# The perfectoid Tate algebra has uncountable Krull dimension

Jack J Garzella

###### Abstract

Let $K$ be a perfectoid field with pseudo-uniformizer $\pi$. We adapt an
argument of Du in [Du20] to show that the perfectoid Tate algebra $K\langle
x^{1/p^{\infty}}\rangle$ has an uncountable chain of distinct prime ideals.
First, we conceptualize Du’s argument, defining the notion of a Newton polygon
formalism on a ring. We prove a version of Du’s theorem in the prescence of a
sufficiently nondiscrete Newton polygon formalism. Then, we apply our
framework to the perfectoid Tate algebra via a "nonstandard" Newton polygon
formalism (roughly, the roles of the series variable $x$ and the pseudo-
uniformizer $\pi$ are switched). We conclude a similar statement for
multivatiate perfectoid Tate algebras using the one-variable case. We also
answer a question of Heitmann, showing that if $R$ is a complete local
noetherian domain of mixed characteristic $(0,p)$, the $p$-adic completion of
it’s absolute integral closure $R^{+}$ has uncountable Krull dimension.

##  1 Introduction

Perfectoid rings have been extremely useful tools since their introduction by
Scholze in [Sch12], leading to many advances in the mixed characteristic
theory of commutative algebra (see [And18], [MS21], [BIM19]), $p$-adic Hodge
theory ([Sch14], [Sch13], [KL15]) and the Langlands program (e.g. [CS17],
[Sch15], [FS21]). However, their study as algebraic objects is limited due to
the fact that any perfectoid ring which is not a field is non-noetherian
[BCKW19, §2, Corollary 2.9.3], so the vast majority of theorems in commutative
algebra do not apply.

Despite their non-noetherianness, perfectoid rings have many nice properties,
such as the tilting equivalence ([KL15] and [Sch12]). Most if not all of these
properties come from the fact that perfectoid rings are geometrically
meaningful, which in this case precisely means they are rings of functions
associated to meaningful geometric objects via Huber’s theory of adic spaces
(see [Hub96], the geometric objects in question are always analytic adic
spaces). One may then ask the question

######  Question 1.1.

Do perfectoid rings form a nice class of non-noetherian rings for which we can
“do commutative algebra”?

This question is of special interest when the perfectoid rings are (quotients
of) a perfectoid Tate algebra over a field, i.e.

$K\langle x_{1}^{1/p^{\infty}},\ldots,x_{n}^{1/p^{\infty}}\rangle.$

Such rings form the so-called perfectoid covers of noetherian rigid analytic
(adic) spaces, and thus are the most closely related to noetherian rings.

The purpose of this work is to show, at least in one sense, the answer to this
question is negative.

######  Theorem 1.2 (Corollary 5.9).

Let $K$ be a perfectoid field. Then the perfectoid Tate algebra in $n$
variables has uncountable Krull dimension.

In [LL21], Lang and Ludwig show that Fontaine’s period ring
$\mathbb{A}_{\inf}$ has infinite Krull dimension. Building on this, Du shows
in [Du20] that $\mathbb{A}_{\inf}$ in fact has uncountable Krull dimension.
Du’s method additionally applies to the ring of formal power series over a
nondiscretely valued ring. Du uses a family of valuations arising from
mutiplicative Gauss norms on $\mathbb{A}_{\inf}$, explicitly calculating some
Newton polygons of elements and estimating the asymptotic behavior of their
Legendre transforms.

In this work, we conceptualize Du’s argument, defining a framework in which
Du’s argument applies, which includes more general rings than are treated in
[Du20]. We also provide a version of Du’s proof which uses basic properties of
little omega notation (a variant of Big O) instead of explicit estimates.
Then, we apply this formalism to the perfectoid Tate algebra over
$\mathcal{O}_{K}$, the ring of integers of $K$, and check that the ideals
which form the uncountable chain do not contain our selected pseudo-
uniformizer $\pi$.

There are two important points–firstly, the roles of $x$ and $\pi$ are swapped
from the “standard” way that one thinks about Newton polygons. This exchange
of the arithmetic and geometric variables is reminiscent of ideas from the
world of varieties over function fields, and reflects the “perfectoid outlook”
that $p$ (or more generally, $\pi$) should be treated as “just another
variable”.

Secondly, unlike many proofs about perfectoid rings, the result in mixed
characteristic does not follow from tilting to characteristic $p$–in
particular, as the tilting map is not additive, there no guarantee that the
untilt of an arbitrary prime ideal should even be closed under addition.
Instead, we prove the theorem in each characteristic separately, in both cases
passing to the spherical completion, and using the explicit description of the
spherical completion described in [Poo93]. In fact, passing to the explicit
description of the spherical completion happens to be just what is needed for
formalizing for Du’s argument, which roughly corresponds to the fact that in
order to use Newton polygon methods, one has to “tell apart the coefficients
from the variables”.

Finally, we apply our same formalism to prove that $\widehat{R^{+}}$, the
$p$-adic completion of the absolute integral closure of a noetherian local
domain $R$ of mixed characteristic and dimension greater than 2, has
uncountable Krull dimension. This answers a question posed in [Hei22].

All of the examples of rings with uncountable Krull dimension via Du’s method
have the one thing in common: they are “nondiscrete” with respect to one
variable, and complete with respect to another. For example, if we consider
$R=\mathcal{O}_{K}[[x]]$, where $K$ is a perfectoid field, $R$ is complete
with respect to $x$, and nondiscrete with respect to $\pi$ (the pseudo-
uniformizer). On the other hand, if $R=\mathcal{O}_{K}\langle
x^{1/p^{\infty}}\rangle$, then $R$ is complete with resepct to $\pi$ and
nondiscrete with respect to $x$ (it is also nondiscrete with respect to $\pi$,
but this is immaterial to the argument). Roughly, these two facts guarantee
the ability to construct Newton polygons of (sufficiently) arbitrary shapes.
Completeness allows for infinite Newton polygons and ensures that a
convergence condition is not a restriction on the possible shapes of Newton
polygons, while nondiscreteness allows these Newton polygons to approach 0 as
the index variable goes to infinity. Uncountability via Du’s method follows
from the ability to construct such Newton polygons of elements in $R$.

Thus, we expect that any perfectoid ring which “looks at least two
dimensional” (for example, has a regular element which is algebraically
independent from $\pi$) to have uncountable Krull dimension. Furthermore, we
expect that Du’s method will apply to such rings, given a sufficently explicit
way of writing down Newton polygons (we make this precise in Section 3).

In Section 2, we give a bit of technical background on Mal’cev-Neumann Series,
the key ingredient in Poonen’s explicit description of spherical completions.
Section 2.1 is devoted to proving the multiplicativity of Gauss norms on
Mal’cev-Neumann series in a very general situation. In this general setting,
we give an example of an element whose Gauss norm is not a maximum and is
instead an honest supremum. Getting around such examples is quite technical
and not strictly necessary for our examples, but it could have independent
interest. Section 2.1 can be skipped for a first reading. In Section 3, we
define the an abstract formalism for Newton polygons of elements of a ring,
and show a few properties. In Section 4, we outline Du’s argument and prove it
for an arbitrary Newton polygon formalism. In Section 5, we apply Du’s theorem
to various perfectoid rings, including the (p-adic completion of) the absolute
integral closure and the perfectoid Tate algebra.

###  1.1 Acknowledgements

The author would like to thank Kiran Kedlaya for providing much support during
the process of writing this paper. Moreover, the author would like to thank
Hanlin Cai, Gabriel Dorfsman-Hopkins, Alexander Mathers, Shravan Patankar,
Karl Schwede, Kevin Tucker, and Peter Wear for helpful conversations and
comments. In addition, the author thanks Shravan Patankar for suggesting the
question of Heitmann.

This material is based on work partially supported by the National Science
Foundation Graduate Research Fellowship Program under Grant No. 2038238, and a
fellowship from the Sloan Foundation.

##  2 Background on Mal’cev-Neumann Series

We define Mal’cev-Neumann rings in a slightly more general setting than they
sometimes appear in the literature, as we will need this later.

######  Definition 2.1.

Let $R$ be a ring, and let $\Gamma$ be a totally ordered group. The Mal’cev-
Neumann series (in one variable) over $R$ with value group $\Gamma$ is the set
of formal sums

$f=\sum_{i\in\Gamma_{\geq 0}}a_{i}t^{i}$

such that the support of $f$, i.e. $\\{i|a_{i}\neq 0\\}$, is well-ordered. It
is denoted $R[[t^{\Gamma_{\geq 0}}]]$.

