left\|\sum_{i}v_{i}\right\|=\max_{i}\|v_{i}\|$

for all $v_{i}\in V_{i}$. Given $v\in V$ and a linear form $\mu\in V^{\vee}$
with $\langle\mu,v\rangle\neq 0$, it is straightforward to check that the
decomposition $V=Kv\oplus\operatorname{Ker}\mu$ is orthogonal for $\|\cdot\|$
if and only if

$\frac{|\langle\mu,w\rangle|}{|\langle\mu,v\rangle|}\leq\frac{\|w\|}{\|v\|}$

for all $w\in V$. Arguing by induction on $\dim V$, we will thus be done if we
prove the existence of $v\in V$ and $\mu\in V^{\vee}$ with
$\langle\mu,v\rangle\neq 0$ such that

$\frac{|\langle\mu,w\rangle|}{|\langle\mu,v\rangle|}\leq\frac{\|w\|}{\|v\|}\leq\frac{\|w\|^{\prime}}{\|v\|^{\prime}}$
(1.3)

for all $w\in V$, since $V=Kv\oplus\ker\mu$ will then be orthogonal for both
$\|\cdot\|$ and $\|\cdot\|^{\prime}$. Let $(e^{\prime}_{i})$ be an orthogonal
basis for $\|\cdot\|^{\prime}$. By Lemma 1.16, we have

$\sup_{w\in
W\setminus\\{0\\}}\frac{\|w\|}{\|w\|^{\prime}}=\frac{\|e^{\prime}_{i}\|}{\|e^{\prime}_{i}\|^{\prime}}$

for some $i$, and $v:=e^{\prime}_{i}$ therefore satisfies
$\|w\|/\|v\|\leq\|w\|^{\prime}/\|v\|^{\prime}$ for all $w\in V$. Let now
$(e_{j})$ be an orthogonal basis for $\|\cdot\|$. Since the dual basis
$(e_{j}^{\vee})$ is orthogonal for the dual norm $\|\cdot\|^{\vee}$, we
similarly get

$\|v\|=\sup_{\mu\in
V^{\vee}\setminus\\{0\\}}\frac{|\langle\mu,v\rangle|}{\|\mu\|^{\vee}}=\frac{|\langle
e_{j}^{\vee},v\rangle|}{\|e_{j}^{\vee}\|^{\vee}}$

for some $j$. It follows that $\mu:=e_{j}^{\vee}$ satisfies

$|\langle\mu,v\rangle|=\|\mu\|^{\vee}\|v\|\geq\frac{|\langle\mu,w\rangle|}{\|w\|}\|v\|$

for all $w\in V\setminus\\{0\\}$, and (1.3) follows. ∎

####  1.6. Ground field extension

In this section, $K$ is assumed to be non-Archimedean. In this context, the
tensor product of two ultrametric seminorms $\|\cdot\|$, $\|\cdot\|^{\prime}$
on (possibly infinite dimensional) $K$-vector spaces $U,U^{\prime}$ is the
seminorm $\|\cdot\|_{U\otimes U^{\prime}}$ defined by setting for each $w\in
U\otimes U^{\prime}$

$\|w\|_{U\otimes U^{\prime}}=\inf_{w=\sum_{i}u_{i}\otimes
u^{\prime}_{i}}\max_{i}\|u_{i}\|\|u^{\prime}_{i}\|^{\prime},$

the infimum ranging over all finite sum decompositions $w=\sum_{i}u_{i}\otimes
u^{\prime}_{i}$ with $u_{i}\in U$, $u^{\prime}_{i}\in U^{\prime}$. As a
special case of this construction, we have:

######  Definition 1.25.

Assume that $K$ is non-Archimedean, and let $K^{\prime}/K$ be a complete field
extension. The _ground field extension_ of an ultrametric seminorm $\|\cdot\|$
on $V$ is the ultrametric seminorm $\|\cdot\|_{K^{\prime}}$ on
$V_{K^{\prime}}=V\otimes_{K}K^{\prime}$ defined by setting for each
$v^{\prime}\in V_{K^{\prime}}$

$\|v^{\prime}\|_{K^{\prime}}=\inf_{v^{\prime}=\sum\beta_{i}v_{i}}\max_{i}|\beta_{i}|\|v_{i}\|,$

the infimum ranging over all decompositions $v^{\prime}=\sum\beta_{i}v_{i}$
with $\beta_{i}\in K^{\prime}$ and $v_{i}\in V$.

Our notation identifies for simplicity $V$ with its image in $V_{K^{\prime}}$
under $v\mapsto v\otimes 1$.

######  Proposition 1.26.

Let $K^{\prime}/K$ be a complete field extension.

  * (i)

For each ultrametric norm $\|\cdot\|$ on $V$, $\|\cdot\|_{K^{\prime}}$ is an
ultrametric norm on $V_{K^{\prime}}$ which coincides with $\|\cdot\|$ on $V$.

  * (ii)

The map $\|\cdot\|\mapsto\|\cdot\|_{K^{\prime}}$ is an isometric embedding
$\mathcal{N}^{\mathrm{ultr}}(V)\hookrightarrow\mathcal{N}^{\mathrm{ultr}}(V_{K^{\prime}})$.

  * (iii)

For all $v^{\prime}\in V_{K^{\prime}}$ we have

$\|v^{\prime}\|_{K^{\prime}}=\sup_{\mu\in
V^{\vee}\setminus\\{0\\}}\frac{|\langle\mu\otimes
1,v^{\prime}\rangle|}{\|\mu\|^{\vee}}.$

This last equality shows the compatibility of our definition with [CMor15,
Definition 1.8].

######  Lemma 1.27.

If $\|\cdot\|$ is a diagonalizable norm with orthogonal basis $(e_{i})$, then
$\|\cdot\|_{K^{\prime}}$ is also a diagonalizable norm with orthogonal basis
$(e_{i})$ (viewed as a basis of $V_{K^{\prime}}$), and
$\|e_{i}\|_{K^{\prime}}=\|e_{i}\|$.

###### Proof.

Write $v^{\prime}=\sum_{j}\alpha^{\prime}_{j}e_{j}$ with
$\alpha^{\prime}_{j}\in K^{\prime}$. By definition, we have

$\|v^{\prime}\|_{K^{\prime}}\leq\max_{i}|\alpha^{\prime}_{i}|\|e_{i}\|.$

Conversely, given a decomposition $v^{\prime}=\sum_{i}\beta_{i}v_{i}$ with
$\beta_{i}\in K^{\prime}$ and $v_{i}\in V$, we have to show that

$\max_{j}|\alpha^{\prime}_{j}|\|e_{j}\|\leq\max_{i}|\beta_{i}|\|v_{i}\|.$

Writing $v_{i}=\sum_{j}\alpha_{ij}e_{j}$ with $\alpha_{ij}\in K$, we have for
each $j$ $\alpha^{\prime}_{j}=\sum_{i}\beta_{i}\alpha_{ij}$, and hence

$|\alpha^{\prime}_{j}|\|e_{j}\|\leq\max_{i}\left(|\beta_{i}||\alpha_{ij}|\right)\|e_{j}\|\leq\max_{i}\left(|\beta_{i}|\max_{k}|\alpha_{ik}|\|e_{k}\|\right)=\max_{i}|\beta_{i}|\|v_{i}\|.$

∎

###### Proof of Proposition 1.26.

Assume first that $\|\cdot\|$ is diagonalizable. By Lemma 1.27,
$\|\cdot\|_{K^{\prime}}$ is an ultrametric norm, and it coincides with
$\|\cdot\|$ on $V$. Denote by $\|v^{\prime}\|^{\prime}$ the right-hand
supremum in (iii). Pick a nonzero $\mu\in V^{\vee}$, and consider a
decomposition $v^{\prime}=\sum_{i}\beta_{i}v_{i}$ with $\beta_{i}\in
K^{\prime}$ and $v_{i}\in V$. Then

$\frac{|\langle\mu\otimes
1,v^{\prime}\rangle|}{\|\mu\|^{\vee}}\leq\max_{i}|\beta_{i}|\frac{|\langle\mu,v_{i}\rangle|}{\|\mu\|^{\vee}}\leq\max_{i}|\beta_{i}|\|\|v_{i}\|.$

Taking the infimum over all decompositions and the supremum over $\mu$ yields
$\|v^{\prime}\|^{\prime}\leq\|v^{\prime}\|_{K^{\prime}}$. Conversely, let
$(e_{i})$ be an orthogonal basis for $\|\cdot\|$, write
$v^{\prime}=\sum_{i}\alpha^{\prime}_{i}e_{i}$, and pick an index $i$ achieving
$\|v^{\prime}\|_{K^{\prime}}=\max_{i}|\alpha^{\prime}_{i}|\|e_{i}\|$. By Lemma
1.21, $\mu:=e_{i}^{\vee}$ satisfies

$\frac{|\langle\mu\otimes
1,v^{\prime}\rangle|}{\|e_{i}^{\vee}\|^{\vee}}=|\alpha^{\prime}_{i}|\|e_{i}\|=\|v^{\prime}\|_{K^{\prime}},$

which proves (iii) for $\|\cdot\|$. In the general case, it is straightforward
to see that $\|\cdot\|\mapsto\|\cdot\|_{K^{\prime}}$ and
$\|\cdot\|\mapsto\|\cdot\|^{\prime}$ are both $1$-Lipschitz continuous, and we
conclude by density of diagonalizable norms in
$\mathcal{N}^{\mathrm{ultr}}(V)$ (Theorem 1.19). ∎

For later use we finally quote the next result from [CMor15, Lemma 1.12 &
p.29].

######  Lemma 1.28.

Assume that $K$ is trivially valued, and let
$(V_{\alpha},\|\cdot\|_{\alpha})_{\alpha\in A}$ be an at most countable family
of ultrametric norms on finite dimensional $K$-vector spaces. We may then find
a nontrivially valued complete field extension $K^{\prime}/K$ such that for
each $\alpha\in A$, $\|\cdot\|_{\alpha,K^{\prime}}$ is the only ultrametric
norm on on $V_{\alpha,K^{\prime}}$ that coincides with $\|\cdot\|_{\alpha}$ on
$V_{\alpha}$.

####  1.7. Lattice norms

In this section, $K$ is non-Archimedean, with associated real-valued valuation
$v_{K}=-\log|\cdot|$. The valuation ring $K^{\circ}$ is a _Prüfer domain_ ,
i.e. every finitely generated ideal of $K^{\circ}$ is principal. This implies
that a $K^{\circ}$-module $M$ is flat if and only if it is torsion-free. If
$M$ is further finitely generated, then it is free (since $K^{\circ}$ is
local).

######  Definition 1.29.

A _lattice_ 111In the densely valued case, the present notion of lattice is
more restrictive than the one used in [CMor15, §1.3.3]. of $V$ is a finite
$K^{\circ}$-submodule $\mathcal{V}$ of $V$ such that
$\mathcal{V}\otimes_{K^{\circ}}K=V$

A lattice is thus of the form $\mathcal{V}=\sum_{i}K^{\circ}e_{i}$ with
$(e_{i})$ a basis of $V$, and $\operatorname{GL}(V)$ therefore acts
transitively on the set of lattices. A lattice $\mathcal{V}$ determines a
_lattice norm_ $\|\cdot\|_{\mathcal{V}}$ on $V$, by setting

$\|v\|_{\mathcal{V}}:=\inf\left\\{|\alpha|\mid\alpha\in
K,\,v\in\alpha\mathcal{V}\right\\}.$

######  Lemma 1.30.

Let $(e_{i})$ be a $K^{\circ}$-basis of a lattice $\mathcal{V}$ of $V$. Then
$(e_{i})$ is an orthonomal basis of $\|\cdot\|_{\mathcal{V}}$. In particular,
$\mathcal{V}$ coincides with the unit ball of $\|\cdot\|_{\mathcal{V}}$.

###### Proof.

Pick $v\in V$, and write $v=\sum_{i}\alpha_{i}e_{i}$ with $\alpha\in K^{N}$.
Given $\alpha\in K$, we then have $v\in\alpha\mathcal{V}$ if and only if
$|\alpha_{i}|\leq|\alpha|$, and hence
$\|v\|_{\mathcal{V}}=\max_{i}|\alpha_{i}|$. This means that $(e_{i})$ is
orthonormal for $\|\cdot\|_{\mathcal{V}}$, and also implies that $\mathcal{V}$
is the unit ball of $\|\cdot\|_{\mathcal{V}}$. ∎

######  Lemma 1.31.

Denote by
$\mathcal{N}^{\mathrm{latt}}(V)\subset\mathcal{N}^{\mathrm{diag}}(V)$ the set
of lattice norms.

  * (i)

A norm is a lattice norm if and only if it is a pure diagonalizable norm, i.e.
it admits an orthonormal basis.

  * (ii

If $K$ is discretely valued, $\mathcal{N}^{\mathrm{latt}}(V)$ is discrete and
closed in $\mathcal{N}^{\mathrm{ultr}}(V)=\mathcal{N}^{\mathrm{diag}}(V)$. If
$K$ is non-trivially valued with uniformizing parameter $\pi_{K}$, the closed
unit ball $B$ of any ultrametric norm $\|\cdot\|$ is a lattice, and the
associated lattice norm $\|\cdot\|_{B}$ satisfies

$\operatorname{\operatorname{d}_{\infty}}(\|\cdot\|,\|\cdot\|_{B})\leq
v_{K}(\pi_{K}).$

  * (iii)

If $K$ is densely valued, $\mathcal{N}^{\mathrm{latt}}(V)$ is dense in
$\mathcal{N}^{\mathrm{diag}}(V)$, and hence also in
$\mathcal{N}(V)^{\mathrm{ultr}}$. Further, the unit ball $B$ of a norm
$\|\cdot\|$ is a lattice if and only $\|\cdot\|$ is a lattice norm.

  * (iv)

Let $\|\cdot\|$ be the lattice norm determined by a lattice $\mathcal{V}$ of
$V$. For each complete field extension $K^{\prime}/K$, the induced norm
$\|\cdot\|_{K^{\prime}}$ on $V_{K^{\prime}}$ is the lattice norm determined by
the lattice $\mathcal{V}_{K^{\prime\circ}}$.

###### Proof.

(i) is a direct consequence of Lemma 1.30. Assume that $K$ is discretely
valued. Let $\|\cdot\|_{\mathcal{V}}\neq\|\cdot\|_{\mathcal{V}}^{\prime}$ be
two distinct lattice norms, and pick a joint orthogonal basis $(e_{i})$ as in
Proposition 1.14. We then have
$\mathcal{V}=\sum_{i}K^{\circ}\pi_{K}^{m_{i}}e_{i}$ and
$\mathcal{V}^{\prime}=\sum_{i}K^{\circ}\pi_{K}^{m^{\prime}_{i}}e_{i}$ for some
integers $m_{i},m^{\prime}_{i}\in\mathbb{Z}$, and hence

$\operatorname{\operatorname{d}_{\infty}}(\|\cdot\|_{\mathcal{V}},\|\cdot\|_{\mathcal{V}^{\prime}})=\max_{i}\left|\log\frac{\|e_{i}\|_{\mathcal{V}}}{\|e_{i}\|_{\mathcal{V}^{\prime}}}\right|$
$=v_{K}(\pi_{K})\max_{i}|m_{i}-m^{\prime}_{i}|\geq v_{K}(\pi_{K}).$

This shows that $\mathcal{N}^{\mathrm{latt}}(V)$ is discrete and closed. Next,
let $\|\cdot\|$ be any diagonalizable norm, pick an orthogonal basis $(e_{i})$
for $\|\cdot\|$, and write a given $v\in V$ as
$v=\sum_{i}u_{i}\pi_{K}^{n_{i}}e_{i}$ with $n_{i}\in\mathbb{Z}$ and $u_{i}$ a
unit. Then $\|v\|\leq 1$ if and only if $|\pi_{K}|^{n_{i}}\|e_{i}\|\leq 1$ for
all $i$, and we infer $B=\sum_{i}K^{\circ}\pi_{K}^{m_{i}}e_{i}$ with
$m_{i}:=\lceil\log\|e_{i}\|/v_{K}(\pi_{K})\rceil$. In particular, $B$ is a
lattice with basis $(\pi_{K}^{m_{i}}e_{i})$, and hence

$\operatorname{\operatorname{d}_{\infty}}(\|\cdot\|,\|\cdot\|_{B})=\max_{i}\left|\log\frac{\|e_{i}\|_{B}}{\|e_{i}\|}\right|=\max_{i}\left|m_{i}v_{K}(\pi_{K})-\log\|e_{i}\|\right|\leq
v_{K}(\pi_{K}).$

Assume finally that $K$ is densely valued. That
$\mathcal{N}^{\mathrm{latt}}(V)$ is dense in $\mathcal{N}^{\mathrm{diag}}(V)$
is easily seen by approximating the values of a given diagonalizable norm
$\|\cdot\|$ on an orthogonal basis $(e_{i})$ by elements of the dense subset
$|K|$ of $\mathbb{R}_{+}$. Similarly, any norm $\|\cdot\|$ is determined by
its closed unit ball $B$, via

$\|v\|=\inf\left\\{|\alpha|\mid v\in\alpha B\text{ with }\alpha\in
K\right\\}.$

As a result, $\|\cdot\|$ is a lattice norm if and only if $B$ is a lattice.
Finally, (iv) is a direct consequence of Lemma 1.30 and Lemma 1.27. ∎

######  Example 1.32.

Let $L/K$ be a finite extension, with induced absolute value $|\cdot|_{L}$.
When $K$ is discretely valued, the unit ball $L^{\circ}$ is always finite over
$K^{\circ}$. In the densely valued case, (iii) shows that $L^{\circ}$ is
finite over $K^{\circ}$ only if $|\cdot|_{L}$ admits a $K$-orthonormal basis,
which implies in particular that $|L|=|K|$, i.e. $L/K$ is unramified.

###  2\. Determinants and successive minima

The goal of this section is to investigate induced norms on the determinant
line, leading to the notion of _relative volume_ of two norms. We relate the
latter to successive minima via a Minkowski-type theorem, and then apply the
results of [BC] to obtain a general convergence result for the relative
volumes of certain sequences of norms on the graded pieces of a graded
algebra.

In this section, $V$ still denotes a finite dimensional vector space over a
field $K$, complete with respect to an absolute value, and we set as before
$N:=\dim V$.

####  2.1. The determinant of a norm

The _determinant line_ of $V$ is $\det V:=\Lambda^{N}V$. We have a natural
isomorphism $\det(V^{\vee})\simeq(\det V)^{\vee}$, induced by the pairing

$\langle v_{1}\wedge\dots\wedge
v_{N},\mu_{1}\wedge\dots\wedge\mu_{N}\rangle=\det\left(\langle
v_{i},\mu_{j}\rangle\right).$

In particular, if $(e_{i})$ is basis of $V$ with dual basis $(e_{i}^{\vee})$,
then

$\left(e_{1}\wedge\dots\wedge
e_{N}\right)^{\vee}=e_{1}^{\vee}\wedge\dots\wedge e_{N}^{\vee}.$

######  Definition 2.1.

To each norm $\|\cdot\|$ on $V$, we associate a norm $\det\|\cdot\|$ on $\det
V$ by setting for $\tau\in\det V$

$\det\|\tau\|=\inf_{\tau=v_{1}\wedge\dots\wedge v_{N}}\prod_{i}\|v_{i}\|,$

where the infimum runs over all decompositions $\tau=v_{1}\wedge\dots\wedge
v_{N}$ with $v_{i}\in V$.

By construction, $\det\|\cdot\|$ is the largest seminorm on $\det V$ with the
submultiplicativity property

$\det\|v_{1}\wedge\dots\wedge v_{N}\|\leq\prod_{i}\|v_{i}\|$ (2.1)

for all $v_{i}\in V$. That it is actually a norm follows from the next result,
which is readily checked.

######  Lemma 2.2.

If we view the dual $\tau^{\vee}\in\det V^{\vee}$ of a nonzero $\tau\in\det V$
as a multilinear form on $V$, then $\left(\det\|\tau\|\right)^{-1}$ coincides
with the operator norm

$\|\tau^{\vee}\|_{\mathrm{op}}:=\sup_{v_{1},\dots,v_{N}\in
V}\frac{|\tau^{\vee}(v_{1},\dots,v_{N})|}{\|v_{1}\|\dots\|v_{N}\|}.$

######  Lemma 2.3.

The map $\det:\mathcal{N}(V)\to\mathcal{N}(\det V)$ is $N$-Lipschitz
continuous with respect to $\operatorname{\operatorname{d}_{\infty}}$-metrics.

###### Proof.

Given $\|\cdot\|,\|\cdot\|^{\prime}\in\mathcal{N}(V)$,
$C:=\exp\operatorname{\operatorname{d}_{\infty}}(\|\cdot\|,\|\cdot\|^{\prime})$
satisfies $C^{-1}\|\cdot\|\leq\|\cdot\|^{\prime}\leq C\|\cdot\|$. From the
definition, we immediately get

$C^{-N}\det\|\cdot\|\leq\det\|\cdot\|^{\prime}\leq C^{N}\det\|\cdot\|,$

i.e.
$\operatorname{\operatorname{d}_{\infty}}(\|\cdot\|,\|\cdot\|^{\prime})\leq
N\log
C=N\,\operatorname{\operatorname{d}_{\infty}}(\|\cdot\|,\|\cdot\|^{\prime})$.
∎

Computing the determinant of a norm is typically a hard problem. As an
illustration we show:

######  Lemma 2.4.

Assume that $K$ is Archimedean. Pick $p\in[1,\infty]$, and denote by
$\|\cdot\|_{p}$ the $\ell^{p}$-norm in a given basis $(e_{i})$. Then
$\det\|e_{1}\wedge\dots\wedge e_{N}\|_{p}=1$ for $p\in[1,2]$, and
$\det\|e_{1}\wedge\dots\wedge e_{N}\|_{p}<1$ for $p>2$.

###### Proof.

Set $\tau:=e_{1}\wedge\dots\wedge e_{N}$. After identifying $V$ with $K^{N}$,
Lemma 2.2 yields

$\left(\det\|e_{1}\wedge\dots\wedge e_{N}\|_{p}\right)^{-1}=\sup_{v_{i}\neq
0}\frac{|\det(v_{1},\dots,v_{N})|}{\|v_{1}\|_{p}\dots\|v_{N}\|_{p}}.$

For $p=2$, the classical Hadamard inequality states that

$|\det(v_{1},\dots,v_{N})|\leq\prod_{i}\|v_{i}\|_{2}$

for all $v_{i}$, i.e. $\det\|e_{1}\wedge\dots\wedge e_{N}\|_{2}=1$. For
$p\in[1,2]$, we have $\|\cdot\|_{2}\leq\|\cdot\|_{p}$, and hence
$\det\|\cdot\|_{2}\leq\det\|\cdot\|_{p}$, and we infer
$\det\|e_{1}\wedge\dots\wedge e_{N}\|_{p}=1$. Assume now $p>2$. The basis
$(v_{i})$ defined by $v_{1}=e_{1}+e_{2}$, $v_{2}=-e_{1}+e_{2}$ and
$v_{i}=e_{i}$ for $i>2$ satisfies $\det(v_{1},\dots,v_{N})=2$,
$\|v_{1}\|_{p}=\|v_{2}\|_{p}=2^{1/p}$ and $\|v_{i}\|_{p}=1$ for $i>2$, which
yields

$\det\|e_{1}\wedge\dots\wedge e_{N}\|_{p}\leq 2^{\tfrac{2}{p}-1}<1.$

∎

######  Example 2.5.

For $p=\infty$, determining the precise value of $\det\|e_{1}\wedge\dots\wedge
e_{N}\|_{\infty}$ amounts to maximizing the determinant of a $N\times
N$-matrix with entries in $\\{\pm 1\\}$, and is known as the _Hadamard maximal
determinant problem_. By [CL65], we have for instance

$N^{-\frac{N}{2}}\leq\det\|e_{1}\wedge\dots\wedge e_{N}\|_{\infty}\leq
N^{-\frac{N}{2}\left(1-\frac{\log(4/3)}{\log N}\right)}.$

The lower bound is achieved if and only if there exists an
$\ell^{2}$-orthogonal basis with entries in $\\{\pm 1\\}$ (which implies that
$N$ is a multiple of $4$), but the exact value $\det\|e_{1}\wedge\dots\wedge
e_{N}\|_{\infty}$ is unknown in the general case.

####  2.2. Determinants of diagonalizable norms

As we shall see in this section, the determinant of a diagonalizable norm is
very well-behaved. By density, this will also be the case for all ultrametric
norms, in the non-Archimedean case.

######  Lemma 2.6.

If $\|\cdot\|$ is diagonalizable, then a basis $(e_{i})$ of $V$ satisfies

$\det\|e_{1}\wedge\dots\wedge e_{N}\|=\prod_{i}\|e_{i}\|$

if and only if $(e_{i})$ is orthogonal for $\|\cdot\|$.

######  Corollary 2.7.

Assume that $K$ is non-Archimedean, and let $\mathcal{V}$ be a lattice of $V$.
Then $\det\|\cdot\|_{\mathcal{V}}=\|\cdot\|_{\det\mathcal{V}}$ is the norm
determined by the lattice $\det\mathcal{V}:=\bigwedge^{N}\mathcal{V}$ of $\det
V$.

###### Proof of Lemma 2.6.

When $K$ is Archimedean, the result is easily seen to be equivalent to the
classical Hadamard inequality (and the equality case thereof) recalled in
Lemma 2.4. Assume now that $K$ is non-Archimedean, and let $(e_{i})$ be an
orthogonal basis for $\|\cdot\|$. We need to show that each basis $(v_{i})$
such that $v_{1}\wedge\dots\wedge v_{N}=e_{1}\wedge\dots\wedge e_{N}$
satisfies $\prod_{i}\|e_{i}\|\leq\prod_{i}\|v_{i}\|$. If we write
$v_{i}=\sum_{j}\alpha_{ij}e_{j}$ with $\alpha_{ij}\in K$, then
$\det(\alpha_{ij})=1$. Expanding out the determinant and using the ultrametric
inequality, we get $\prod_{i}|\alpha_{i\sigma(i)}|\geq 1$ for some permutation
$\sigma$. Since $\|\cdot\|$ is diagonalized in $(e_{i})$, we have

$\|v_{i}\|=\max_{j}|\alpha_{ij}|\|e_{j}\|\geq\left|\alpha_{i\sigma(i)}\right|\left\|e_{\sigma(i)}\right\|,$

and we obtain as desired

$\prod_{i}\|v_{i}\|\geq\prod_{i}\left|\alpha_{i\sigma(i)}\right|\left\|e_{\sigma(i)\|}\right\|=\left(\prod_{i}|\alpha_{i\sigma(i)}|\right)\left(\prod_{i}\|e_{\sigma(i)}\right)\geq\prod_{i}\|e_{i}\|.$

Conversely, any basis $(e_{i})$ satisfying $\det\|e_{1}\wedge\dots\wedge
e_{N}\|=\prod_{i}\|e_{i}\|$ is orthogonal for $\|\cdot\|$, as a direct
consequence of Lemma 1.20. ∎

######  Lemma 2.8.

If $\|\cdot\|$ is a diagonalizable norm on $V$, then
$\det\left(\|\cdot\|^{\vee}\right)=\left(\det\|\cdot\|\right)^{\vee}$ under
the canonical isomorphism $\det\left(V^{\vee}\right)\simeq\left(\det
V\right)^{\vee}$.

###### Proof.

Let $(e_{i})$ be an orthogonal basis for $\|\cdot\|$. By Lemma 1.21, the dual
basis $(e_{i}^{\vee})$ is orthogonal for $\|\cdot\|^{\vee}$, and
$\|e_{i}^{\vee}\|^{\vee}=\|e_{i}\|^{-1}$. By Lemma 2.6, we infer

$\det\left\|e_{1}^{\vee}\wedge\dots\wedge
e_{N}^{\vee}\right\|^{\vee}=\prod_{i}\left\|e_{i}^{\vee}\right\|^{\vee}=\left(\prod_{i}\|e_{i}\|\right)^{-1}=\left(\det\|e_{1}\wedge\dots\wedge
e_{N}\|\right)^{-1},$

hence the result. ∎

######  Lemma 2.9.

Let $\|\cdot\|$ be a diagonalizable norm on $V$, and consider an exact
sequence of vector spaces

$0\to V^{\prime}\to V\to V^{\prime\prime}\to 0,$

with induced norms $\|\cdot\|^{\prime},\|\cdot\|^{\prime\prime}$ on
$V^{\prime},V^{\prime\prime}$. Under the canonical isomorphism

$\det V\simeq\det V^{\prime}\otimes\det V^{\prime\prime},$

we then have

$\det\|\cdot\|=\det\|\cdot\|^{\prime}\otimes\det\|\cdot\|^{\prime\prime}.$

######  Corollary 2.10.

If $K$ is non-Archimedean, Lemma 2.8 and 2.9 hold for any ultrametric norm.