One can define addition and multiplication as with standard power series,
using the well-ordered property to guarentee that the ring is closed under
multiplication. Note that since we only allow positive exponents,
$R[[t^{\Gamma_{\geq 0}}]]$ is the “integral” version of the Mal’cev-Neumann
series. We can analogously define a “rational” version, which is a field when
$R=K$, and we denote this by $K((t^{\Gamma}))$.

######  Definition 2.2.

Let $R$ be a valued ring. Then the Gauss norm of radius $\rho$ on
$R[[t^{\Gamma_{\geq 0}}]]$ is given by

$\sup_{i}\\{|a_{i}|\rho^{i}\\}$

for $0<\rho<1$.

Note that we have not yet proven that the Gauss norm is a norm, we will do
this in a bit. We also make use of the following (slightly generalized)
$p$-adic version of the Mal’cev-Neumann series, as defined in [Poo93].

######  Definition 2.3.

Let $R$ be a $W(k)$-algebra, for $k$ a field of characteristic $p$. Let
$\Gamma$ be a totally ordered group with a fixed embedding
$\mathbb{Z}\subset\Gamma$.

We say a series $\sum_{i}a_{i}t^{i}$ in $R[[t^{\Gamma_{\geq 0}}]]$ is a null
series if $\sum_{j\in i+\mathbb{Z}}a_{j}p^{j}=0$ in $R$. The set of null
series is an ideal $N$ (use the proof of [Poo93], Prop 4.3), and we define the
$p$-adic Mal’cev-Neumann series as

$R[[p^{\Gamma_{\geq 0}}]]\colonequals R[[t^{\Gamma_{\geq 0}}]]/N$

As before, we can define a “rational” version for a field $K$ which contains
the fraction field of $W(k)$, with similar operations. We denote this by
$K((p^{\Gamma}))$. We record the following folklore lemma, due to Kedlaya:

######  Lemma 2.4 (Kedlaya).

Let $k$ be a perfect field of characteristic $p$. Let $R$ be a $p$-complete
$W(k)$-algebra which embeds into $W(R/p)$. Then

$R[[p^{\Gamma_{\geq 0}}]]\cong\widehat{R[[t^{\Gamma_{\geq 0}}]]}/(t-p)$

where the completion is $p$-adic.

###### Proof.

First, consider the natural projection from $R[[t^{\Gamma_{\geq 0}}]]$ to
$R[[p^{\Gamma_{\geq 0}}]]$ from the definition of $R[[p^{\Gamma_{\geq 0}}]]$.
Now, take the induced map on $p$-adic completions. This map is surjective as
completions preserve surjections, and as $R[[p^{\Gamma_{\geq 0}}]]$ is
$p$-adically complete, we have

$\pi:\widehat{R[[t^{\Gamma_{\geq 0}}]]}\to R[[p^{\Gamma_{\geq 0}}]].$

So it is enough to show that the kernel of this map is $(t-p)$.

First, however, we show that the kernel is contained in $(t,p)$. Consider the
$p$-adic norm on $R$ (restrict the $p$-adic norm from $W(R/p)$). Then let
$|\cdot|$ be the Gauss norm on $R[[t^{\Gamma_{\geq 0}}]]$ of radius
$\epsilon=|p|$ which extends the $p$-adic norm on $R$. Suppose
$f=\sum_{i}a_{i}t^{i}$ is a null series, but not in the ideal $(t,p)$. Then
there exists an index $i$ with $i<1$ such that $\epsilon<|a_{i}|$. Then,
consider the sum

$f^{\prime}=\sum_{j\in i+\mathbb{N}\cup\\{0\\}}a_{j}t^{j},$

which is zero in $R$ because $f$ is a null series. Thus,

$a_{i}p^{i}=\sum_{j\in i+\mathbb{N}}a_{j}t^{j}.$

However, the norm of the first term of $f^{\prime}$ has

$\epsilon^{1+i}=\epsilon(\epsilon^{i})=\epsilon|t^{i}|<|a_{i}t^{i}|$

every other term has $1+i<j$, so

$|a_{j}t^{j}|<|t^{j}|<\epsilon^{1+i}.$

By the strong triangle inequality, the left hand side of Equation (*). must
have greater norm than the right hand side, a contradiction.

Finally, we show that $f$ is indeed in $(t-p)$, though here we need to use the
fact that we took the $p$-adic completion. Since $\\{t-p,p\\}$ is a generating
set for the ideal $(t,p)$, we may write $f=a(t-p)+bp$. We now show that
$b\in\operatorname{Ker}\pi$: as $t-p\in\operatorname{Ker}\pi$, $bp$ certainly
is. Now, it is clear from the definition of null series that $N$ is a
$p$-saturated ideal, that is $bp\in N\implies b\in N$. So $b\in N$, and

$f=a(t-p)+p(c(t-p)+dp)=a(t-p)+pc(t-p)+p^{2}d.$

And we see that mod $p^{2}$, $f$ is divisible by $t-p$. Continuing by
induction, f is divisible by $t-p$ mod $p^{n}$ for all $n$, so it is divisible
by $t-p$ in the completion. ∎

We also have the following lemma, which is a modified version of [Poo93,
Proposition 4.4], but adjusted to our context.

######  Lemma 2.5.

Let $R$ be a $p$-complete $W(k)$-algebra which embeds into $W(R/p)$. Then any
element in $R[[p^{\Gamma_{\geq 0}}]]$ can be written uniquely as a sum
$f=\sum_{i}a_{i}p^{i}$ with $a_{i}\in R$ such that $p\nmid a_{i}$.

###### Proof.

We follow [Poo93, Proposition 4.4], in particular choose the set of
representatives $\Gamma^{\prime}=\Gamma\cap[0,1]$ of $\Gamma/\mathbb{Z}$, and
let $f=\sum_{i}a_{i}p^{i}$. Then we can consider for any
$\gamma\in\Gamma^{\prime}$,
$f_{\gamma}=\sum_{n\in\mathbb{N}}a_{\gamma+n}p^{n}$, which is an element of
$\widehat{R}$. Since $R$ embeds into $W(R/p)$, we have that $f_{\gamma}$ has a
unique representation in $\widehat{R}$. Then, finish the proof with exactly
the same logic as [Poo93]:

$\sum_{\gamma\in\Gamma^{\prime}}f_{\gamma}$

is an element of $R[[p^{\Gamma_{\geq 0}}]]$ which is equivalent to $f$. ∎

We would like to define the $p$-adic version of the Gauss norm, but we must be
slightly careful about what the “norm on the base ring” precisely is:

######  Definition 2.6.

Let $k$ be a perfect field of characteristic $p$, and let $R$ be a
$W(k)$-algebra which embeds into $W(R/p)$. We say that $R$ has a family of
$p$-adic norms if $R$ has a (continuous) family of norms/valuations
$|\cdot|_{\rho}^{\text{base}}$ for all $\rho\in[0,1]$, subject to the
conditions that

  * •

$|p|_{\rho}^{\text{base}}=\rho$

  * •

$|\cdot|_{\rho}^{\text{base}}$ agree for any element which is not divisible by
$p$.

######  Example 2.7.

Suppose that $R/p$ is perfect and that $R$ can be embedded in $W(R/p)$. Then
$R$ has a family of $p$-adic norms by restricting from the Witt vectors. In
this case, we can write any element $f$ of $R$ uniquely as a sum
$f=\sum_{i}a_{i}p^{i}$ where $a_{i}$ is in the image of the multiplicative
lift in $W(R/p)$.

######  Definition 2.8.

Suppose $R$ is a $W(k)$-algebra which has a family of $p$-adic valuations and
can be embedded into $W(R/p)$. Then the $p$-adic Gauss norm of radius $\rho$
on $R[[p^{\Gamma_{\geq 0}}]]$ is defined to be

$|f|_{\rho}:=\sup_{i}\\{|a_{i}|_{\rho}^{\text{base}}\rho^{i}\\}.$

One of the main applications of Mal’cev-Neumann rings is constructing the
spherical completion of a field of characteristic $p$.