###### Proof of Lemma 2.9.

Set $N^{\prime}:=\dim V^{\prime}$, $N^{\prime\prime}:=\dim V^{\prime\prime}$,
and denote by $\pi:V\to V^{\prime\prime}$ the given surjection. Pick nonzero
$\tau^{\prime}\in\det V^{\prime}$, $\tau^{\prime\prime}\in\det
V^{\prime\prime}$ and $\varepsilon>0$. Definition 2.1, we may then find
$v^{\prime}_{1},\dots,v^{\prime}_{N^{\prime}}\in V^{\prime}$ and
$v^{\prime\prime}_{1},\dots,v^{\prime\prime}_{N^{\prime\prime}}\in V^{\prime}$
such that $\tau^{\prime}=v^{\prime}_{1}\wedge\dots\wedge
v^{\prime}_{N^{\prime}}$,
$\tau^{\prime\prime}=\pi(v^{\prime\prime}_{1})\wedge\dots\wedge\pi(v^{\prime\prime}_{N^{\prime\prime}})$,

$\prod_{i}\|v^{\prime}_{i}\|\leq(1+\varepsilon)\det\|\tau^{\prime}\|$

and

$\prod_{i}\|v^{\prime\prime}_{i}\|\leq(1+\varepsilon)\det\|\tau^{\prime\prime}\|$

The isomorphism $\det V^{\prime}\otimes\det V^{\prime\prime}\simeq\det V$ maps
$\tau^{\prime}\otimes\tau^{\prime\prime}$ to

$v^{\prime}_{1}\wedge\dots\wedge v^{\prime}_{N^{\prime}}\wedge
v^{\prime\prime}_{1}\wedge\dots\wedge v^{\prime\prime}_{N^{\prime\prime}},$

which satisfies

$\det\|\tau^{\prime}\otimes\tau^{\prime\prime}\|=\det\|v^{\prime}_{1}\wedge\dots\wedge
v^{\prime}_{N^{\prime}}\wedge v^{\prime\prime}_{1}\wedge\dots\wedge
v^{\prime\prime}_{N^{\prime\prime}}\|$
$\leq\prod_{i}\|v^{\prime}_{i}\|\prod_{i}\|v^{\prime\prime}_{j}\|\leq(1+\varepsilon)^{2}(\det\|\tau^{\prime}\|^{\prime})(\det\|\tau^{\prime\prime}\|^{\prime\prime}),$

hence
$\det\|\cdot\|\leq\det\|\cdot\|^{\prime}\otimes\det\|\cdot\|^{\prime\prime}$.
By Lemma 1.22, we dually have

$\det(\|\cdot\|^{\vee})\leq\det(\|\cdot\|^{\prime\prime\vee})\otimes\det(\|\cdot\|^{\prime\vee}).$

Since $\|\cdot\|$ is diagonalizable, so are $\|\cdot\|^{\prime}$ and
$\|\cdot\|^{\prime\prime}$, by Lemma 1.13. By Lemma 2.8, we thus have
$(\det\|\cdot\|)^{-1}\leq(\det\|\cdot\|^{\prime\prime})^{-1}\otimes(\det\|\cdot\|^{\prime})^{-1}$,
hence the result. ∎

###### Proof of Corollary 2.10.

By Theorem 1.19, diagonalizable norms are dense in the set of ultrametric
norms, and we conclude by continuity of $\det$ (Lemma 2.3).

∎

In the Archimedean case, both Lemma 2.6 and Lemma 2.8 fail in general for non-
diagonalizable norms.

######  Example 2.11.

Assume that $K$ is Archimedean, and let $\|\cdot\|$ be the
$\ell^{\infty}$-norm on $K^{N}$. The dual norm $\|\cdot\|^{\vee}$ is the
$\ell^{1}$-norm, and Lemma 2.4 thus shows that
$\det(\|\cdot\|^{\vee})\neq(\det\|\cdot\|)^{\vee}$. Also, the exact sequence
$0\to V^{\prime}\to V\to V^{\prime\prime}\to 0$ with $V^{\prime}=Ke_{1}$,
$V^{\prime\prime}=Ke_{2}$ shows that
$\det\|\cdot\|<(\det\|\cdot\|^{\prime})\otimes(\det\|\cdot\|^{\prime\prime})$.

Recall from Section 1.3 that each basis ${\bf e}=(e_{i})$ of $V$ defines an
apartment $\mathbb{A}_{\bf e}=\iota_{\bf e}(\mathbb{R}^{N})$ in
$\mathcal{N}^{\mathrm{diag}}(V)$ and a Gram-Schmidt projection $\rho_{\bf
e}:\mathcal{N}(V)\to\mathbb{A}_{\bf e}$. For later use, we show:

######  Lemma 2.12.

For each diagonalizable norm $\|\cdot\|$, we have $\det\|\cdot\|=\det\rho_{\bf
e}(\|\cdot\|)$.

###### Proof.

Denote by $W_{i}=\operatorname{Vect}(e_{1},\dots,e_{i})$ the complete flag
defined by ${\bf e}$. By Lemma 2.9, we have

$\det\|\cdot\|=\bigotimes_{i}\det\|\cdot\|_{W_{i}/W_{i-1}},$

under the identification $\det V\simeq\bigotimes_{i}\det(W_{i}/W_{i-1})$. By
definition of $\|\cdot\|_{\bf e}:=\rho_{\bf e}(\|\cdot\|)$, we infer

$\det\|e_{1}\wedge\dots\wedge e_{N}\|=\prod_{i}\|e_{i}\|_{\bf
e}=\det\|e_{1}\wedge\dots\wedge e_{N}\|_{\bf e},$

where the second equality follows from Lemma 2.6. ∎

####  2.3. Relative volume

We introduce the following ’additive version’ of [Tem14, 2.4.3].

######  Definition 2.13.

The _relative volume_ of two norms $\|\cdot\|,\|\cdot\|^{\prime}$ on $V$ is
defined as the real number

$\operatorname{vol}(\|\cdot\|,\|\cdot\|^{\prime}):=\log\left(\frac{\det\|\cdot\|^{\prime}}{\det\|\cdot\|}\right).$

######  Proposition 2.14.

The relative volume satisfies the following properties.

  * (i)

$\operatorname{vol}(\|\cdot\|,\|\cdot\|^{\prime})=\operatorname{vol}(\|\cdot\|,\|\cdot\|^{\prime\prime})+\operatorname{vol}(\|\cdot\|^{\prime\prime},\|\cdot\|^{\prime})$.

  * (ii)

$\operatorname{vol}(\|\cdot\|,e^{c}\|\cdot\|^{\prime})=\operatorname{vol}(\|\cdot\|,\|\cdot\|^{\prime})+Nc$
for $c\in\mathbb{R}$.

  * (iii)

$\|\cdot\|_{1}\leq\|\cdot\|_{2}\Longrightarrow\operatorname{vol}(\|\cdot\|,\|\cdot\|_{1})\leq\operatorname{vol}(\|\cdot\|,\|\cdot\|_{2})$.

  * (iv)

$\operatorname{vol}(\|\cdot\|,\|\cdot\|^{\prime})$ is $N$-Lipschitz continuous
in each variable.

  * (v)

If $K^{\prime}/K$ is a non-Archimedean extension and
$\|\cdot\|_{1},\|\cdot\|_{2}$ are two ultrametric norms on $V$ with ground
field extensions $\|\cdot\|^{\prime}_{1},\|\cdot\|^{\prime}_{2}$ to
$V_{K^{\prime}}$, then
$\operatorname{vol}(\|\cdot\|^{\prime}_{1},\|\cdot\|^{\prime}_{2})=\operatorname{vol}(\|\cdot\|_{1},\|\cdot\|_{2})$.

###### Proof.

The first three properties are obvious, and imply the fourth one as a formal
consequence. By Lemma 1.27 and Lemma 2.6, the last property holds when
$\|\cdot\|_{1},\|\cdot\|_{2}$ are diagonalizable, and the general ultrametric
case follows by density. ∎

As we next show, in the Archimedean case, the relative volume is equivalent to
the (logarithmic) volume ratio. The non-Archimedean case will be analyzed in
the next section.

######  Proposition 2.15.

Assume that $K$ is Archimedean (i.e. $K=\mathbb{R}$ or $\mathbb{C}$), and let
$\|\cdot\|,\|\cdot\|^{\prime}$ be two norms on $V$, with unit balls
$B,B^{\prime}$.

  * (i)

If $\|\cdot\|,\|\cdot\|^{\prime}$ are diagonalizable, then

$\operatorname{vol}(\|\cdot\|,\|\cdot\|^{\prime})=\frac{1}{[K:\mathbb{R}]}\log\left(\frac{\operatorname{vol}(B)}{\operatorname{vol}(B^{\prime})}\right),$

where $\operatorname{vol}$ is any choice of Haar measure on $V$.

  * (ii)

In the general case, we have

$\operatorname{vol}(\|\cdot\|,\|\cdot\|^{\prime})=\frac{1}{[K:\mathbb{R}]}\log\left(\frac{\operatorname{vol}(B)}{\operatorname{vol}(B^{\prime})}\right)+O(N\log
N).$ (2.2)

The constant in $O(N\log N)$ is purely numerical ($10$ would do).

###### Proof.

If $\|\cdot\|$, $\|\cdot\|^{\prime}$ are diagonalizable, we can pick an
orthonormal basis $(e_{i})$ for $\|\cdot\|$ in which $\|\cdot\|^{\prime}$ is
diagonalized, and the change-of-variable formula yields

$\operatorname{vol}(\|\cdot\|,\|\cdot\|^{\prime})=\log\prod_{i}\|e_{i}\|^{\prime}=\frac{1}{[K:\mathbb{R}]}\log\left(\frac{\operatorname{vol}(B)}{\operatorname{vol}(B^{\prime})}\right),$

which proves (i). To prove (ii), we may assume that $\|\cdot\|$ is
diagonalizable, by the cocycle property. By Theorem 1.19 (or the John
ellipsoid theorem), we can find a diagonalizable norm
$\|\cdot\|^{\prime\prime}$ with
$\operatorname{\operatorname{d}_{\infty}}\left(\|\cdot\|^{\prime},\|\cdot\|^{\prime\prime}\right)\leq\log
N$. Since $\operatorname{vol}$ is $N$-Lipschitz continuous, we infer
$|\operatorname{vol}(\|\cdot\|^{\prime},\|\cdot\|^{\prime\prime})|\leq N\log
N$. We similarly have

$\left|\frac{1}{[K:\mathbb{R}]}\log\left(\frac{\operatorname{vol}(B^{\prime})}{\operatorname{vol}(B^{\prime\prime})}\right)\right|\leq
N\log N,$

and we conclude thanks to (i) and the cocycle property. ∎

Here again, the error term is generally nonzero in the non-diagonalizable
case.

######  Example 2.16.

Let $\|\cdot\|_{p}$ be the $\ell^{p}$-norm in a given basis $(e_{i})$ By Lemma
2.4, we have $\operatorname{vol}(\|\cdot\|_{1},\|\cdot\|_{2})=0$ for $p\leq
2$, while the volume of the unit ball of $\|\cdot\|_{1}$ is strictly smaller
than that of $\|\cdot\|_{2}$.

####  2.4. The content of a torsion module

We assume in this section that $K$ is non-Archimedean and non-trivially
valued, with associated valuation $v_{K}=-\log|\cdot|$. The next result was is
proved for instance in [Sch13, Proposition 2.10] (see also [Tem14, Corollary
2.3.8]).

######  Lemma 2.17.

Every finitely presented torsion $K^{\circ}$-module $M$ is isomorphic to a
finite direct sum of cyclic modules

$M\simeq\bigoplus_{i=1}^{r}K^{\circ}/\pi_{i}K^{\circ}.$

with $\pi_{i}\in K^{\circ\circ}$. The sequence
$v_{K}(\pi_{1}),\dots,v_{K}(\pi_{r})$ is further uniquely determined by $M$,
up to permutation.

###### Proof.

Observe first that $M$ is necessarily of the form
$M\simeq\mathcal{V}/\mathcal{V}^{\prime}$ with
$\mathcal{V}^{\prime}\subset\mathcal{V}$ two lattices in a $K$-vector space
$V$. Indeed, pick a finite free $K^{\circ}$-module $\mathcal{V}$ with a
surjection $\mathcal{V}\twoheadrightarrow M$. Since $M$ is finitely presented,
the kernel $\mathcal{V}^{\prime}$ is finitely generated, and hence a lattice
of $V:=\mathcal{V}_{K}$ since $M$ is torsion. We claim that $V$ admits a basis
$(e_{i})$ such that $\mathcal{V}=\bigoplus_{i}K^{\circ}e_{i}$ and
$\mathcal{V}^{\prime}=\bigoplus_{i}K^{\circ}\pi_{i}e_{i}$ for some $\pi_{i}\in
K^{\circ}$, which will yield as desired
$M\simeq\bigoplus_{i=1}^{r}K^{\circ}/\pi_{i}K^{\circ}$ with $v_{K}(\pi_{i})>0$
(since $K^{\circ}/\pi_{i}K^{\circ}=0$ when $v_{K}(\pi_{i})=0$). Indeed,
Proposition 1.14 yields a basis $(e_{i})$ that jointly diagonalizes
$\|\cdot\|_{\mathcal{V}}$ and $\|\cdot\|_{\mathcal{V}^{\prime}}$. As lattice
norms take values in $|K|$, we can arrange that $\|e_{i}\|_{\mathcal{V}}=1$
after multiplying each $e_{i}$ by a scalar. Since
$\mathcal{V}^{\prime}\subset\mathcal{V}$, we then have
$\|e_{i}\|_{\mathcal{V}^{\prime}}=|\pi_{i}^{-1}|$ for some $\pi_{i}\in
K^{\circ}$, and we get the claim by Lemma 1.30. Uniqueness is proved as in the
usual structure theorem for torsion modules over PID’s. ∎

######  Definition 2.18.

The _content_ of a finitely presented torsion module $M$ is defined as

$\operatorname{cont}(M):=\sum_{i}\ v_{K}(\pi_{i})\in(0,+\infty).$

Note that this $-\log$ of the content as defined in [Tem14, 2.6.1].
Alternatively, $\operatorname{cont}(M)$ is obtained by applying $v_{K}$ to the
fractional ideal sheaf $\\{\alpha\in K\mid\alpha\cdot\det M=0\\}$.