######  Theorem 2.9 (Poonen, [Poo93], Corollary 5.5-5.6).

Let $K$ be a nonarchimedean algebraically closed field with valuation $v$,
residue field $k$ and value group $\Gamma$. Then there exists a unique (up to
non-unique isomorphism) spherically complete extension of $K$, which is also
maximally complete ([Kap42], [Kap45]). We denote this as $K^{sp}$. Being a
field, $K$ contains either $\mathbb{Q}$ or $\mathbb{F}_{p}$.

  * •

If the restriction of $v$ to $\mathbb{Q}$ or $\mathbb{F}_{p}$ is the trivial
valuation, then $K^{sp}\cong k((t^{\Gamma}))$.

  * •

If $K$ has characteristic zero and the restriction of $v$ to $\mathbb{Q}$ is
the $p$-adic valuation, then
$K^{sp}\cong\operatorname{Frac}(W(k))((p^{\Gamma}))$.

######  Remark 2.10.

If we have a $W(k)$-algebra $R$ of mixed characteristic, we may choose a
pseudo-uniformizer $\pi$ with $\pi^{p}|p$ and choosing an element $f\in
R[[t^{\Gamma_{\geq 0}}]]$ whose $t$-adic valuation is equal to $\pi$, we can
adapt Poonen’s construction to use a chosen pseudo-uniformizer, and our
expansions will be $\pi$-adic instead of $p$-adic. All of our proofs about
$p$-adic Mal’cev-Neumann series apply to the $\pi$-adic variant with the
proofs simply replacing $p$ with $\pi$.

###  2.1 Gauss Norms on Mal’cev-Neumann Series

We show that the Gauss norm is a norm, and that it is multiplicative. We will
prove multiplicativity for the full ring of Mal’cev-Neumann series. However,
most concrete applications require multiplicativity only for various subrings
with finiteness conditions, whose proofs are much easier. We will do the
general case, skipping the simpler ad hoc cases. This will require a bit of
extra work.

We now investigate an obstacle to proving the multiplicativity of Gauss norms
on Mal’cev-Neumann rings (and working with these rings in general). For many
proofs of multiplicativity of Gauss norms on simpler rings, one uses that the
$\sup$ in the definition of the Gauss norm is a maximum. However, this is not
true in general for Mal’cev-Neumann rings.

######  Definition 2.11.

The Puiseux series over the ring $R$ is the power series ring $R[[x]]$
adjoined with all $n$-th roots of $x$, that is

$\bigcup_{n\in\mathbb{N}}R[[x^{1/n}]].$

We will denote this ring by $R[[x^{1/\mathbb{N}}]]$.

######  Example 2.12.

Let $R=k[[x^{1/\mathbb{N}}]]$. Fix $\mu$ between zero and one, and equip $R$
with the valuation associated to the Gauss norm of radius $\mu$. Now, we will
construct an element of $R[[x^{1/\mathbb{N}}]][[t^{\Gamma_{\geq 0}}]]$ whose
Gauss norm is not a maximum. Pick $\rho$ between zero and one. Pick a sequence
$\delta_{n}\to 0$ with $\delta_{n}\in\Gamma\cap[0,1]$ such that

$\delta_{n}<\frac{1}{n}\log_{\rho}(\mu).$

Then, consider the element

$f=\sum_{n=1}^{\infty}x^{1+\frac{1}{n}}t^{2-\delta_{n}}.$

Note that the support of $f$, the set $\\{2-\delta_{n}\\}$, is well-ordered.
We have

$|f|_{\rho}=\sup\\{\mu^{1+\frac{1}{n}}\rho^{2-\delta_{n}}\\}=\mu\rho^{2},$

yet the norm of every term of $f$ is less than $\mu\rho^{2}$.

Example 2.12 is not necessarily an obstacle to the Gauss (semi-)norms on
$R[[t^{\Gamma_{\geq 0}}]]$ and $R[[p^{\Gamma_{\geq 0}}]]$ being
multiplicative. However, it is an obstacle to the standard proofs of
multiplicativity. Later in the proof, we will see that the real difficulty is
not the fact that the norm is not achieved, but that there are “coefficients
with arbitrarily close norm” to the norm of $f$. To reason about such
elements, we introduce the following notion.

For the following, let $f=\sum_{i}a_{i}r^{i}$ be an element of the $t$\- or
$p$-adic Mal’cev-Neumann series. If one wants, one may only think about the
$p$-adic series, the proofs for $t$-adic are strictly easier.

######  Definition 2.13.

The $\operatorname{argnorm}$ of an element $f$ is the smallest value $i$ for
which one of the following happens.

  * •

The Gauss norm of $f$ is achieved at index $i$.

  * •

There exists a sequence $i_{n}\to i$ which achieves the Gauss norm of $i$.

Such a smallest $i$ exists by the well-ordered property.

In addition, we fix a $\rho$ and fix the following notation: let
$\operatorname{res}_{I}f$ be the element with coefficients of $f$ supported on
$I$. Let $\operatorname{res}_{I}^{N\leq}f$ be the element whose coefficients
correspond to those of $f$ such that $i\in I$ and $N\leq|a_{i}|\rho^{i}$.

We record two lemmas

######  Lemma 2.14.

[Box Lemma] There exist numbers $\epsilon_{a},\delta_{a}$, depending only on
$f$, for which

$\operatorname{res}_{I}^{|f|-\delta_{a}\leq}f=0$

for $I=(\operatorname{argnorm}f,\operatorname{argnorm}f+\epsilon_{a})$.

###### Proof.

If there did not exist such numbers, we would have that for all $\epsilon$
greater than $0$, there exists a nonzero coefficient (i.e. an $i$ for which
$a_{i}\neq 0$), between $\operatorname{argnorm}f$ and
$\operatorname{argnorm}f+\epsilon$, which contradicts well-orderedness of the
support of $f$. ∎

######  Lemma 2.15.

[Bar Lemma] Given an $\epsilon$ small enough, there exists a
$\delta_{b,\epsilon}$ for which

$\operatorname{res}_{I}^{|f|-\delta_{b}\leq}f=0$

for $I=[0,\operatorname{argnorm}f-\delta_{b,\epsilon})$. If $\epsilon$ is a
fixed $\epsilon_{b}$, then we shorten the notation to
$\delta_{b}:=\delta_{b,\epsilon_{b}}$.

###### Proof.

If the negation of this lemma holds, then one can find a sequence $i_{n}$ such
that $i_{n}<\operatorname{argnorm}f-\epsilon$ for all $n$, but
$\sup{|a_{i}|\rho^{i}}=|f|$. This contradicts the fact that
$\operatorname{argnorm}f$ is the argument of the norm (e.g. pass to a
convergent subsequence). ∎

Figure 1: A norm graph of an element in the Mal’cev-Neumann ring with Gauss
norm not achieved at a maximum. There are no coefficients in the black box by
the Box Lemma (2.14) and there are no coefficients in the blue rectangle by
the Bar Lemma (2.15).

######  Hypothesis 2.16.

For the rest of the section, assume $R$ is $p$-complete if we are in the
$p$-adic case (i.e. $f\in R[[p^{\Gamma_{\geq 0}}]]$). This is used for the
invocation of Lemmas 2.4 and 2.5.

Our proofs of the properties of the Gauss norm in the $t$-adic case rest on
using properties of the standard formula for a product

$\left(\sum_{i}a_{i}t^{i}\right)\left(\sum_{j}b_{j}t^{j}\right)=\sum_{k}\sum_{i+j=k}a_{i}b_{j}t^{k}.$

However, for the $p$-adic case we don’t have this formula, as we must account
for “$p$-adic carrying”. Thus, we use the following slightly weaker property
which applies in the $p$-adic case (and of course still holds for the $t$-adic
Mal’cev-Neumann series).

######  Lemma 2.17.

Let

$f=\sum_{i}a_{i}r^{i},g=\sum_{j}b_{j}r^{j}$

be elements of the $t$-adic or $p$-adic Mal’cev-Neumann series
($R[[t^{\Gamma_{\geq 0}}]]$ or $R[[p^{\Gamma_{\geq 0}}]]$), with $r=t$ or $p$
respectively. Then we have the following property:

$fg=\sum_{k}\sum_{k^{\prime}\leq k}\left(\sum_{i+j=k,(i,j)\in
S_{f,g,k^{\prime}}}a_{i}b_{j}r^{k}\right)$

where $S_{f,g,k^{\prime}}$ is a finite subset of $\Gamma_{\geq
0}\times\Gamma_{\geq 0}$ such that $\bigcup_{k^{\prime}\leq
k}S_{f,g,k^{\prime}}$ is finite.

Roughly, $S_{f,g,k^{\prime}}$ is the set of indices that, after $r$-adic
carrying has been done, end up in the $r^{k}$-th place.

###### Proof.

For $t$-adic series, just take $S_{f,g,k^{\prime}}$ to be zero for
$k^{\prime}\leq k$ and use the standard formula for products stated above.

For $p$-adic series, we can use Lemma 2.4 to see that we may first multiply
elements as $t$-adic series, and then quotient by $t-p$. we can “collect in
terms of $p$” (by the uniqueness of the representation in $R[[p^{\Gamma_{\geq
0}}]]$, Lemma 2.5). As the quotient is canonically isomorphic to Poonen’s
definition, the elments $f$ and $g$ must have well-ordered support. However,
all possible $p$-adic carry operations happen in the set $k\mathbb{Z}$, and
thus $k\mathbb{Z}\cap\operatorname{supp}(f)$ must have a minimum element, and
respectively for $g$. Thus, the sum over all the $S_{f,g,k^{\prime}}$ must be
finite. ∎

######  Lemma 2.18.

Let $f,g$ be two elements in $R[[t^{\Gamma_{\geq 0}}]]$ or $R[[p^{\Gamma_{\geq
0}}]]$. Then $\operatorname{supp}fg$ is contained in
$(\operatorname{supp}f+\operatorname{supp}g)\mathbb{N}$.

###### Proof.

If $f$ and $g$ are in $R[[t^{\Gamma_{\geq 0}}]]$, then by the product identity
the support of $fg$ is in $\operatorname{supp}f+\operatorname{supp}g$. On the
other hand if they are in $R[[p^{\Gamma_{\geq 0}}]]$, $p$-adic carry
operations may cause an index in the support to increase by an integer. ∎

For the following, fix $\rho$ with $0<\rho\leq 1$.

######  Lemma 2.19.

Let $R$ be a valued ring. Then the Gauss norm on $R[[t^{\Gamma_{\geq 0}}]]$
(resp. $R[[p^{\Gamma_{\geq 0}}]]$) satisfies the strong triangle inequality

###### Proof.

In the $t$-adic case, this follows from the formula for the sum of two
elements with the strong triangle inequality on $R$.

In the $p$-adic case, by Lemma 2.4 we may first do the addition using the
simpler $t$-adic formula, and then do “$p$-adic carrying”. However, any
$p$-adic carry cannot increase the norm as $\rho\leq 1$. ∎

######  Corollary 2.20.

The Gauss norm on $R[[t^{\Gamma_{\geq 0}}]]$ (resp. $R[[p^{\Gamma_{\geq
0}}]]$) is submultiplicative.

###### Proof.

Using the above formulas for the product of two elements
$f=\sum_{i}a_{i}r^{i}$ and $g=\sum_{j}b_{j}r^{j}$ we see that each coordinate
is a sum of elements of the form $a_{i}b_{j}r^{i+j+n}$, where $n\neq 0$ only
if there is a $p$-adic carry operation. Now, use the strong triangle
inequality and the fact that $\rho\leq 1$ to conclude that the norm of $fg$
must be less than or equal to the greatest of these coordinates, but by
submultiplicativity on $R$ this norm is less than or equal to $|f||g|$. ∎

######  Lemma 2.21.

The Gauss norms on $R[[p^{\Gamma_{\geq 0}}]]$ for $0<\rho\leq 1$ form a
continuous family.

###### Proof.

The function $\rho\mapsto|a_{i}|\rho^{i}$ is continuous, and a sup of
continuous functions is continuous. Note here that in the $p$-adic setting,
$|a_{i}|=|a_{i}|^{base}_{\rho}$ as in Definition 2.6. Since we may take
$a_{i}$ such that $p\nmid a_{i}$, $|a_{i}|$ does not depend on the norm which
is on $R$. ∎

To show that the Gauss norm is multiplicative, we need one key lemma as an
input. Essentially, the idea is that one may reduce to the case where all
elements $f$ are supported in a neighborhood of the $\operatorname{argnorm}$.

######  Lemma 2.22.

Let $f,g$ be in $R[[t^{\Gamma_{\geq 0}}]]$ or $R[[p^{\Gamma_{\geq 0}}]]$, and
let $|\cdot|$ be the Gauss norm of radius $\rho$. Let
$I^{h}_{\epsilon}=(\operatorname{argnorm}h-\epsilon,\operatorname{argnorm}h]$
for $h\in R[[t^{\Gamma_{\geq 0}}]]$ (resp. $R[[p^{\Gamma_{\geq 0}}]]$). If the
$h$ is clear from context, we will omit it.

Then there exists an $\epsilon_{1},\epsilon_{2}$, and a $\delta$ greater than
zero with the property that

$\left|\operatorname{res}_{I_{\epsilon_{1}}}^{|f|-\delta\leq}f\right|\left|\operatorname{res}_{I_{\epsilon_{2}}}^{|g|-\delta\leq}g\right|=\left|\left(\operatorname{res}_{I_{\epsilon_{1}}}^{|f|-\delta\leq}f\right)\left(\operatorname{res}_{I_{\epsilon_{2}}}^{|g|-\delta\leq}g\right)\right|\implies|f||g|=|fg|$

Figure 2: A picture of $\operatorname{res}_{I_{\epsilon}}^{|f|-\delta\leq}f$.
One should think of removing all of the coefficients of $f$ except those which
attain a norm which is very close to the norm of $f$. If the set of norms of
terms of $f$, (i.e. $\\{|a_{i}|\rho^{i}\\}$) does not accumulate at $|f|$ as
in Example 2.12, then for sufficiently small $\epsilon$ and $\delta$,
$\operatorname{res}_{I_{\epsilon}}^{|f|-\delta\leq}f$ has at most one term,
i.e. the orange box in the picture would have only a single filled-in dot
which attains the norm, and the classical proof works. Lemma 2.22 shows that
to show multiplicativity, while one cannot reduce to a single point as in the
classical proof, one can reduce to the (arbitrarily small) orange box of width
$\epsilon$ and height $\delta$.

###### Proof.

We will show that every coefficient of
$\operatorname{res}_{I_{\epsilon^{\prime}}}^{|f||g|-\delta\leq}fg$ is a sum of
products of coefficients of $\operatorname{res}_{I_{\epsilon_{1}}}f$ and
$\operatorname{res}_{I_{\epsilon_{2}}}g$ for some $\epsilon^{\prime}$, from
which the claim follows.

By Lemma 2.17, any such coefficient of $fg$ is a sum of products of
coefficients $a_{i}$ and $b_{j}$ such that
$i+j\leq\operatorname{argnorm}f+\operatorname{argnorm}g$.

Choose $\epsilon^{\text{box}}_{f}$ and $\epsilon^{\text{box}}_{g}$ as in the
Box Lemma (Lemma 2.14). Then, choose
$\epsilon^{\text{bar}}_{f}<\epsilon^{\text{box}}_{g}$ and
$\epsilon^{\text{bar}}_{g}<\epsilon^{\text{box}}_{f}$ as in the Bar Lemma
(Lemma 2.15). From these lemmas, we have corresponding
$\delta^{\text{box}}_{f},\delta^{\text{bar}}_{f},\delta^{\text{box}}_{g},\delta^{\text{bar}}_{g}$.
Finally, choose $\delta$ so that
$\delta<\min\\{|f|\delta^{\text{box}}_{f},|f|\delta^{\text{bar}}_{f},|g|\delta^{\text{box}}_{g},|g|\delta^{\text{bar}}_{g}\\}$.
Then, for any product of coefficients of $f$ and $g$ in
$\operatorname{res}_{I_{\epsilon^{\prime}}}^{|f||g|-\delta\leq}fg$, we have

$|f|-\frac{\delta}{|g|}<|a_{i}|\frac{|b_{i}|}{|g|}<|a_{i}|,$

which gives by the Bar Lemma that

$\operatorname{argnorm}f-\epsilon_{1}<i\implies
j<\operatorname{argnorm}g+\epsilon_{1},$

using that $i+j\leq\operatorname{argnorm}f+\operatorname{argnorm}g$. Now, by
the Box Lemma,

$j<\operatorname{argnorm}g+\epsilon_{1}\implies j<\operatorname{argnorm}g.$

We can use a completely symmetric argument to deduce that
$\operatorname{argnorm}g-\epsilon_{2}<j$ and $i<\operatorname{argnorm}f$, and
these four statements taken together are what we wanted to show. ∎

######  Remark 2.23.

The previous proof shows that we may also choose our $\epsilon$’s less than
$\epsilon_{1}$ and $\epsilon_{2}$ and the statement still applies.