######  Example 2.19.

If $K$ is discretely valued with uniformizing parameter $\pi_{K}$, then
$\operatorname{cont}(M)=v_{K}(\pi_{K})\ell(M)$ with $\ell(M)$ the length of
$M$.

The content is closely related to the relative volume. Indeed, as we saw
during the proof of Lemma 2.17, every finitely presented torsion module is the
quotient of two lattices in the same vector space, and we have:

######  Lemma 2.20.

If $\mathcal{V}^{\prime}\subset\mathcal{V}$ are lattices in a given $K$-vector
space $V$, then

$\operatorname{cont}(\mathcal{V}/\mathcal{V}^{\prime})=\operatorname{vol}\left(\|\cdot\|_{\mathcal{V}},\|\cdot\|_{\mathcal{V}^{\prime}}\right).$

Assuming now that $K$ is discretely valued, we conclude this section with an
analogue of Proposition 2.15, relating the relative volume to the virtual
length used in [BG+16]. Recall that the virtual length
$\ell(\mathcal{V}/\mathcal{V}^{\prime})\in\mathbb{Z}$ of two lattices
$\mathcal{V},\mathcal{V}^{\prime}$ in a $K$-vector space is defined as

$\ell(\mathcal{V}/\mathcal{V}^{\prime}):=\ell(\mathcal{V}/\mathcal{V}^{\prime\prime})-\ell(\mathcal{V}^{\prime}/\mathcal{V}^{\prime\prime})$

for any lattice $\mathcal{V}^{\prime\prime}$ contained in both $\mathcal{V}$
and $\mathcal{V}^{\prime}$ (cf. [BG+16, Definition 4.1.1], [Ser, III,§1]).

######  Proposition 2.21.

Assume that $K$ is discretely valued with uniformizing parameter $\pi_{K}$,
and let $\|\cdot\|,\|\cdot\|^{\prime}$ be two ultrametric norms on $V$. Denote
by $B,B^{\prime}$ their unit balls, and by
$\|\cdot\|_{B},\|\cdot\|_{B^{\prime}}$ the associated lattice norms. Then

$\operatorname{vol}\left(\|\cdot\|_{B},\|\cdot\|_{B^{\prime}}\right)=v_{K}(\pi_{K})\ell(B/B^{\prime})=\operatorname{vol}(\|\cdot\|,\|\cdot\|^{\prime})+O(N).$

Note that the absolute value of $K$ is normalized by $v_{K}(\pi_{K})=1$ in
[BG+16].

###### Proof of Proposition 2.21.

The first equality follows from Example 2.19. By Lemma 1.30, we further have
$\operatorname{\operatorname{d}_{\infty}}(\|\cdot\|,\|\cdot\|_{B})\leq
v_{K}(\pi_{K})$ and
$\operatorname{\operatorname{d}_{\infty}}(\|\cdot\|^{\prime},\|\cdot\|_{B^{\prime}})\leq
v_{K}(\pi_{K})$, and hence
$\operatorname{vol}(\|\cdot\|,\|\cdot\|^{\prime})=\operatorname{vol}(\|\cdot\|_{B},\|\cdot\|_{B^{\prime}})+O(N)$
by $N$-Lipschitz continuity of $\operatorname{vol}$. ∎

####  2.5. Relative successive minima

As mentioned in Example 1.5, an ultrametric norm $\|\cdot\|_{0}$ on $V$ with
respect to the trivial absolute $|\cdot|_{0}$ on $K$ is equivalent to the data
of a filtration of $V$. Following the convention of [BC], the latter will be
understood in this section as a decreasing, left-continuous, separating and
exhaustive $\mathbb{R}$-filtration $F^{\bullet}V$, i.e. a family
$(F^{\lambda}V)_{\lambda\in\mathbb{R}}$ of $K$-subspaces of $V$ such that

  * (i)

$F^{\lambda}V\subset F^{\lambda^{\prime}}V$ when
$\lambda\geq\lambda^{\prime}$;

  * (ii)

$F^{\lambda}V=\bigcap_{\lambda^{\prime}<\lambda}F^{\lambda^{\prime}}V$;

  * (iii)

$F^{\lambda}V=0$ for $\lambda\gg 0$;

  * (iii)

$F^{\lambda}V=V$ for $\lambda\ll 0$.

The correspondence is given by

$-\log\|v\|_{0}=\sup\left\\{\lambda\in\mathbb{R}\mid v\in
F^{\lambda}V\right\\},\,\,\,\,\,F^{\lambda}V=\left\\{v\in V\mid\|v\|_{0}\leq
e^{-\lambda}\right\\},$

We define the _jumping values_ $\lambda_{1}\geq\dots\geq\lambda_{N}$ of a
filtration $(F^{\lambda}V)_{\lambda\in\mathbb{R}}$ by setting

$\lambda_{i}:=\sup\left\\{\lambda\in\mathbb{R}\mid\dim F^{\lambda}V\geq
i\right\\}.$

We then have

$-\frac{d}{d\lambda}\dim F^{\lambda}V=\sum_{i=1}^{N}\delta_{\lambda_{i}}$

in the sense of distributions.

######  Definition 2.22.

Let $\|\cdot\|,\|\cdot\|^{\prime}\in\mathcal{N}(V)$ be two norms on $V$ (with
respect to the given absolute value of $K$). The _relative successive minima_
of $\|\cdot\|$ with respect to $\|\cdot\|^{\prime}$ are defined as the jumping
values $\lambda_{i}=\lambda_{i}(\|\cdot\|,\|\cdot\|^{\prime})$ of the
filtration

$F^{\lambda}V:=\operatorname{Vect}\left\\{v\in V\mid\|v\|\leq
e^{-\lambda}\|v\|^{\prime}\right\\}.$

As one immediately sees, the relative successive minima can be alternatively
characterized via a min-max principle, as follows:

$\lambda_{i}(\|\cdot\|,\|\cdot\|^{\prime})=\sup_{W\subset V,\,\dim W\geq
i}\left(\inf_{w\in W\setminus\\{0\\}}\log\frac{\|w\|^{\prime}}{\|w\|}\right).$
(2.3)

In particular,

$\lambda_{1}(\|\cdot\|,\|\cdot\|^{\prime})=\sup_{v\in
V\setminus\\{0\\}}\log\left(\frac{\|v\|^{\prime}}{\|v\|}\right),$
$\lambda_{N}(\|\cdot\|,\|\cdot\|^{\prime})=\inf_{v\in
V\setminus\\{0\\}}\log\left(\frac{\|v\|^{\prime}}{\|v\|}\right)=-\lambda_{1}(\|\cdot\|^{\prime},\|\cdot\|)$

and

$\operatorname{\operatorname{d}_{\infty}}(\|\cdot\|,\|\cdot\|^{\prime})=\max\left\\{\lambda_{1}(\|\cdot\|,\|\cdot\|^{\prime}),\lambda_{1}(\|\cdot\|^{\prime},\|\cdot\|)\right\\}.$
(2.4)

######  Lemma 2.23.

Each $\lambda_{i}(\|\cdot\|,\|\cdot\|^{\prime})$ is a $1$-Lipschitz continuous
function of $\|\cdot\|,\|\cdot\|^{\prime}\in\mathcal{N}(V)$.

###### Proof.

As already observed before, this follows formally from the fact that
$\|\cdot\|^{\prime}\mapsto\lambda_{i}(\|\cdot\|,\|\cdot\|^{\prime})$ is non-
decreasing and satisfies
$\lambda_{i}(\|\cdot\|,e^{t}\|\cdot\|^{\prime})=\lambda_{i}(\|\cdot\|,\|\cdot\|^{\prime})+t$
for $t\in\mathbb{R}$, both of which are straigthforward from (2.3). ∎

In the diagonalizable case, we can mimick the usual min-max characterization
of the eigenvalues of a Hermitian matrix and prove:

######  Proposition 2.24.

Assume that $\|\cdot\|,\|\cdot\|^{\prime}\in\mathcal{N}^{\mathrm{diag}}(V)$
are diagonalizable, and choose a basis $(e_{i})$ of $V$ in which both norms
are diagonalized, as in Proposition 1.14, ordered so that

$\frac{\|e_{1}\|^{\prime}}{\|e_{1}\|}\geq\dots\geq\frac{\|e_{N}\|^{\prime}}{\|e_{N}\|}.$

Then

$\lambda_{i}(\|\cdot\|,\|\cdot\|^{\prime})=\log\frac{\|e_{i}\|^{\prime}}{\|e_{i}\|}.$

###### Proof.

Set $W_{i}:=\operatorname{Vect}(e_{1},\dots,e_{i})$,
$W^{\prime}_{i}:=\operatorname{Vect}(e_{i},\dots,e_{N})$, and observe that

$\log\frac{\|e_{i}\|^{\prime}}{\|e_{i}\|}=\inf_{w\in
W_{i}\setminus\\{0\\}}\log\frac{\|w\|^{\prime}}{\|w\|}=\sup_{w\in
W^{\prime}_{i}\setminus\\{0\\}}\log\frac{\|w\|^{\prime}}{\|w\|}.$

By (2.3), the first equality yields
$\log\frac{\|e_{i}\|^{\prime}}{\|e_{i}\|}\leq\lambda_{i}(\|\cdot\|,\|\cdot\|^{\prime})$.
On the other hand, each subspace $W\subset V$ with $\dim W\geq i$ satisfies
$W\cap W^{\prime}_{i}\neq\\{0\\}$ for dimension reason, and hence

$\inf_{w\in W\setminus\\{0\\}}\log\frac{\|w\|^{\prime}}{\|w\|}\leq\sup_{w\in
W^{\prime}_{i}\setminus\\{0\\}}\log\frac{\|w\|^{\prime}}{\|w\|}=\log\frac{\|e_{i}\|^{\prime}}{\|e_{i}\|},$

and using (2.3) again yields
$\lambda_{i}(\|\cdot\|,\|\cdot\|^{\prime})\leq\log\frac{\|e_{i}\|^{\prime}}{\|e_{i}\|}$.
∎

We are now in a position to prove the following analogue of Minkowski’s second
theorem.

######  Theorem 2.25.

For any two norms $\|\cdot\|,\|\cdot\|^{\prime}$ on $V$, we have

$\operatorname{vol}(\|\cdot\|,\|\cdot\|^{\prime})=\sum_{i}\lambda_{i}(\|\cdot\|,\|\cdot\|^{\prime})+O(N\log
N).$

If $\|\cdot\|$ and $\|\cdot\|^{\prime}$ are diagonalizable (or merely
ultrametric, in the non-Archimedean case), then

$\operatorname{vol}(\|\cdot\|,\|\cdot\|^{\prime})=\sum_{i}\lambda_{i}(\|\cdot\|,\|\cdot\|^{\prime}).$

For non-diagonalizable norms, the term in $O(N\log N)$ is nonzero in general:

######  Example 2.26.

Assume that $K$ is Archimedean, and denote by $\|\cdot\|_{p}$ the standard
$\ell^{p}$-norm on $K^{2}$. Then
$\operatorname{vol}(\|\cdot\|_{2},\|\cdot\|_{1})=0$, while
$\lambda_{1}(\|\cdot\|_{2},\|\cdot\|_{1})=\log\sqrt{2}$ and
$\lambda_{2}(\|\cdot\|_{2},\|\cdot\|_{1})=0$.

###### Proof of Theorem 2.25.

Note that $\operatorname{vol}(\|\cdot\|,\|\cdot\|^{\prime})$ and
$\sum_{i}\lambda_{i}(\|\cdot\|,\|\cdot\|^{\prime})$ are both $N$-Lipschitz
continuous in $\|\cdot\|,\|\cdot\|^{\prime}$, by Proposition 2.14 and Lemma
2.23, respectively. When both norms are diagonalizable, Lemma 2.6 and
Proposition 2.24 yield

$\operatorname{vol}(\|\cdot\|,\|\cdot\|^{\prime})=\sum_{i}\lambda_{i}(\|\cdot\|,\|\cdot\|^{\prime}).$

In the general case, any norm lies at distance $O(\log N)$ of
$\mathcal{N}^{\mathrm{diag}}(V)$, by Theorem 1.19, and we conclude by
$N$-Lipschitz continuity. Finally, if $K$ is non-Archimedean,
$\mathcal{N}^{\mathrm{diag}}(V)$ is dense in $\mathcal{N}^{\mathrm{ultr}}(V)$
by Theorem 1.19, and we infer

$\operatorname{vol}(\|\cdot\|,\|\cdot\|^{\prime})=\sum_{i}\lambda_{i}(\|\cdot\|,\|\cdot\|^{\prime}).$

for any two ultrametric norms, by continuity. ∎

######  Remark 2.27.

When $K$ is discretely valued, the results of this section can be traced back
to Mahler’s paper [Mah41].

####  2.6. Graded norms

In this final section, we deduce from [BC] a general existence result for
limits of relative volumes, which will be crucial later on. Let
$R=\bigoplus_{m\in\mathbb{N}}R_{m}$ be a graded $K$-algebra with
$N_{m}:=\dim_{K}R_{m}$ finite for all $m$.

######  Definition 2.28.

A _graded norm_ $\|\cdot\|_{\bullet}$ on $R$ is defined as a sequence of norms
$\|\cdot\|_{m}\in\mathcal{N}_{m}:=\mathcal{N}(R_{m})$ on the graded pieces
$R_{m}$. A graded norm $\|\cdot\|_{\bullet}$ is _submultiplicative_ (resp.
_multiplicative_) if

$\|s\cdot
s^{\prime}\|_{m+m^{\prime}}\leq\|s\|_{m}\cdot\|s^{\prime}\|_{m^{\prime}}\text{
(resp. }\|s\cdot
s^{\prime}\|_{m+m^{\prime}}=\|s\|_{m}\cdot\|s^{\prime}\|_{m^{\prime}})$

for all $m,m^{\prime}\in\mathbb{N}$, $s\in R_{m}$, $s^{\prime}\in
R_{m^{\prime}}$.