######  Corollary 2.24.

The Gauss norms on $R[[t^{\Gamma_{\geq 0}}]]$ and $R[[p^{\Gamma_{\geq 0}}]]$
are multiplicative.

###### Proof.

Let $f,g$ be elements of $R[[t^{\Gamma_{\geq 0}}]]$ or $R[[p^{\Gamma_{\geq
0}}]]$. Assume for the sake of contradiction that $|fg|<|f||g|$ strictly. Pick
$\epsilon_{1}$, $\epsilon_{2}$ as in the previous lemma so that
$|fg|<(|f|-\delta)(|g|-\delta)$, where $\delta$ is given by the previous
lemma. We can assume that
$f=\operatorname{res}_{I_{\epsilon_{1}}}^{|f|-\delta\leq}f$ and likewise for
$g$. Then there exists a unique smallest element $i$ of $\operatorname{supp}f$
by well-orderedness. Likewise, we have a smallest index $j$ for $g$ in
$I_{\epsilon_{2}}$. Then, $a_{i}b_{j}$ must be a coefficient in the product,
and there can be no cancellation by minimality. But then,
$|fg|<|a_{i}||b_{i}|=|a_{i}b_{i}|$, which is a contradiction by
multiplicativity downstairs. ∎

##  3 Formalisms

###  3.1 Valuative Maps

We first define a notion which axiomatizes the idea of a “collection of
valuations”:

######  Definition 3.1.

Let $R$ be a ring, and let $(C,\oplus,\otimes,\leq)$ be a partially ordered
tropical semiring. Then a map of sets $\mathscr{L}\\!\mathscr{N}:R\to C$ is
valuative if

  1. (i)

$\mathscr{L}\\!\mathscr{N}$ is superadditive, that is
$\mathscr{L}\\!\mathscr{N}(a)\oplus\mathscr{L}\\!\mathscr{N}(b)\leq\mathscr{L}\\!\mathscr{N}(a+b)$

  2. (ii)

$\mathscr{L}\\!\mathscr{N}$ is multiplicative, or
$\mathscr{L}\\!\mathscr{N}(a)\otimes\mathscr{L}\\!\mathscr{N}(b)=\mathscr{L}\\!\mathscr{N}(ab)$

for all $a,b\in R$.

As a first example, taking $\mathscr{L}\\!\mathscr{N}$ to be a valuation $v$
of $R$ on a totally ordered group $\Gamma$, we see that $v$ is valuative
(here, tropical addition is taking the $\min$). Also, if one takes any two
valuations $v,v^{\prime}$, the natural map
$R\to\Gamma_{v}\times\Gamma_{v^{\prime}}$ is valuative. We will cheifly be
concerned with the case when our tropical semiring $C$ is
$C([0,1],\mathbb{R}_{\geq 0})$, and our map “comes from a family of Gauss
norms”.

The significance of the notion valuative is the fact that it is just enough to
pull back prime ideals.

######  Definition 3.2.

Let $C$ a tropical semiring. Then an ideal $I$ of $C$ is prime if the
complement of $I$ is (tropically) multiplicatively closed.

######  Proposition 3.3.

Let $\mathscr{L}\\!\mathscr{N}:R\to C$ be a valuative map. Let $\mathfrak{p}$
be a prime (tropical) ideal which is closed under the partial order of $C$
(i.e. if $F\in\mathfrak{p},F\leq G\implies G\in\mathfrak{p}$). Then
$\mathscr{L}\\!\mathscr{N}^{-1}(\mathfrak{p})$ is a prime ideal in $R$.

###### Proof.

Unpack the previous two definitions. ∎

If $C=\Gamma_{v}\cup\\{\infty\\}$ for some valuation $v$ on $R$, this says the
familiar fact that the support of a valuation is prime. Our most important
example of a tropical prime which we will pull back is the prime ideal defined
by the little omega notation $\omega(G)$ for $G$ some continuous function. We
will delay our definition of little omega notation until section 4, where we
will use it to prove our version of Du’s theorem.

###  3.2 Newton Polygon Formalisms

For the rest of this paper, the Legendre Transform always refers to the
infimum Legendre transform of a decreasing convex function $F$ on the positive
real numbers, that is

$\mathscr{L}(F)(t)=\inf_{x}\\{F(x)+xt\\}.$

######  Definition 3.4.

Let $R$ be a ring, and let $U\subset\mathbb{R}_{\geq 0}$ be an interval. A
Newton polygon formalism with domain $U$ on $R$ is a diagram of sets

${R}$${C(\mathbb{R}_{\geq 0},\mathbb{R}_{\geq 0})}$${C(U,\mathbb{R}_{\geq
0})}$$\scriptstyle{\mathscr{N}}$$\scriptstyle{\mathscr{L}\\!\mathscr{N}}$$\scriptstyle{\mathscr{L}}$

where $\mathscr{L}$ is the Legendre transform (restricted to
$C(U,\mathbb{R}_{\geq 0})$) such that

  * •

$\mathscr{L}\\!\mathscr{N}$ is superadditive and multiplicative

  * •

$\mathscr{N}(f)$ is convex for every $f\in R$.

$\mathscr{N}(f)$ is called the Newton polygon of $f$ under the formalism.

######  Remark 3.5.

Note that the Newton polygon of $f$ in the definition has no requirement to be
a polygon! We refrain from calling it the “Newton lower convex hull” for
consistency and recognizability.

######  Remark 3.6.

Since $\mathscr{L}\\!\mathscr{N}$ must be the Legendre transform of
$\mathscr{N}$, the data of a Newton polygon formalism is only the map
$\mathscr{N}$.

Examples of Newton polygon formalisms include standard examples of rings which
have Newton polygons, for example take $R=K[x],R=K\langle x\rangle,$ or
$R=K[[x]]$ for $K$ a valued field. The main theorems in this paper will follow
from the existence of “nonstandard” Newton polygon formalisms on various
perfectoid rings $R$. However, all of the rings we consider are alike in that
the Newton polygon formalism “comes from a continuous family of multiplicative
Gauss norms”. The multiplicative norms correspond to valuations, and the
continuous family of valuations becomes the map $\mathscr{L}\\!\mathscr{N}$.
Below, we give a technical lemma which shows that Mal’cev-Neumann rings have
Newton polygon formalisms in the presense of reasonable Gauss norms.

######  Proposition 3.7.

Let $R$ be $V[[p^{\Gamma_{\geq 0}}]]$ for $V$ a ring with a family of $p$-adic
valuations such that $V$ embeds into $W(V/p)$. Respectively, assume
$R=V[[t^{\Gamma_{\geq 0}}]]$ for $V$ some valued ring.

Suppose that the Gauss norms on $R$ of radius $s$ are multiplicative norms for
$s\in U$. Then the associated valuations $v_{s}:R\to\mathbb{R}$ give a Newton
polygon formalism with $\mathscr{L}\\!\mathscr{N}(f)=s\mapsto v_{s}(f)$

###### Proof.

Let $r=p$ or $t$, respectively.

We need to define a map $\mathscr{N}$, then we need to show that for any
$f=\sum_{i}a_{i}r^{i}$, $\mathscr{L}\\!\mathscr{N}(f)$ has the above form, and
further that $\mathscr{L}\\!\mathscr{N}(f)$ is a continuous valuative map.

We define $\mathscr{N}(f)$ to by its graph: let $\Gamma(\mathscr{N}(f))$be the
nonincreasing lower convex hull of the points $(i,v(a_{i}))$ in
$\mathbb{R}^{2}$. Thus $\mathscr{N}(f)$ is a continuous positive function, and
so lies in $C(\mathbb{R}_{\geq 0},\mathbb{R}_{\geq 0})$.

We can write $\mathscr{L}\\!\mathscr{N}(f)$, the legendre transform of
$\mathscr{N}(f)$, as

$s\mapsto\inf_{x\in\mathbb{R}_{\geq 0}}\\{\mathscr{N}(f)(x)+sx\\}.$

On the other hand, the valuation associated to the Gauss norm of radius $s$ is

$\inf_{i\in I}\\{v(a_{i})+si\\}$

where $I$ is the set of nonzero indices of $f$, i.e. $I=\\{i|a_{i}\neq 0\\}$.
We will show that in $\mathscr{L}\\!\mathscr{N}(f)$, the infimum can be taken
over $I$. Then the two expressions match.