######  Definition 2.29.

We say that two graded norms $\|\cdot\|_{\bullet}$,
$\|\cdot\|^{\prime}_{\bullet}$ on $R$ are

  * (i)

_linearly close_ if
$\operatorname{\operatorname{d}_{\infty}}(\|\cdot\|_{m},\|\cdot\|^{\prime}_{m})=O(m)$;

  * (ii)

_asymptotically equal_ if
$\operatorname{\operatorname{d}_{\infty}}(\|\cdot\|_{m},\|\cdot\|^{\prime}_{m})=o(m)$.

In other words, $\|\cdot\|$ and $\|\cdot\|^{\prime}$ are linearly close if the
distortion between $\|\cdot\|_{m},\|\cdot\|^{\prime}_{m}$ has at most
exponential growth, and they are asymptotically equal if the distortion has
subexponential growth. The following lemma is a direct consequence of the
Lipschitz continuity property of Proposition 2.14.

######  Lemma 2.30.

If two graded norms $\|\cdot\|_{\bullet},\|\cdot\|^{\prime}_{\bullet}$ on $R$
are asymptotically equal, then

$\frac{1}{mN_{m}}\operatorname{vol}(\|\cdot\|_{m},\|\cdot\|^{\prime}_{m})\to
0.$

Relying on the Okounkov body technique developed in [BC], we next show:

######  Theorem 2.31.

Assume that the graded $K$-algebra $R$ is integral and of finite type. Let
$\|\cdot\|_{\bullet},\|\cdot\|^{\prime}_{\bullet}$ be two linearly close
submultiplicative graded norms on $R$, and assume that they are linearly close
to some multiplicative graded norm. Then

$\frac{1}{mN_{m}}\operatorname{vol}(\|\cdot\|_{m},\|\cdot\|^{\prime}_{m})$

admits a limit in $\mathbb{R}$.

###### Proof.

Let $\|\cdot\|^{\prime\prime}_{\bullet}$ be a multiplicative graded norm on
$R$ linearly close to $\|\cdot\|_{\bullet}$ and
$\|\cdot\|^{\prime}_{\bullet}$. Since

$\operatorname{vol}(\|\cdot\|_{m},\|\cdot\|^{\prime}_{m})=\operatorname{vol}(\|\cdot\|_{m},\|\cdot\|^{\prime\prime}_{m})-\operatorname{vol}(\|\cdot\|^{\prime}_{m},\|\cdot\|^{\prime\prime}_{m}),$

it is enough to prove the result when $\|\cdot\|^{\prime}$ itself is
multiplicative. The relative successive minima
$\lambda_{i}(\|\cdot\|_{m},\|\cdot\|^{\prime}_{m})$ are the jumping values of
the

$F^{\lambda}R_{m}=\left\\{s\in R_{m}\mid\|s\|_{m}\leq
e^{-\lambda}\|s\|^{\prime}_{m}\right\\},$ (2.5)

and they satisfy

$\operatorname{vol}(\|\cdot\|_{m},\|\cdot\|^{\prime}_{m})=\sum_{i=1}^{N_{m}}\lambda_{i}(\|\cdot\|_{m},\|\cdot\|^{\prime}_{m})+O(N_{m}\log
N_{m})$ (2.6)

by Theorem 2.25. Since $\|\cdot\|$ is submultiplicative and
$\|\cdot\|^{\prime}$ is multiplicative, the filtration $F^{\bullet}R$ is
multiplicative in the sense of [BC], i.e. it satisfies

$F^{\lambda}R_{m}\cdot F^{\lambda^{\prime}}R_{m^{\prime}}\subset
F^{\lambda+\lambda^{\prime}}R_{m+m^{\prime}}.$

Since $\|\cdot\|_{\bullet}$ and $\|\cdot\|^{\prime}_{\bullet}$ are linearly
close, there exists $C>0$ with

$e^{-Cm}\|\cdot\|_{m}\leq\|\cdot\|^{\prime}_{m}\leq e^{Cm}\|\cdot\|_{m},$

and hence $F^{\lambda}R_{m}=\\{0\\}$ for $\lambda>Cm$,
$F^{\lambda}R_{m}=R_{m}$ for $\lambda<-Cm$. The filtration $F^{\bullet}R$ is
thus linearly bounded in the terminology of [BC], and [BC, Theorem A] implies
that the scaled successive minima
$(m^{-1}\lambda_{i}(\|\cdot\|_{m},\|\cdot\|^{\prime}_{m}))$ equidistribute. In
particular,

$\frac{1}{mN_{m}}\sum_{i=1}^{N_{m}}\lambda_{i}(\|\cdot\|_{m},\|\cdot\|^{\prime}_{m})$

admits a limit in $\mathbb{R}$. Since $R$ is finitely generated, $N_{m}$ has
at most polynomial growth by Hilbert’s theorem, hence $\log N_{m}=o(m)$, and
we are thus done by (2.6). ∎

###  3\. Alternative metric structures on spaces of norms

The goal of this section, which stands somewhat appart from the rest of the
paper, is to exploit the properties of determinants of norms to endow the
space of diagonalizable norms with natural metric structures, recovering in
particular the Bruhat-Tits metric in the non-Archimedean case.

####  3.1. The triangle inequality

The relative successive minima of two norms
$\|\cdot\|,\|\cdot\|^{\prime}\in\mathcal{N}(V)$ define a point
$\lambda\left(\|\cdot\|,\|\cdot\|^{\prime}\right)$ in the rational polyhedral
cone

$\mathcal{C}:=\left\\{\lambda\in\mathbb{R}^{N}\mid\lambda_{1}\geq\dots\geq\lambda_{N}\right\\}\simeq\mathbb{R}^{N}/\mathfrak{S}_{N},$

which is a Weyl chamber for the Weyl group $\mathfrak{S}_{N}$ of
$\operatorname{GL}_{N}$. Given an $\mathfrak{S}_{N}$-invariant norm $\chi$ on
$\mathbb{R}^{N}$, set

$\operatorname{d}_{\chi}\left(\|\cdot\|,\|\cdot\|^{\prime}\right):=\chi\left(\lambda\left(\|\cdot\|,\|\cdot\|^{\prime}\right)\right).$

The resulting function
$\operatorname{d}_{\chi}:\mathcal{N}(V)\times\mathcal{N}(V)\to\mathbb{R}_{+}$
is symmetric, with $\operatorname{d}_{\chi}(\|\cdot\|,\|\cdot\|^{\prime})=0$
if and only if $\|\cdot\|=\|\cdot\|^{\prime}$. The following result is
inspired by Gerardin’s proof of [Ger81, 2.4.7, Corollaire 2].

######  Theorem 3.1.

For each $\mathfrak{S}_{N}$-invariant norm $\chi$ on $\mathbb{R}^{N}$,
$\operatorname{d}_{\chi}$ satisfies the triangle inequality on
$\mathcal{N}^{\mathrm{diag}}(V)$, and is characterized as the unique metric on
$\mathcal{N}^{\mathrm{diag}}(V)$ for which

$\iota_{\bf
e}:(\mathbb{R}^{N},\chi)\hookrightarrow(\mathcal{N}^{\mathrm{diag}}(V),\operatorname{d}_{\chi})$

is an isometric embedding for each basis ${\bf e}$ of $V$.

By Proposition 1.17, the Goldman-Iwahori metric
$\operatorname{\operatorname{d}_{\infty}}$ on $\mathcal{N}^{\mathrm{diag}}(V)$
corresponds to the $\ell^{\infty}$-norm on $\mathbb{R}^{N}$. By equivalence of
norms on $\mathbb{R}^{N}$, any metric $\operatorname{d}_{\chi}$ produced by
Theorem 3.1 is Lipschitz equivalent to
$\operatorname{\operatorname{d}_{\infty}}$. Besides the latter, the most
important case is the Euclidian metric $\operatorname{d}_{2}$ induced by the
$\ell^{2}$-norm:

######  Example 3.2.

When $K$ is Archimedean, $\operatorname{d}_{2}$ coincides with the Riemannian
metric of the symmetric space
$\mathcal{N}^{\mathrm{diag}}(V)\simeq\operatorname{GL}_{N}(K)/U_{N}(K)$ (see
Theorem 3.7 below). When $K$ is non-Archimedean,
$(\mathcal{N}^{\mathrm{diag}}(V),\operatorname{d}_{2})$ is a realization of
the Bruhat-Tits building of $\operatorname{GL}_{N}(K)$ with its Euclidian
metric, see for instance [Par00, Chapter III]. In both cases,
$(\mathcal{N}^{\mathrm{diag}}(V),\operatorname{d}_{2})$ is a $\mathrm{CAT}(0)$
metric space.

######  Corollary 3.3.

If $K$ is non-Archimedean, the space $\mathcal{N}^{\mathrm{ultr}}(V)$ of
ultrametric norms is complete metric with respect to
$\operatorname{d}_{\chi}$, which is characterized as the unique compatible
metric such that $\iota_{\bf
e}:(\mathbb{R}^{N},\chi)\hookrightarrow(\mathcal{N}^{\mathrm{ultr}}(V),\operatorname{d}_{\chi})$
is an isometric embedding for all bases ${\bf e}$. For $\chi=\ell^{2}$,
$(\mathcal{N}^{\mathrm{ultr}}(V),\operatorname{d}_{2})$ is a $\mathrm{CAT}(0)$
metric space.

###### Proof.

On $\mathcal{N}^{\mathrm{diag}}(V)$, $\operatorname{d}_{\chi}$ is equivalent
to $\operatorname{d}_{\infty}$ and satisfies the triangle inquality. This is
also the case on $\mathcal{N}^{\mathrm{ultr}}(V)$, by density of
$\mathcal{N}^{\mathrm{diag}}(V)$. Since $\mathcal{N}^{\mathrm{ultr}}(V)$ is
complete for $d_{\infty}$, it is also complete for $d_{\chi}$, and is thus the
completion of $(\mathcal{N}^{\mathrm{diag}}(V),d_{\chi})$. Conversely, any
metric on $\mathcal{N}^{\mathrm{ultr}}(V)$ with the stated property must
coincide with $\operatorname{d}_{\chi}$ on the dense subset
$\mathcal{N}^{\mathrm{diag}}(V)$, hence everywhere. For $\chi=\ell^{2}$, the
Bruhat-Tits building $(\mathcal{N}^{\mathrm{diag}}(V),\operatorname{d}_{2})$
is a $\mathrm{CAT}(0)$ metric space, and this property is preserved under
completion. ∎

As in the construction of the Euclidian metric on any Euclidian building, the
key to the proof of Theorem 3.1 is to show that the Gram-Schmidt projections
introduced in Definition 1.18 are distance-decreasing.

######  Lemma 3.4.

For each basis ${\bf e}$ of $V$, the Gram-Schmidt projection $\rho_{\bf
e}:\mathcal{N}^{\mathrm{diag}}(V)\to\mathbb{A}_{\bf e}$ satisfies

$\operatorname{d}_{\chi}\left(\rho_{\bf e}(\|\cdot\|),\rho_{\bf
e}(\|\cdot\|^{\prime})\right)\leq\operatorname{d}_{\chi}(\|\cdot\|,\|\cdot\|^{\prime})$
(3.1)

for all $\|\cdot\|,\|\cdot\|^{\prime}\in\mathcal{N}^{\mathrm{diag}}(V)$.

We first recall some elementary facts.

######  Definition 3.5.

Given $\lambda,\lambda^{\prime}\in\mathcal{C}$, one says that $\lambda$ is
_majorized by $\lambda^{\prime}$_, written $\lambda\preceq\lambda^{\prime}$,
if
$\lambda_{1}+\dots+\lambda_{i}\leq\lambda^{\prime}_{1}+\dots+\lambda^{\prime}_{i}$
for all $i$, with equality for $i=N$.

######  Lemma 3.6.

We have $\lambda\preceq\lambda^{\prime}$ if and only $\lambda$ belongs to the
convex envelope of the $\mathfrak{S}_{N}$-orbit of $\lambda^{\prime}$, and
then $\chi(\lambda^{\prime})\leq\chi(\lambda)$ for any
$\mathfrak{S}_{N}$-invariant norm $\chi$ on $\mathbb{R}^{N}$.