We see that $\mathscr{N}(f)$ has the following property. For every
$x\in\mathbb{R}_{\geq 0}$, there exists a neighborhood $U_{x}$ of $x$ with
either

  1. (1)

$\mathscr{N}(f)$ is piecewise-linear on $U_{x}$, or

  2. (2)

The set $S^{\prime}\colonequals\\{i|(i,v(a_{i}))\in\Gamma(\mathscr{N}(f))\\}$
accumulates to $x$.

Suppose $x$ is the point where the infimum is achieved in
$\mathscr{L}\\!\mathscr{N}(f)(s)$ (this point must be finite, as
$\mathscr{N}(f)$ is nonincreasing and positive and so will be dominated by the
linear term eventually). Then, in case (1), such a minimum can only be
attained at a node of a piecewise-linear function, so we conclude that $x$ is
a node, and that $x\in I$. In case (2), we choose an accumulating set
$x_{j}\to x$ with $x_{j}\in I$, and by continuity the limit of this sequence
must be the infimum. Thus, we may take the inf over $x\in I$, as desired.

Finally, the map $\mathscr{L}\\!\mathscr{N}$ is valuative because $v_{s}$ are
multiplicative valuations (as the corresponding Gauss norms are
multiplicative), and form a continuous family. ∎

######  Example 3.8.

Let $R=\mathbb{A}_{\inf}$. Let $r=p$, and let $S$ be the image of the
multiplicative lift of $\mathcal{O}_{K}^{\flat}$. Then the Gauss norms are
multiplicative in this situation [FF18, Chapter 3]. By the same argument as
Proposition 3.7, there is a Newton polygon formalism on $\mathbb{A}_{\inf}$.
These are the Newton polygons on $\mathbb{A}_{\inf}$ used in [Du20] and
[FF18].

We also record a few basic lemmas:

######  Lemma 3.9.

Let $R\xhookrightarrow{}R^{\prime}$ be an inclusion, and let $\mathscr{N}$ be
a Newton polygon formalism on $R^{\prime}$. Then $\mathscr{N}$ is a Newton
polygon formalism on $R$.

###### Proof.

Straightforward verification from definitions. ∎

######  Lemma 3.10.

Let $r=p$ or $t$, respectively. If $\mathscr{N}$ is a Newton polygon formalism
on $R$ coming from a multiplicative Gauss norm (Proposition 3.7), then this
can be extended to a Newton polygon formalism on $\widehat{R}$ for the
$r$-adic topology on $R$.

###### Proof.

One can extend the Gauss norms by the limit, and $r$-adic convergence
guarantees that the limit exists for any $f\in\widehat{R}$. ∎

##  4 Du’s Argument

###  4.1 Statements and Background

In this section, we will prove Theorem 4.6, using the properties of big O
notation and its variants from computer science. For completeness, we define
them here and state some properties.

######  Definition 4.1.

Let $F$ and $G$ be functions $U\to\mathbb{R}_{\geq 0}$ for some subset
$U\subset\mathbb{R}_{\geq 0}$. We consider $\lim\frac{F}{G}$, where all limits
are taken as $t\to a\in[0,\infty]$ such that $U$ accumulates to $a$. We say
that

  * •

$F\in o(G)$ if $\lim\frac{F}{G}=0$

  * •

$F\in O(G)$ if $\lim\sup\frac{F}{G}<\infty$

  * •

$F\sim G$ if $\lim\frac{F}{G}=1$

  * •

$F\in\Omega(G)$ if $0<\lim\inf\frac{F}{G}$

  * •

$F\in\omega(G)$ if $\lim\frac{F}{G}=\infty$

  * •

$F\in\Theta(G)$ if $F\in O(G)$ and $F\in\Omega(G)$

If there is ambiguity, we may add the “approaches” into the notation, i.e.
$O_{t\to 0}(G)$.

We remind the reader that $F\in\Theta(G)$ is a symmetric notion, as is $F\sim
G$. We list a few properties, mostly concerning little omega, which we will
use later:

######  Proposition 4.2.

Let $F,G,H\colon U\to\mathbb{R}$

  1. (1)

(Exclusivity of big theta) $F\in\Theta(G)\implies F\notin\omega(G)$

  2. (2)

(Asymptotically arbitrarily close implies asymptotically equal) $H\in o(G)$,
assume $|F-G|<H$ for all $x$ sufficiently close to $a$. Then $F\sim G$.

  3. (3)

(Existence of arbitrarily small functions) For all $G$, there exists some
(continuous) function in $o(G)$.

  4. (4)

(Partial-order-reversing property of little omega) $F\leq
G\implies\omega(G)\subset\omega(F)$

  5. (5)

(Theta-invariance of little omega) $F\in\omega(H)\implies G\in\omega(H)$.

  6. (6)

(Closure of little omega under tropical addition)
$F,G\in\omega(H)\implies\min(F,G)\in\omega(H)$

  7. (7)

(Closure of little omega under arbitrary tropical mulitplication)
$F\in\omega(H)$ $\implies$ $F+G\in\omega(H)$

  8. (8)

(Complement of little omega is tropically multiplicatively closed)
$F,G\notin\omega(H)\implies F+G\notin\omega(G)$

  9. (9)

(Little omega is closed under order) If $F\in\omega(H)$ and $F\leq G$, then
$G\in\omega(H)$.

All of these facts can be proved straightforwardly from the previous
definitions. The last four parts show that $\omega(G)$ is a tropical semiring
prime ideal which is closed under order in $C(U,\mathbb{R}_{\geq 0})$.

In order to state Du’s Theorem, we will now switch gears, back to Newton
polygon formalisms. In much of Newton polygon theory, theorems are deduced by
analyzing individual Newton polygons. Our approach will be slightly different,
we aim to deduce facts about the ring $R$ from the possible Newton polygons
from a given formalism.

######  Definition 4.3.

The discrete approximation of a function $G\in C(\mathbb{R}_{\geq
0},\mathbb{R}_{\geq 0})$ is the piecewise-linear function with nodes
$(n,G(n))$ for $n\in\mathbb{N}$.

######  Definition 4.4.

A Newton polygon formalism $\mathscr{N}$ discretely approximates a function
$H\in C(\mathbb{R}_{\geq 0},\mathbb{R}_{\geq 0})$ if there exists an element
$r\in R$ whose Newton polygon $\mathscr{N}(r)$ has

  * •

$\mathscr{N}(r)$ and $G$ are asymptotically arbitrarily close in the sense of
Proposition 4.2, that is there exists some $H\in o(G)$ for which
$|\mathscr{N}(r)-G|<H$.

  * •

The slopes of secant lines of $\mathscr{N}(r)$ and $G$ are asymptotically
arbitrarily close, that is given $x\in\mathbb{R}_{\geq 0}$, for any
$x_{1},x_{2}\in(x-\delta,x+\delta)$ for some small $\delta$, we have that
$|m_{G}-m_{\mathscr{N}(r)}|<h^{\prime}\in o(G)$ where $m_{G}$ is the slope of
the secant line connecting $(x_{1},G(x_{1}))$ and $(x_{2},G(x_{2}))$, and
likewise for $m_{\mathscr{N}(r)}$.

where $G$ is the discrete approximation of the function $H$.

######  Remark 4.5.

Let $G$ be a function whose discrete approximation is nonconvex. Then it is
impossible for the standard Newton polygon formalism (say, on $V[[x]]$ for a
valued ring $V$) to discretely approximate $G$.

Next, we have enough to state Du’s main theorem ([Du20]), in our
conceptualized form.

######  Theorem 4.6 (after Du).

Let $R$ be a ring and $\mathscr{N}$ a Newton polygon formalism with domain
$(0,1]$ on $R$. If $\mathscr{N}$ discretely approximates any (convex) function
$G\in C(\mathbb{R}_{\geq 0},\mathbb{R}_{\geq 0})$ which has
$\lim_{i\to\infty}g(i)=0$, then $R$ has uncountable Krull dimension.

###  4.2 Proof of Du’s Theorem

######  Proposition 4.7.

Let $F$ and $G$ be asymptotically arbitrarily close, in the sense of the
previous section. Suppose further that the slopes of secant lines of $F$ and
$G$ are asymptotically arbitrarily close as in Definition 4.4. Then we have
$\mathscr{L}(F)\sim\mathscr{L}(G)$ as $t\to 0$, where $\mathscr{L}$ is the
Legendre transform.