###### Proof.

It is straightforward to see that any $\lambda$ in the convex envelope of the
$\mathfrak{S}_{N}$-orbit of $\lambda^{\prime}$ satisfies
$\lambda\preceq\lambda^{\prime}$. As observed in [Rad52], the converse is a
simple consequence of the Hahn-Banach theorem. Assuming indeed that
$\lambda\preceq\lambda^{\prime}$, it is enough to show that for each
$\mu\in\mathbb{R}^{N}$ there exists $\sigma\in\mathfrak{S}_{N}$ with

$\sum_{i}\mu_{i}\lambda_{i}\leq\sum_{i}\mu_{i}\lambda^{\prime}_{\sigma(i)}.$

Choose $\sigma$ such that $\mu_{\sigma(1)}\geq\dots\geq\mu_{\sigma(N)}$. Then

$\sum_{i}\mu_{i}\lambda_{i}=\sum_{i}\mu_{\sigma(i)}\lambda_{\sigma(i)}=\sum_{i<N}\left(\mu_{\sigma(1)}-\mu_{\sigma(i+1)}\right)\left(\lambda_{\sigma(1)}+\dots+\lambda_{\sigma(i)}\right)+\mu_{\sigma(N)}\left(\lambda_{\sigma(1)}+\dots+\lambda_{\sigma(N)}\right)$
$\leq\sum_{i<N}\left(\mu_{\sigma(1)}-\mu_{\sigma(i+1)}\right)\left(\lambda^{\prime}_{1}+\dots+\lambda^{\prime}_{i}\right)+\mu_{\sigma(N)}\left(\lambda^{\prime}_{1}+\dots+\lambda^{\prime}_{N}\right)=\sum_{i}\mu_{\sigma(i)}\lambda^{\prime}_{i}=\sum_{i}\mu_{i}\lambda^{\prime}_{\sigma^{-1}(i)}.$

Assume finally that $\lambda\preceq\lambda^{\prime}$, and hence
$\lambda=\sum_{j}t_{j}\mu_{j}$ with $\mu_{i}$ $\mathfrak{S}_{N}$-equivalent to
$\lambda^{\prime}$, $t_{j}\in\mathbb{R}_{+}$ and $\sum_{j}t_{j}=1$. By
$\mathfrak{S}_{N}$-invariance of $\chi$, we infer

$\chi(\lambda)\leq\sum_{j}t_{j}\chi(\mu_{i})=\sum_{j}t_{j}\chi(\lambda^{\prime})=\chi(\lambda^{\prime}).$

∎

###### Proof of Lemma 3.4.

By Lemma 3.6, it will be enough to show that the relative successive minima
$\lambda$ (resp. $\mu$) of $\rho_{\bf e}(\|\cdot\|^{\prime})$ with respect to
$\rho_{\bf e}(\|\cdot\|^{\prime})$ (resp. $\|\cdot\|^{\prime}$ with respect to
$\|\cdot\|$) satisfy $\lambda_{1}+\dots+\lambda_{i}\leq\mu_{1}+\dots+\mu_{i}$
for all $i$, with equality for $i=N$. Set
$W:=\operatorname{Vect}(e_{1},\dots,e_{i})$, and observe that the restriction
$\rho_{\bf e}(\|\cdot\|)_{W}$ of $\rho_{\bf e}(\|\cdot\|)$ to $W$ satisfies by
definition

$\rho_{\bf e}(\|\cdot\|)_{W}=\rho_{{\bf e}_{W}}(\|\cdot\|_{W})$

with ${\bf e}_{W}=(e_{1},\dots,e_{i})$. By Lemma 2.12, we thus have
$\det(\rho_{\bf e}(\|\cdot\|)_{W})=\det(\|\cdot\|_{W})$, and Theorem 2.25
yields

$\lambda_{1}+\dots+\lambda_{i}=\operatorname{vol}\left(\rho_{\bf
e}(\|\cdot\|)_{W},\rho_{\bf e}(\|\cdot\|^{\prime})_{W}\right)$
$=\operatorname{vol}\left(\|\cdot\|_{W},\|\cdot\|^{\prime}_{W}\right)=\sum_{j\leq
i}\lambda_{j}(\|\cdot\|_{W},\|\cdot\|^{\prime}_{W})$ $\leq\sum_{j\leq
i}\lambda_{j}(\|\cdot\|,\|\cdot\|^{\prime})=\mu_{1}+\dots+\mu_{i},$

where the last inequality follows directly from the min-max principle (2.3),
and is an equality when $i=N$, i.e. $W=V$. ∎

###### Proof of Theorem 3.1.

By construction, $\iota_{\bf e}$ is an isometric embedding with respect to
$\chi$ and $\operatorname{d}_{\chi}$, and the latter therefore satisfies the
triangle inequality on each apartment $\mathbb{A}_{\bf e}=\iota_{\bf
e}(\mathbb{R}^{N})$. Pick three diagonalizable norms
$\|\cdot\|_{i}\in\mathcal{N}^{\mathrm{diag}}(V)$, $i=1,2,3$. We may then
choose a basis ${\bf e}$ with $\|\cdot\|_{1},\|\cdot\|_{2}\in\mathbb{A}_{\bf
e}$. Since $\operatorname{d}_{\chi}$ satisfies the triangle inequality on
$\mathbb{A}_{\bf e}$, we have

$\operatorname{d}_{\chi}\left(\|\cdot\|_{1},\|\cdot\|_{2}\right)=\operatorname{d}_{\chi}\left(\rho_{\bf
e}(\|\cdot\|_{1}),\rho_{\bf e}(\|\cdot\|_{2})\right)$
$\leq\operatorname{d}_{\chi}\left(\rho_{\bf e}(\|\cdot\|_{1}),\rho_{\bf
e}(\|\cdot\|_{3})\right)+\operatorname{d}_{\chi}\left(\rho_{\bf
e}(\|\cdot\|_{3}),\rho_{\bf e}(\|\cdot\|_{2})\right),$

which shows that $\operatorname{d}_{\chi}$ satisfies the triangle inequality
on $\mathcal{N}^{\mathrm{diag}}(V)$, by (3.1). ∎

####  3.2. The Archimedean case: Finsler metrics

Assume that $K$ is Archimedean, i.e. $K=\mathbb{R}$ or $\mathbb{C}$, and
denote by $H(V)$ the real vector space of quadratic/Hermitian forms $h$ on
$V$. Each diagonalizable norm $\|\cdot\|\in\mathcal{N}^{\mathrm{diag}}(V)$ is
then associated to a positive definite form $\gamma(v):=\|v\|^{2}$, thereby
defining an embedding of $\mathcal{N}^{\mathrm{diag}}(V)$ as an open convex
subset of $H(V)$. Further, $\mathcal{N}^{\mathrm{diag}}(V)$ is diffeomorphic
to the symmetric space $\operatorname{GL}(N,K)/U(N,K)$, with $U(N,K)$ the
unitary/orthogonal group.

As we now show, the metric $\operatorname{d}_{\chi}$ constructed above is
induced by a natural Finsler metric on $\mathcal{N}^{\mathrm{diag}}(V)$.
Recall first that for each $h,\gamma\in H(V)$ with $\gamma$ positive definite,
one can find a basis ${\bf e}=(e_{i})$ of $V$ which is orthonormal for
$\gamma$ and orthogonal for $h$, i.e.

$\gamma\left(\sum_{i}\alpha_{i}e_{i}\right)=\sum_{i}|\alpha_{i}|^{2}$

and

$h\left(\sum_{i}\alpha_{i}e_{i}\right)=\sum_{i}\lambda_{i}|\alpha_{i}|^{2}$

for all $\alpha_{i}\in K$. The spectrum $(\lambda_{i})$ is independent of the
choice of ${\bf e}$ up to ordering, hence defines a point
$\lambda_{\gamma}(h)\in\mathcal{C}$, and we then have for any two
$h,h^{\prime}\in H(V)$

$\lambda_{\gamma}(h+h^{\prime})\preceq\lambda_{\gamma}(h)+\lambda_{\gamma}(h^{\prime}).$

This is indeed a simple consequence of the min-max principle, known as the _Ky
Fan inequality_. Given a symmetric norm $\chi$ on $\mathbb{R}^{N}$, it follows
as in Lemma 3.6 that setting for each
$\gamma\in\mathcal{N}^{\mathrm{diag}}(V)$

$|h|_{\chi,\gamma}:=\chi\left(\lambda_{\gamma}(h)\right)$

defines a norm on $H(V)$, and we thus get a continuous Finsler norm
$|\cdot|_{\chi}$ on the tangent bundle of $\mathcal{N}$.

######  Theorem 3.7.

The metric $\operatorname{d}_{\chi}$ on $\mathcal{N}^{\mathrm{diag}}(V)$ in
Theorem 3.1 coincides with the length metric defined by the Finsler norm
$|\cdot|_{\chi}$. In other words, for any two
$\gamma,\gamma^{\prime}\in\mathcal{N}^{\mathrm{diag}}(V)$,
$\operatorname{d}_{\chi}(\gamma,\gamma^{\prime})$ is the infimum over all
smooth paths $(\gamma_{t})_{t\in[0,1]}$ in $\mathcal{N}$ joining $\gamma$ to
$\gamma^{\prime}$ of the corresponding length

$\ell_{\chi}(\gamma):=\int_{0}^{1}|\dot{\gamma}_{t}|_{\chi,\gamma_{t}}dt.$

######  Lemma 3.8.

For each basis ${\bf e}$ of $V$, the Gram-Schmidt projection $\rho_{\bf
e}:\mathcal{N}^{\mathrm{diag}}(V)\to\mathbb{A}_{\bf e}$ is a smooth map, and
it satisfies $\rho_{\bf e}^{*}|\cdot|_{\chi}\leq|\cdot|_{\chi}$.

###### Proof.

The smoothness of $\rho_{\bf e}$ follows from its description in terms of the
Gram-Schmidt orthogonalization process. Pick
$\gamma\in\mathcal{N}^{\mathrm{diag}}(V)$, $h\in H(V)$. As in the proof of
Lemma 3.4, it will be enough to show that $\lambda:=\lambda_{\rho_{\bf
e}(\gamma)}(d_{\gamma}\rho_{\bf e}(h))$ is majorized by
$\mu:=\lambda_{\gamma}(h)$. Differentiating the identity

$\operatorname{vol}(\gamma,\gamma^{\prime})=\operatorname{vol}(\rho_{\bf
e}(\gamma),\rho_{\bf e}(\gamma^{\prime}))$

with respect to $\gamma^{\prime}$ shows that the trace
$\operatorname{Tr}_{\gamma}(h)$ satisfies

$\operatorname{Tr}_{\gamma}(h)=\operatorname{Tr}_{\rho_{\bf
e}(\gamma)}(d_{\gamma}\rho_{\bf e}(h)),$

i.e. $\lambda_{1}+\dots+\lambda_{N}=\mu_{1}+\dots+\mu_{N}$. Arguing as in
Lemma 3.4, we apply this fact to the restrictions of $\gamma$ and $h$ to the
span $W$ of $(e_{1},\dots,e_{i})$ for a given $i$, and get

$\lambda_{1}+\dots+\lambda_{i}=\operatorname{Tr}_{\gamma|_{W}}(h|_{W})\leq\mu_{1}\dots+\mu_{i}$

thanks to the min-max principle. ∎

###### Proof of Theorem 3.7.

Pick a basis $e$, and observe that the differential $d_{\lambda}\iota_{\bf
e}:\mathbb{R}^{N}\to H(V)$ at $\lambda\in\mathbb{R}^{N}$ satisfies for all
$\mu\in\mathbb{R}^{N}$

$\lambda_{\iota_{\bf e}(\lambda)}(d_{\lambda}\iota_{\bf
e}(\mu))=\mu\mod\mathfrak{S}_{N}.$

As a result, $\iota_{\bf e}^{*}|\cdot|_{\chi}$ is the constant Finsler norm
$\chi$ on $\mathbb{R}^{N}$, and the $\chi$-length of any smooth path
$\gamma:[0,1]\to\mathbb{A}_{\bf e}$ joining $\gamma=\iota_{\bf e}(\lambda)$ to
$\gamma^{\prime}=\iota_{\bf e}(\lambda^{\prime})$ thus satisfies

$\ell_{\chi}(\gamma)\geq\chi(\lambda^{\prime}-\lambda)=\operatorname{d}_{\chi}(\gamma,\gamma^{\prime}),$

with equality when $\gamma$ is the image of the line segment
$[\lambda,\lambda^{\prime}]$. If $\gamma:[0,1]\to\mathcal{N}$ is now a smooth
path joining $\gamma$ to $\gamma^{\prime}$ in $\mathcal{N}^{\mathrm{diag}}(V)$
only, the previous case applies to $\rho_{\bf
e}\circ\gamma:[0,1]\to\mathbb{A}_{\bf e}$, which combines with (ii) of Lemma
3.8 to give $\ell_{\chi}(\gamma)\geq\ell_{\chi}(\rho_{\bf
e}\circ\gamma)\geq\operatorname{d}_{\chi}(\gamma,\gamma^{\prime})$. ∎

##  Part II Models and metrics

###  4\. Analytification and models

This section reviews some well-known facts on Berkovich analytifications and
models, with an emphasis on the reduced fiber condition. We provide in
particular a direct proof of the relevant version of the Bosch-Lütkebohmert-
Raynaud reduced fiber theorem.

As before, $K$ denotes a field equipped with an arbitrary (Archimedean or non-
Archimedean) complete absolute value. All schemes over $K$ (or $K^{\circ}$ if
$K$ is non-Archimedean) considered below are _separated_ , unless otherwise
specified.

####  4.1. Analytification

To each scheme $X$ of finite type over $K$, Berkovich associates in [Ber,
§3.4] an analytification $X^{\mathrm{an}}$. In the present paper, we only need
the underlying topological space, which is Hausdorff and locally compact. As
such, the analytifications of $X$ and of the reduced scheme
$X_{\operatorname{red}}$ coincide, but it will nevertheless be useful for
inductive arguments to allow $X$ to be non-reduced.

Assume first that $X$ is affine, i.e. $X=\operatorname{Spec}A$ with $A$ a
finite type $K$-algebra. The topological space $X^{\mathrm{an}}$ is defined as
the set of all multiplicative seminorms $A$ extending the given absolute value
on $K$, endowed with the topology of pointwise convergence. The multiplicative
seminorm associated to $x\in X^{\mathrm{an}}$ is denoted by $f\mapsto|f(x)|$.
The set of $f\in A$ with $|f(x)|=0$ is a prime ideal, thereby defining a
natural continuous map $\rho:X^{\mathrm{an}}\to X$. We thus have $|f(x)|=0$ if
and only if $f$ vanishes at $\xi=\rho(x)$, and $f\mapsto|f(x)|$ defines a norm
on the residue field $\kappa(\xi)$.

######  Lemma 4.1.

A function $f\in A$ satisfies $|f|\equiv 0$ on $X^{\mathrm{an}}$ if and only
if $f$ is nilpotent.

###### Proof.

For each closed point $\xi\in X$, the absolute value on $K$ (uniquely) extends
to the finite field extension $\kappa(\xi)$ of $K$, and
$\rho(X^{\mathrm{an}})$ therefore contains the set of closed points of $X$ (in
fact, $\rho$ injects $X^{\mathrm{an}}$ as the set of closed points of $X$ if
$K$ is Archimedean, while $\rho$ maps $X^{\mathrm{an}}$ onto $X$ when $K$ is
non-Archimedean). As a consequence, a function $f\in A$ with $|f|\equiv 0$
vanishes at all closed points of $X$, and hence is nilpotent. ∎

Consider now an arbitrary $K$-scheme of finite type $X$, and cover it with
finitely many affine open subschemes $U_{i}$. Since $X$ is separated, each
$U_{ij}=U_{i}\cap U_{j}$ is affine, and $U_{ij}^{\mathrm{an}}$ is homeomorphic
to the inverse image of $U_{ij}$ in both $U_{i}^{\mathrm{an}}$ and
$U_{j}^{\mathrm{an}}$. We can thus glue $U_{i}^{\mathrm{an}}$ and
$U_{j}^{\mathrm{an}}$ together along their common open subset
$U_{ij}^{\mathrm{an}}$ to define the topological space $X^{\mathrm{an}}$ with
its continuous map $\rho:X^{\mathrm{an}}\to X$. The GAGA theorem [Ber, 3.4.8,
3.5.3] guarantees that $X^{\mathrm{an}}$ is Hausdorff (since we always assume
$X$ separated), locally compact, and $X^{\mathrm{an}}$ is compact if and only
if $X$ is proper.

######  Example 4.2.

If $K$ is Archimedean, the Gelfand-Mazur theorem yields the following
description of $X^{\mathrm{an}}$. When $K=\mathbb{C}$, $X^{\mathrm{an}}$ is
the usual analytification of $X$, i.e. $X^{\mathrm{an}}=X(\mathbb{C})$ with
its Euclidian topology. When $K=\mathbb{R}$, $X^{\mathrm{an}}$ is identified
with the set of closed points of $X$, i.e. the quotient of
$(X\otimes\mathbb{C})^{\mathrm{an}}=X(\mathbb{C})$ by complex conjugation.

######  Example 4.3.

When $K$ is non-Archimedean with valuation $v_{K}=-\log|\cdot|$,
$X^{\mathrm{an}}$ can be seen as a space of semivaluations on $X$, i.e. real
valuations on the residue fields of points of $X$. More precisely, the
bijective map $x\mapsto(\rho(x),v_{x})$ with $v_{x}(f):=-\log|f(x)|$ describes
$X^{\mathrm{an}}$ as the set of pairs $(\xi,v)$ where $\xi\in X$ is a scheme
point and $v:\kappa(\xi)^{*}\to\mathbb{R}$ is a rank $1$ valuation on the
residue field $\kappa(\xi)$ extending $v_{K}$.

####  4.2. Models and reduction

In what follows, $K$ is non-Archimedean (possibly trivially valued). Recall
that the valuation ring $K^{\circ}$ is then noetherian if and only if $K$ is
discretely valued. Let $X$ be a $K$-scheme of finite type.

######  Definition 4.4.

A _model_ of $X$ is a (separated) flat, finite type $K^{\circ}$-scheme
$\mathcal{X}$ together with an identification of $K$-schemes
$\mathcal{X}_{K}\simeq X$.

If $\mathcal{X}^{\prime}$, $\mathcal{X}$ are two models of $X$, we say that
$\mathcal{X}^{\prime}$ _dominates_ $\mathcal{X}$ if there is a _proper_
morphism $\mathcal{X}^{\prime}\to\mathcal{X}$ compatible with the
identifications with the general fiber.

The _special fiber_ of a model $\mathcal{X}$ is the $\widetilde{K}$-scheme of
finite type $\mathcal{X}_{s}:=\mathcal{X}_{\widetilde{K}}$. In the trivially
valued case, $\mathcal{X}=X$ is the only model, and $\mathcal{X}_{s}=X$.

######  Lemma 4.5.

A model $\mathcal{X}$ is automatically finitely presented over $K^{\circ}$.

This follows from a general result of Raynaud-Gruson [RG71, Théorème 3.4.6],
and goes back to Nagata [Nag66, Theorem 3]. For the sake of completeness, we
reproduce here a simple argument due to Antoine Ducros [Duc], which is
basically equivalent to that of Nagata.

###### Proof.

We claim that it is enough to prove the result when $\mathcal{X}$ is
projective over $K^{\circ}$. Indeed, arguing locally, we may first assume that
$\mathcal{X}$ is affine, and we get the claim by choosing a closed embedding
in an affine space and passing to the schematic closure in the corresponding
projective space. Pick a closed embedding
$\mathcal{X}\hookrightarrow\mathbb{P}^{N}_{K^{\circ}}$, and denote by $I$ the
corresponding homogeneous ideal of $R:=K^{\circ}[t_{0},\dots,t_{N}]$. Since
both $(R/I)\otimes K$ and $(R/I)\otimes\widetilde{K}$ are noetherian, we may
choose a finitely generated homogeneous ideal $I^{\prime}\subset I$ such that
$R/I^{\prime}\to R/I$ becomes an isomorphism after tensoring with either $K$
or $\widetilde{K}$. This means that the finitely presented closed subscheme
$\mathcal{X}^{\prime}\subset\mathbb{P}^{N}_{K^{\circ}}$ defined by
$I^{\prime}$ has the same special fiber and generic fiber as $\mathcal{X}$. If
we can show that $\mathcal{X}^{\prime}$ is flat over $K^{\circ}$, it will
coincide with the schematic closure of its generic fiber, which will prove
that $\mathcal{X}=\mathcal{X}^{\prime}$ is finitely presented. But
$\mathcal{X}^{\prime}$ is flat over $K^{\circ}$ if and only if the finite type
$K^{\circ}$-module $\mathcal{V}_{m}:=(R/I^{\prime})_{m}$ is free for all
$m\in\mathbb{N}$ large enough, which is indeed the case since
$\dim_{K}\mathcal{V}_{m}\otimes
K=\dim_{\widetilde{K}}\mathcal{V}_{m}\otimes\widetilde{K}$, by choice of
$I^{\prime}$. ∎

A model $\mathcal{X}$ of $X$ determines a compact subset $\mathcal{X}^{\beth}$
222The letter $\beth$ (’bet’) is the second letter of the Hebrew alphabet. The
chosen notation follows the lead of [Thu07]. of $X^{\mathrm{an}}$ and a
_reduction map_

$\operatorname{red}_{\mathcal{X}}:\mathcal{X}^{\beth}\to\mathcal{X}_{s},$

as follows. If $\mathcal{X}$ is affine, i.e.
$\mathcal{X}=\operatorname{Spec}(\mathcal{A})$ with $\mathcal{A}$ a flat (i.e.
torsion-free) finite type $K^{\circ}$-algebra, then

$\mathcal{X}^{\beth}=\left\\{x\in X^{\mathrm{an}}\mid|f(x)|\leq 1\text{ for
all }f\in\mathcal{A}\right\\},$

and the reduction $\operatorname{red}_{\mathcal{X}}(x)$ of
$x\in\mathcal{X}^{\beth}$ is the point of $\mathcal{X}_{s}$ induced by the
prime ideal $\\{f\in\mathcal{A}\mid|f(x)|<1\\}$. In the general case,
$\mathcal{X}$ is covered by finitely many affine open subschemes
$\mathcal{U}_{i}$, whose generic fibers $U_{i}$ give an affine open cover of
$X$, and

$\mathcal{X}^{\beth}=\bigcup_{i}\mathcal{U}_{i}^{\beth}\subset\bigcup_{i}U_{i}^{\mathrm{an}}=X^{\mathrm{an}}.$

In the language of Example 4.3, $\mathcal{X}^{\beth}$ consists of those
semivaluations on $X$ that admit a center on $\mathcal{X}$, and
$\operatorname{red}_{\mathcal{X}}$ maps a semivaluation to its center, which
is necessarily on $\mathcal{X}_{s}$. By the valuative criterion of properness,
we thus have:

######  Lemma 4.6.

For any model $\mathcal{X}^{\prime}$ dominating $\mathcal{X}$, we have
$(\mathcal{X}^{\prime})^{\beth}=\mathcal{X}^{\beth}$, and
$\mathcal{X}^{\beth}=X^{\mathrm{an}}$ if $\mathcal{X}$ is proper over
$K^{\circ}$ (and hence $X$ proper over $K$).

######  Example 4.7.

In the trivially valued case, $X$ is the only model, and $X^{\beth}$ coincides
with the construction of [Thu07].

The compact set $\mathcal{X}^{\beth}$ associated to a model $\mathcal{X}$ can
also be understood as (the underlying topological space of) the generic fiber
in the sense of [Ber94, §1] of the formal completion $\widehat{\mathcal{X}}$
of $\mathcal{X}$. As above, it is enough to consider the case where
$X=\operatorname{Spec}(A)$ and $\mathcal{X}=\operatorname{Spec}(\mathcal{A})$
are affine. For any two nonzero $\pi,\pi^{\prime}\in K^{\circ\circ}$, there
exists $n\gg 1$ with $\pi^{n}\in K^{\circ}\pi^{\prime}$, and the formal
completion $\widehat{\mathcal{A}}$ of $\mathcal{A}$ with respect to $\pi$ is
thus independent of the choice of $\pi$ (set
$\widehat{\mathcal{A}}=\mathcal{A}=A$ in the trivially valued case). The
$K^{\circ}$-algebra $\widehat{\mathcal{A}}$ is flat and topologically of
finite type, and
$\widehat{\mathcal{X}}=\operatorname{Spf}(\widehat{\mathcal{A}})$ is thus an
admissible formal $K^{\circ}$-scheme, whose generic fiber
$\widehat{\mathcal{X}}_{\eta}$ is defined as the set of bounded multiplicative
seminorms on the $K$-affinoid algebra
$\widehat{A}:=\widehat{\mathcal{A}}\otimes K$. Composing such a seminorm with
the canonical map $A\to\widehat{A}$ defines a continuous map
$\widehat{\mathcal{X}}_{\eta}\to\mathcal{X}^{\beth}$, which is easily see to
be bijective by density of the image of $A$ in $\widehat{A}$, and hence a
homeomorphism.

The following well-known result holds for the reduction map of any admissible
formal scheme (see for instance [GRW15, §2.13]).

######  Lemma 4.8.

The reduction map
$\operatorname{red}_{\mathcal{X}}:\mathcal{X}^{\beth}\to\mathcal{X}_{s}$ of
any model $\mathcal{X}$ is anticontinuous and surjective. Further, the
preimage $\Gamma(\mathcal{X})$ of the set of generic points of
$\mathcal{X}_{s}$ is a finite set.

We shall call $\Gamma(\mathcal{X})$ the set of _Shilov points_. When
$\mathcal{X}$ (and hence $X$) is affine, $\Gamma(\mathcal{X})$ is exactly the
Shilov boundary of the $K$-affinoid domain $\mathcal{X}^{\beth}$ in the sense
of [Ber, 2.4.4]. Covering a general model with affine open subschemes, we get:

######  Lemma 4.9.

For any model $\mathcal{X}$ of $X$ and $f\in\mathcal{O}(X)$, the sup-seminorm
on $\mathcal{X}^{\beth}$ satisfies

$\|f\|_{\mathcal{X}^{\beth}}:=\sup_{\mathcal{X}^{\beth}}|f|=\max_{\Gamma(\mathcal{X})}|f|.$

######  Example 4.10.

If $K$ is trivially valued and $\eta$ is a generic point of $X=X_{s}$, the
point of $X^{\beth}$ corresponding to the trivial valuation on
$\kappa(\eta)=\mathcal{O}_{X,\eta}$ is the unique Shilov point mapping to
$\eta$.

######  Example 4.11.

Assume that $K$ is nontrivially valued and $\mathcal{X}$ is normal, i.e.
integrally closed in $X$ with $X$ normal. For each generic point $\eta$ of
$\mathcal{X}_{s}$, the local ring $\mathcal{O}_{\mathcal{X},\eta}$ is a rank
one valuation ring. This is of course well-known when $\mathcal{X}$ is
noetherian (i.e. $K$ discretely valued), while the general case is proved in
[Kna00, Theorem 2.6.1]. As observed in [GS13, Proposition 2.3], one easily
checks that the corresponding point of $\mathcal{X}^{\beth}$ is the unique
Shilov point of $\mathcal{X}$ mapping to $\eta$.

######  Example 4.12.

If we merely assume that $\mathcal{X}$ is integrally closed in $X$, it is
still true that there is a unique Shilov point mapping to any given generic
point $\eta$ of $\mathcal{X}_{s}$. In the discretely valued case, this is
proved in [CLT09, Lemme 2.1]. In the densely valued case, the assumption
implies that $\mathcal{X}_{s}$ is reduced by Theorem 4.19 below, which also
implies that
$\operatorname{red}_{\mathcal{X}}:\mathcal{X}^{\beth}\to\mathcal{X}_{s}$
coincides with the affinoid reduction map, and we conclude by [Ber,
Proposition 2.4.4].

####  4.3. Sup-seminorm, integral closure and reduced fiber

The goal of this section is to review the relation between sup-seminorm,
integral closure and reduced fiber.

Assume first that $K$ is non-Archimedean and nontrivially valued,
$X=\operatorname{Spec}(A)$ is affine and
$\mathcal{X}=\operatorname{Spec}(\mathcal{A})$ is an affine model of $X$.
Denote as above by $\widehat{\mathcal{A}}$ the formal completion of
$\mathcal{A}$ with respect to any nonzero $\pi\in K^{\circ\circ}$, by
$\widehat{A}:=\widehat{\mathcal{A}}\otimes K$ the associated $K$-affinoid
algebra, and write $f\mapsto\widehat{f}$ for the canonical map
$A\to\widehat{A}$ (which is not injective in general, cf. Example 4.14 below).
The affinoid algebra $\widehat{A}$ is equipped with the _sup-seminorm_
$\|\cdot\|_{\sup}$, defined by setting for $g\in\widehat{A}$

$\|g\|_{\sup}:=\sup_{\mathcal{X}^{\beth}}|g|=\max_{\Gamma(\mathcal{X})}|g|,$

where the second equality holds by [Ber, 2.4.4]. The sup-seminorm on
$\mathcal{X}^{\beth}$ of $f\in A$ as in Lemma 4.9 can thus be written as

$\|f\|_{\mathcal{X}^{\beth}}=\|\widehat{f}\|_{\sup}.$

By [BGR, 6.2.1/4] and [Ber, 3.4.3], we have:

######  Lemma 4.13.

The sup-seminorm on $\widehat{A}$ is a norm if and only if $\widehat{A}$ is
reduced. This holds in particular if $A$ is reduced.

While $A\to\widehat{A}$ has dense image, it is not injective in general, even
when $A$ is reduced:

######  Example 4.14.

If $K$ is discretely valued, $\mathcal{A}:=K$ is of finite type of
$K^{\circ}$, and hence a model of $A=K$, for which $\widehat{A}=\\{0\\}$
(thanks to Antoine Ducros for this simple example).

######  Proposition 4.15.

The unit ball of $\|\cdot\|_{\sup}$ coincides with the integral closure
$\widehat{\mathcal{A}}^{\prime}$ of $\widehat{\mathcal{A}}$ in $\widehat{A}$.
Similarly, the unit ball of $\|\cdot\|_{\mathcal{X}^{\beth}}$ coincides with
the integral closure $\mathcal{A}^{\prime}$ of $\mathcal{A}$ in $A$, and the
induced map $\mathcal{A}^{\prime}\to\widehat{\mathcal{A}}^{\prime}$ further
has dense image.