######  Remark 4.8.

If the functions $F$ and $G$ are smooth, one may replace the “secant lines”
condition with the derivatives being asymptotically arbitrarily close.
However, the case of interest is piecewise-linear continuous functions, and we
need control on all possible tangent lines, so we use the more technical
condition in the proposition.

###### Proof.

Every tangent line can be written as a limit of secant lines, so the condition
on secant lines implies that the slopes of lines tangent to $f$ and $g$ are
arbitrarily close. This guarantees (based on the slope interpretation of the
Legendre transform) that the $\mathscr{L}(F)$ is asymptotically arbitrarily
close to $\mathscr{L}(G)$ as $t\to 0$. We are then done by Proposition 4.2,
part (2). ∎

We emphasize that the next proof, while quite different in word to Proposition
4 of [Du20], is quite similar in deed.

###### Proof of Theorem 4.6.

Let $\mathscr{N}$ be the Newton polygon formalism on $R$.

First, we see that $\omega(t^{\lambda})$ for $0<\lambda<1$ is an uncountable
chain of distinct semiring prime ideals. Indeed,
$t^{\lambda}\in\omega(t^{\mu})$ precisely when $\mu<\lambda$, by exclusivity,
the preorder-reversing property, and a direct calculation of the limit of the
quotient. Thus, by Theta-invariance of little omega, we need only find an
element $f_{\lambda}\in R$ such that
$\mathscr{L}\\!\mathscr{N}(f)\in\Theta(t^{\lambda})$.

By Proposition 4.7, it is enough to show that there exists $f_{\lambda}$ such
that $\mathscr{N}(f_{\lambda})$ is asymptotic to
$\mathscr{L}^{-1}(t^{\lambda})$. By a standard Legendre transformation
calculation $\mathscr{L}^{-1}(t^{\lambda})=c_{\lambda}x^{-r_{\lambda}}$ for
positive constants $c_{\lambda}$ and $r_{\lambda}$. Then, as $\mathscr{N}$
discretely approximates all convex functions with limit zero, it discretely
approximates $\mathscr{L}^{-1}(t^{\lambda})$. By a short calculation, we can
see that the piecewise-linear function with integral values at $f$ is
asymptotic to $f$ for any $f$. Thus, we have such an $f_{\lambda}$ and the
theorem is proved. ∎

##  5 Uncountability of the Krull Dimension of Various Rings

######  Corollary 5.1 (Du).

Let $V$ be a nondiscrete valuation ring. Then $V[[x]]$ has uncountable Krull
dimension.

###### Proof.

We have the obvious Newton polygon formalism. To use Theorem 4.6 (Du’s
argument), we need to show that this Newton polygon formalism discretely
approximantes (see 4.4) any (convex) function with limit zero. This follows
quickly from nondiscreteness of the valution on $V$, we will spell the
argument out a bit here as all other results of this type will follow by a
verbatim argument.

Take a (convex) function $f$ in $C(\mathbb{R}_{\geq 0},\mathbb{R}_{\geq 0})$
which has limit zero as $t\to\infty$. Let $g$ be the discrete approximation of
this function, we wish to construct a Newton polygon which is asymptotically
arbitrarily close to $g$, which has secant lines which are also asymptotically
arbitrarily close to $g$. Since the valuation on $V$ is nondiscrete, one may
simply choose coefficients $a_{i}$ which are arbitrarily close to
$f(i)(=g(i))$. By choosing close enough $a_{i}$, one verifies that Newton
polygon of the element $r=\sum_{i}a_{i}x^{i}$ has the desired properties. ∎

Note that $r$ in the previous proof may be an infinite sum, so we are using
the fact that $V[[x]]$ is $x$-adically complete.

######  Corollary 5.2 (Du).

$\mathbb{A}_{\inf}$ has uncountable Krull dimension.

###### Proof.

The Gauss norms are multiplicative (again see [FF18, Chapter 3]) and the norm
is nondiscrete on the image of the multiplicative lift. Thus, we have a Newton
polygon formalism on $\mathbb{A}_{\inf}$, which discretely approximates any
function with limit zero (by the nondiscreteness of the valuation on the image
of the multiplicative lift and the fact that $\mathbb{A}_{\inf}$ is
$\pi$-complete). And we conclude by Theorem 4.6. ∎

We have now recreated the results from [Du20], so we move on to new results:

######  Lemma 5.3.

Let $V=W(\kappa)[x^{1/p^{\infty}}]$ (resp. $\kappa[x^{1/p^{\infty}}]$) for
some perfect field $\kappa$ of characteristic $p$. Then the $p$-adic
completion (resp. $t$-adic completion) of $V[[p^{\Gamma_{\geq 0}}]]$ (resp.
$V[[t^{\Gamma_{\geq 0}}]]$) has uncountable Krull dimension.

###### Proof.

$V$ has an obvious family of $p$-adic valuations (resp. has an obvious
valuation). Furthermore, in the $p$-adic case $V$ embeds into $W(V/p)$, so by
Corollary 2.24 the Gauss norm on $\widehat{V}[[p^{\Gamma_{\geq 0}}]]$ (resp.
$V[[t^{\Gamma_{\geq 0}}]]$) is multiplicative. Note that to use Corollary 2.24
we must use $p$-adic completion of $V$ in the $p$-adic case. We may replace
$V$ with $\widehat{V}$ without loss of generality as

$\widehat{V[[p^{\Gamma_{\geq 0}}]]}\cong\widehat{\widehat{V}[[p^{\Gamma_{\geq
0}}]]}.$

Next, by Proposition 3.7, $\widehat{V}[[p^{\Gamma_{\geq 0}}]]$ (resp.
$V[[t^{\Gamma_{\geq 0}}]]$) has a Newton polygon formalism. This extends to
the $p$-adic completion (resp. $t$-adic completion) by Lemma 3.10. As we have
arbitrary $p$-th roots of $x$, this valuation is clearly nondiscrete on
elements of the image of the multiplicative lift (resp. the valuation is
clearly nondiscrete on $V$). Thus, this formalism discretely approximates any
function with limit zero as $s\to\infty$ and we conclude by Theorem 4.6. ∎

We will use the previous lemma to show that the perfectoid Tate algebra of
dimension one has uncountable Krull dimension, by embedding into the ring
described in the lemma. For the absolute integral closure of a noetherian
domain of mixed characteristic (of dimension 2), we will do a similar trick
with a different ring:

######  Lemma 5.4.

Let $V=W(\kappa)[[x^{1/\mathbb{N}}]]$. Then the $p$-adic completion of
$V[[p^{\Gamma_{\geq 0}}]]$ has uncountable Krull dimension.

###### Proof.

We use the same proof as Lemma 5.3. Again, $V$ has an obvious $p$-adic family
of valuations, and embeds into $W(V/p)$ by the obvious map. Again, the
valuation is nondiscrete, so the formalism discretely approximates any
function with limit zero. ∎

###  5.1 The perfectoid Tate algebra over $\mathcal{O}_{K}$

Let $K$ be a perfectoid field of characteristic $p$. Suppose to begin that $K$
is spherically complete. Then $K\cong\kappa[[t^{\Gamma_{\geq 0}}]]$. Pick $t$
as our pseudo-uniformizer. Then the completion of the ring
$\kappa[[t^{\Gamma_{\geq 0}}]][x^{1/p^{\infty}}]$ is isomorphic to
$\mathcal{O}_{K}\langle x^{1/p^{\infty}}\rangle$. We consider the obvious map

$i\colon\kappa[[t^{\Gamma_{\geq
0}}]][x^{1/p^{\infty}}]\xhookrightarrow{}\kappa[x^{1/p^{\infty}}][[t^{\Gamma_{\geq
0}}]].$

In the latter ring, the “roles of $x$ and $t$ are swapped”. We can thus use
Lemma 3.9 to conclude that the ring on the left has a Newton polygon
formalism, as we know that the one on the right does from the proof of 5.3. We
need only show that this formalism discretely approximates any function with
limit zero. We see that as the Mal’cev-Neumann series variable $t$ has a
preimage in $\kappa[[t^{\Gamma_{\geq 0}}]][x^{1/p^{\infty}}]$, and that the
valuation remains nondiscrete on the preimage $\kappa[x^{1/p^{\infty}}]$, this
indeed holds and we may conclude that $\mathcal{O}_{K}\langle
x^{1/p^{\infty}}\rangle$ has uncountable Krull dimension.