######  Corollary 4.16.

A given $f\in A$ is integral over $\mathcal{A}$ if and only if $\widehat{f}$
is integral over $\widehat{\mathcal{A}}$, and $\mathcal{A}$ is integrally
closed in $A$ if and only if $\widehat{\mathcal{A}}$ is integrally closed in
$\widehat{A}$.

We start with a useful observation.

######  Lemma 4.17.

For each $f\in A$, we have
$f\in\mathcal{A}\Longleftrightarrow\widehat{f}\in\widehat{\mathcal{A}}$.

###### Proof.

We imitate [BL86, Lemma 1.4]. Pick a non-zero $\pi\in K^{\circ\circ}$ such
that $\pi f\in\mathcal{A}$, and note that the canonical map
$\mathcal{A}\to\widehat{\mathcal{A}}$ induces an isomorphism
$\mathcal{A}/\pi\mathcal{A}\simeq\widehat{\mathcal{A}}/\pi\widehat{\mathcal{A}}$.
If $\widehat{f}\in\widehat{\mathcal{A}}$, we thus have $\pi f=\pi g$ for some
$g\in\mathcal{A}$, and hence $f=g\in\mathcal{A}$ after multiplying by
$\pi^{-1}$ in $A$. ∎

###### Proof of Proposition 4.15.

That the unit ball of $\|\cdot\|_{\sup}$ is the integral closure of
$\widehat{\mathcal{A}}$ is a reformulation of [BGR, 6.3.4/1] (and the remark
that follows). Before dealing with the unit ball of
$\|\cdot\|_{\mathcal{X}^{\beth}}$, we first establish the density of the image
of $\mathcal{A}^{\prime}$ in $\widehat{\mathcal{A}}^{\prime}$.

Let thus $g\in\widehat{\mathcal{A}}^{\prime}$, so that
$g^{n}+\sum_{i=1}^{n-1}b_{i}g^{n-i}\in\widehat{\mathcal{A}}$ for some
$b_{i}\in\widehat{\mathcal{A}}$. Since $\mathcal{A}\to\widehat{\mathcal{A}}$
and $A\to\widehat{A}$ have dense images, we can pick sequences $f_{j}\in A$
and $a_{ij}\in\mathcal{A}$ with $\widehat{f}_{j}\to g$ and
$\widehat{a}_{ij}\to b_{i}$ as $j\to\infty$. As $\widehat{\mathcal{A}}$ is
open in $\widehat{A}$ (for instance by Lemma 4.18 below), it follows that
$\widehat{f}_{j}^{n}+\sum_{i=1}^{n-1}\widehat{a}_{ij}\widehat{f}_{j}^{n-i}\in\widehat{\mathcal{A}}$
for all $j\gg 1$, i.e.
$f_{j}^{n}+\sum_{i=1}^{n-1}a_{ij}f_{j}^{n-i}\in\mathcal{A}$, by Lemma 4.17. As
a result, $f_{j}$ is integral over $\mathcal{A}$, i.e.
$f_{j}\in\mathcal{A}^{\prime}$, which proves that
$\mathcal{A}^{\prime}\to\widehat{\mathcal{A}}^{\prime}$ has dense image.

It remains to show that an element $f\in A$ with
$\|f\|_{\mathcal{X}^{\beth}}\leq 1$ is integral over $\mathcal{A}$. Since
$\|\widehat{f}\|_{\sup}\leq 1$, we already know that $\widehat{f}$ belongs to
$\widehat{\mathcal{A}}^{\prime}$. Since
$\mathcal{A}^{\prime}\to\widehat{\mathcal{A}}^{\prime}$ has dense image and
$\widehat{\mathcal{A}}$ is open in $\widehat{A}$, we find
$f^{\prime}\in\mathcal{A}^{\prime}$ with
$\widehat{f}-\widehat{f}^{\prime}\in\widehat{\mathcal{A}}$, i.e.
$f-f^{\prime}\in\mathcal{A}$, and hence $f\in\mathcal{A}^{\prime}$ (see also
[CMor15, Theorem 2.10]) for a direct proof). ∎

Besides the sup-seminorm $\|\cdot\|_{\mathcal{X}^{\beth}}$, $A$ is also
equipped with a ’lattice seminorm’ $\|\cdot\|_{\mathcal{A}}$, defined by

$\|f\|_{\mathcal{A}}:=\inf\left\\{|\alpha|\mid\alpha\in
K,\,f\in\alpha\mathcal{A}\right\\}.$

By Example 4.14, this is again not a norm in general. However, similarly
setting for $g\in\widehat{A}$

$\|g\|_{\widehat{\mathcal{A}}}:=\inf\left\\{|\alpha|\mid\alpha\in
K,\,g\in\alpha\widehat{\mathcal{A}}\right\\},$

does yield a norm on $\widehat{A}$:

######  Lemma 4.18.

For each $g\in\widehat{A}$, the infimum defining
$\|g\|_{\widehat{\mathcal{A}}}$ is achieved. In particular,
$\widehat{\mathcal{A}}$ is the closed unit ball of
$\|\cdot\|_{\widehat{\mathcal{A}}}$, and $K^{\circ\circ}\widehat{\mathcal{A}}$
is its open unit ball.

###### Proof.

The result is true for the polynomial ring
$\mathcal{B}:=K^{\circ}[t_{1},\dots,t_{r}]$, since
$\|\cdot\|_{\widehat{\mathcal{B}}}$ is then the Gauss norm on the Tate algebra
$\widehat{B}=K\\{t_{1},\dots,t_{r}\\}$. In the general case, choose a
surjection
$\mathcal{B}:=K^{\circ}[t_{1},\dots,t_{r}]\twoheadrightarrow\mathcal{A}$ for
some $r\geq 1$, and observe that $\|\cdot\|_{\widehat{\mathcal{A}}}$ is the
quotient seminorm of $\|\cdot\|_{\widehat{\mathcal{B}}}$ with respect to the
induced surjection $\rho:\widehat{B}\twoheadrightarrow\widehat{A}$. The kernel
of $\rho$, being an ideal in a Tate algebra, is strictly closed [BGR,
5.2.7/8]. By definition, this means that for each $g\in\widehat{A}$, there
exists $h\in\widehat{B}$ such that $\rho(h)=g$ and
$\|h\|_{\widehat{\mathcal{B}}}=\|g\|_{\widehat{\mathcal{A}}}$. Since the
desired result holds for $\widehat{\mathcal{B}}$, we can then find $\alpha\in
K$ with
$|\alpha|=\|h\|_{\widehat{\mathcal{B}}}=\|g\|_{\widehat{\mathcal{A}}}$, which
implies that $h\in\alpha\widehat{\mathcal{B}}$, and hence
$g=\rho(h)\in\alpha\widehat{\mathcal{A}}$. ∎

######  Theorem 4.19.

As above, let $\mathcal{X}=\operatorname{Spec}\mathcal{A}$ be an affine model
of $X=\operatorname{Spec}A$. We then have
$\|\cdot\|_{\sup}\leq\|\cdot\|_{\widehat{\mathcal{A}}}$ on $\widehat{A}$, and
hence $\|\cdot\|_{\mathcal{X}^{\beth}}\leq\|\cdot\|_{\mathcal{A}}$ on $A$.
Consider further the following properties:

  * (i)

$\mathcal{X}_{s}$ is reduced;

  * (ii)

$\|\cdot\|_{\sup}=\|\cdot\|_{\widehat{\mathcal{A}}}$ on $\widehat{A}$;

  * (iii)

$\|\cdot\|_{\mathcal{X}^{\beth}}=\|\cdot\|_{\mathcal{A}}$ on $A$;

  * (iv)

$\widehat{\mathcal{A}}$ is integrally closed in $\widehat{A}$;

  * (v)

$\mathcal{A}$ is integrally closed in $A$.

Then
$(i)\Longleftrightarrow(ii)\Longleftrightarrow(iii)\Longrightarrow(iv)\Longleftrightarrow(v)$.
If $K$ is densely valued, we also have $(v)\Longrightarrow(i)$, so that
$(i)$–$(v)$ are then equivalent.

###### Proof.

Pick a nonzero $g\in\widehat{\mathcal{A}}$. By Lemma 4.18,
$\|g\|_{\widehat{\mathcal{A}}}$ is in the value group $|K^{*}|$, and we may
thus assume that $\|g\|_{\widehat{\mathcal{A}}}=1$ after multiplying $g$ by a
nonzero scalar. We then have $g\in\widehat{\mathcal{A}}$, hence $|g(x)|\leq 1$
for all $x\in\mathcal{X}^{\beth}$, which proves that $\|g\|_{\sup}\leq 1$, and
hence $\|\cdot\|_{\sup}\leq\|\cdot\|_{\widehat{\mathcal{A}}}$.

Suppose now that $\mathcal{X}_{s}$ is reduced, and assume by contradiction
that $g$ as above satisfies $\|g\|_{\sup}<1$. By [BGR, 6.2.3/2], $g$ is
topologically nilpotent, i.e. $g^{n}\to 0$. For $n\gg 1$, we thus have
$\|g^{n}\|_{\widehat{\mathcal{A}}}<1$, i.e. $g^{n}\in
K^{\circ\circ}\widehat{\mathcal{A}}$; since
$\mathcal{A}\otimes\widetilde{K}\simeq\widehat{\mathcal{A}}\otimes\widetilde{K}$
is reduced, this implies $g\in K^{\circ\circ}\widehat{\mathcal{A}}$, which
contradicts $\|g\|_{\widehat{\mathcal{A}}}=1$. We have thus proved
(i)$\Longrightarrow$(ii), which trivially implies (iii) by composing with
$A\to\widehat{A}$. If (iii) holds, then $\|\cdot\|_{\mathcal{A}}$ is power-
multiplicative, i.e. $\|f^{n}\|_{\mathcal{A}}=\|f\|_{\mathcal{A}}^{n}$ for
each $f\in A$ and $n\in\mathbb{N}$. In particular, $f^{n}\in
K^{\circ\circ}\mathcal{A}\Longleftrightarrow\|f^{n}\|_{\mathcal{A}}<1\Longleftrightarrow
f\in K^{\circ\circ}\mathcal{A}$, which means that
$\mathcal{A}\otimes\widetilde{K}\simeq\mathcal{A}/K^{\circ\circ}\mathcal{A}$
is reduced.

Since $\widehat{\mathcal{A}}$ (resp. $\mathcal{A}$) is the unit ball of
$\|\cdot\|_{\widehat{\mathcal{A}}}$ (resp. $\|\cdot\|_{\mathcal{A}})$,
Proposition 4.15 shows that (ii) and (iii) respectively imply (iv) and (v),
while Corollary 4.16 shows that (iv) and (v) are equivalent.

Assume finally that $K$ is densely valued and that (v) holds. To prove (i), we
need to show that each $f\in\mathcal{A}$ such that $f^{n}\in\pi\mathcal{A}$
for some $n\geq 1$ and $\pi\in K^{\circ\circ}$ actually satisfies $f\in
K^{\circ\circ}\mathcal{A}$. Since $K$ is densely valued, we can find
$\pi^{\prime}\in K^{\circ\circ}$ with $|\pi|^{1/n}\leq|\pi^{\prime}|<1$, and
hence $\pi/\pi^{\prime n}\in K^{\circ}$. As a result, $g:=\pi^{\prime-1}f\in
A$ satisfies $g^{n}\in\mathcal{A}$, and hence $g\in\mathcal{A}$, since
$\mathcal{A}$ is integrally closed in $A$. We have thus shown as desired that
$f=\pi^{\prime}g\in K^{\circ\circ}\mathcal{A}$ (we are grateful to Walter
Gubler for his help with this argument). ∎

We conclude this section with the following rather special case of the scheme-
theoretic version of the Bosch-Lütkebohmert-Raynaud reduced fiber theorem
[BLR95, Theorem 2.1’].

######  Theorem 4.20.

Assume that $K$ is non-Archimedean and algebraically closed, and let $X$ be a
reduced $K$-scheme of finite type. For each model $\mathcal{X}$ of $X$, the
integral closure $\mathcal{X}^{\prime}$ of $\mathcal{X}$ in $X$ is finite over
$\mathcal{X}$, and hence a model of $X$ as well. Further,
$\mathcal{X}^{\prime}_{s}$ is reduced.

Note conversely that the existence of a model with reduced special fiber
implies that $X$ is reduced, by [EGA, IV,12.1.1].

###### Proof of Theorem 4.20.

In the trivially valued case, there is nothing to prove, and we thus assume
that $K$ is nontrivially valued. Being algebraically closed, it is then
densely valued, and Theorem 4.19 thus shows that $\mathcal{X}^{\prime}_{s}$
will automatically be reduced.

The finiteness of $\mathcal{X}^{\prime}$ over $\mathcal{X}$ being local, we
assume that $\mathcal{X}=\operatorname{Spec}(\mathcal{A})$ is affine and use
the above notation. We will reduce the result to the Grauert-Remmert
finiteness theorem, basically arguing as in [BL86, Proposition 1.5] and
[Tem10, Theorem 3.5.5, Step 3].

Since $A$ is reduced, $\widehat{A}$ is reduced as well by Lemma 4.13, and
[BGR, 6.4.1/5] thus shows that $\widehat{\mathcal{A}}^{\prime}$ is finite over
$\widehat{\mathcal{A}}$, i.e.
$\widehat{\mathcal{A}}^{\prime}=\sum_{i}\widehat{\mathcal{A}}g_{i}$ for a
finite set $g_{i}\in\widehat{A}$, in which we include $1$ for convenience. As
in the proof of Lemma 4.17, we can find for each $i$ some $f_{i}\in A$ with
$g_{i}-\widehat{f}_{i}\in\widehat{\mathcal{A}}$, and hence
$\widehat{\mathcal{A}}^{\prime}=\sum_{i}\widehat{\mathcal{A}}\widehat{f}_{i}$.
By Corollary 4.16, an element $f\in A$ belongs to the integral closure
$\mathcal{A}^{\prime}$ of $\mathcal{A}$ in $A$ if and only $\widehat{f}$
belongs to
$\widehat{\mathcal{A}}^{\prime}=\sum_{i}\widehat{\mathcal{A}}\widehat{f}_{i}$,
which is also equivalent to $f\in\sum_{i}\mathcal{A}f_{i}$ by Lemma 4.17. We