Now, let $K$ be an arbitrary perfectoid field of characteristic $p$, use the
following chain of inclusions:

$K\xhookrightarrow{}K^{alg}\xhookrightarrow{}K^{sp}\cong\kappa((t^{\Gamma})).$

By [KP05], Theorem 3.3, there always exists an embedding from $K^{alg}\to
K^{sp}$ which takes a given pseudo-uniformizer $\pi$ to the variable $t$
(simply post-compose with an automorphism of $\kappa((t^{\Gamma}))$), where
$\kappa$ is the residue field of $K^{alg}$. Moreover, if we choose a pseudo-
uniformizer $\pi\in K$, it remains a pseudo-uniformizer in a fixed algebraic
closure.

######  Theorem 5.5.

Let $K$ be a perfectoid field of characteristic $p$. Then
$\mathcal{O}_{K}\langle x^{1/p^{\infty}}\rangle$ has uncountable Krull
dimension.

###### Proof.

Let $\kappa$ be the residue field of $K^{alg}$, and let $\kappa_{K}\in\kappa$
be the residue field of $K$. Then, by the spherically complete case above,
$\mathcal{O}_{K^{sp}}[x^{1/p^{\infty}}]$ has a Newton polgon formalism, and we
see that along the above mentioned embedding
$\mathcal{O}_{K}[x^{1/p^{\infty}}]$ has a Newton polygon formalism for which
$\pi$ has a preimage. Since the Gauss norm remains nondiscrete on elements
which $p$ does not divide, and $\mathcal{O}_{K}\langle
x^{1/p^{\infty}}\rangle$ is the completion of
$\mathcal{O}_{K}[x^{1/p^{\infty}}]$, we conclude by Theorem 4.6.

∎

If $K$ has characteristic zero, then we can run a similar argument:

$K\xhookrightarrow{}K^{alg}\xhookrightarrow{}K^{sp}\cong
W(\kappa)[[p^{\Gamma_{\geq 0}}]],$

similarly to the characteristic $p$ case. Notice that $p$ is always a pseudo-
uniformizer.

######  Theorem 5.6.

Let $K$ be a perfectoid field of characteristic zero. Then
$\mathcal{O}_{K}\langle x^{1/p^{\infty}}\rangle$ has uncountable Krull
dimension.

###### Proof.

Same argument as in characteristic $p$, just replace $t$ with $p$, $\kappa$
with $W(\kappa)$, and $\kappa_{K}$ with $W(\kappa_{K})$. Since $p$ is always a
pseudo-uniformizer, we do not need to compose with an automorphism of
$K^{sp}$. ∎

Finally, we extend the proof to higher dimensional Tate algebras:

######  Corollary 5.7.

Let $K$ be a perfectoid field. Then the perfectoid Tate algebra in $n$
variables has uncountable Krull dimension.

###### Proof.

We use the surjective ring map

$\mathcal{O}_{K}\langle
x_{1}^{1/p^{\infty}},\ldots,x_{n}^{1/p^{\infty}}\rangle\to\mathcal{O}_{K}\langle
x^{1/p^{\infty}}\rangle,$

which, say, evaluates $x_{2},\ldots,x_{n}$ at zero and pull back the ideals. ∎

###  5.2 The perfectoid Tate algebra over $K$

Let $R=\mathcal{O}_{K}\langle x^{1/p^{\infty}}\rangle$ be the perfectoid Tate
algebra in one variable over $\mathcal{O}_{K}$.

Let us fix the following notation: Let $\mathfrak{p}_{\lambda}\subset R$ be
the ideal $\mathscr{L}\\!\mathscr{N}^{-1}(\omega(s^{\lambda}))$. Our previous
work has shown that the family $\mathfrak{p}_{\lambda}$ for $0<\lambda<1$ is
the family which shows the uncountability of $\dim R$.

######  Lemma 5.8.

Let $\pi$ be the chosen pseudo-uniformizer from the previous section (i.e.
whichever arbitrarily chosen one in characteristic $p$, always $p$ in
characteristic $0$). Any polynomial in $\pi$ that has a coefficient of
valuation 0 is not contained in $\mathfrak{p}_{1}$, and thus by theta-
invariance of little omega and exclusivity of big theta, is not contained in
any other of the $\mathfrak{p}_{\lambda}$ for $0<\lambda\leq 1$.

The basic reason for the previous lemma is that the ideals
$\mathfrak{p}_{\lambda}$ measure how fast coefficients in $p$ are diverging
$x$-adically, so we don’t expect them to interact with the chain of ideals.

###### Proof.

Any polynomial has finitely many nodes and slopes, and if it has a coefficient
of valuation zero, then the Newton polygon (which is actually a polygon in
this case) meets the $x$-axis. Then, the Legendre transform is a linear
function as $s$ goes to zero, so by theta-invariance and theta-exclusivity
this transform is not in the ideal $\omega(t)$ of $C([0,1],\mathbb{R}_{\geq
0})$ by definition of little omega. Finally, as $\omega(t)$ contains all of
the $\omega(t^{\lambda})$ by the partial-order-reversing property, our
polynomial is not in any of these ideals either. ∎

The following corollary is an immediate specialization:

######  Corollary 5.9.

The element $\pi$ is not in any of the $\mathfrak{p}_{\lambda}$. Thus, the
perfectoid ring $R[1/p]$ has uncountable Krull dimension. So the perfectoid
Tate algebra over $K$ in $n$ variables has uncountable Krull dimension,
regardless of characteristic.

###  5.3 The absolute integral closure in mixed characteristic

Let $(A,\mathfrak{m},k)$ be a noetherian local domain of mixed characteristic
which is $\mathfrak{m}$-adically complete. We are concerned with the absolute
integral closure $A^{+}$ of $A$.

######  Theorem 5.10.

Let $\dim A=2$. Then the Krull dimension of the $p$-adic completion of $A^{+}$
is uncountable.

###### Proof.

By the Cohen Structure Theorem, $A$ is an integral extension of
$\mathbb{Z}_{p}[[x]]$, so we may assume that $A\cong\mathbb{Z}_{p}[[x]]$ as
the absolute integral closures will be the same.

The integral closure of $A$ in the algebraic closure of its fraction field
$\mathbb{Q}_{p}[[x]]$ is $\mathcal{O}_{\mathbb{C}_{p}}[[x^{1/\mathbb{N}}]]$.
As before, we can embed into a spherical completion, letting
$\kappa=\overline{\mathbb{F}_{p}}$ and $\Gamma=\mathbb{Q}$, we get

$\mathcal{O}_{\mathbb{C}_{p}}[[x^{1/\mathbb{N}}]]\xhookrightarrow{}\kappa[[p^{\Gamma_{\geq
0}}]][[x^{1/\mathbb{N}}]]$

We may now use the same argument as for the perfectoid Tate algebra, with
Lemma 5.3 replaced by Lemma 5.4. We need only to show that the Mal’cev-Neumann
series variable $p$ pulls back and that the norm remains nondiscrete on
coefficients. But these both follow as before, and these two things with
$p$-adic completeness together imply that the $p$-adic completion of the ring
on the left has a Newton polygon formalism which discretely approximates any
function with limit zero. By Theorem 4.6, we conclude.

∎

######  Corollary 5.11.

Let $2\leq\dim A$. Then the Krull dimension of the $p$-adic completion of
$A^{+}$ is uncountable.

###### Proof.

WLOG $A\cong W(\kappa)[[x_{1},\ldots,x_{n}]]$ where $n=\dim A-1$. Pick a prime
ideal $\mathfrak{p}$ in $A^{+}$ lying over $(x_{2},\ldots,x_{n})$ (which
exists as the extension is integral). Then, using e.g. [Hun11] (see Discussion
2.5), we have that $A^{+}/\mathfrak{p}\cong A_{2}^{+}$ where
$A_{2}:=W(\kappa)[[x_{1}]]$. So we have a surjective ring map $A^{+}\to
A_{2}^{+}$. Now, as $p$-adic completion preserves surjections, we get a
surjection $\widehat{A^{+}}\to\widehat{A_{2}^{+}}$. By the above theorem,
$\widehat{A_{2}^{+}}$ has uncountable Krull dimension, pull back the ideals. ∎

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