conjAthmAconjA propAthmApropA corAthmAcorA lemmathmlemma corthmcor propthmprop
dfthmdf remthmrem exthmex

# Strong congruence spaces and dimension in ${\mathbb{F}}_{1}$-geometry

Manoel Jarra  Manoel Jarra, University of Groningen, the Netherlands, and
IMPA, Rio de Janeiro, Brazil<EMAIL_ADDRESS>
###### Abstract.

We introduce strong congruence spaces, which are topological spaces that
provide a useful concept of dimension for monoid schemes. We study their
properties and show that, given a toric monoid scheme over an algebraically
closed basis, its strong congruence space and the complex toric variety
associated to its fan have the same dimension.

###### Contents

  1. 1 Preliminaries
  2. 2 Congruence spaces
  3. 3 The strong congruence space
  4. 4 Relation to classical scheme theory
  5. 5 Strong congruence space of toric monoid schemes
  6. 6 Examples: The torus and affine space

## Introduction

Dimension theory for ${\mathbb{F}}_{1}$-geometry has not yet been developed in
a satisfactory way. The main problem is that the underlying topological space
of a monoid scheme can have too few points, so it fails to reflect the
expected dimension. For example, if ${\mathbb{F}}$ is a pointed group, the
underlying topological space of the $n$-torus
${\mathbb{T}}^{n}_{{\mathbb{F}}}$ is a singleton, so its Krull dimension (in
the sense of [Sta23, Def. 5.10.1]) is zero, while the dimension of the complex
variety ${\mathbb{T}}^{n}_{\mathbb{C}}$ is $n$. We address this issue by
attaching a new topological space to a monoid scheme $X$ over a basis $k$
whose Krull dimension behaves as expected if $k$ is an algebraically closed
pointed group.

Throughout this text all monoids are assumed to be commutative. A congruence
on a pointed monoid $A$ is an equivalence relation $\mathfrak{c}\subseteq
A\times A$ satisfying $(\mylambda a,\mylambda b)\in\mathfrak{c}$ whenever
$(a,b)\in\mathfrak{c}$ and $\mylambda\in A$, which implies that the projection
$A\twoheadrightarrow A/\mathfrak{c}$ induces a natural structure of pointed
monoid on the quotient. We say that a congruence $\mathfrak{c}$ is prime if
$A/\mathfrak{c}$ is integral. We refer the reader to Section 1 for basic
facts. Spaces of prime congruences were considered in [Ber12] for monoids, in
[Dei13] for sesquiads and more recently in [LR23] for monoid schemes. The set
of prime congruences of a pointed monoid $A$ endowed with the topology
generated by the open sets

$U_{a,b}:=\\{\mathfrak{c}\mid(a,b)\notin\mathfrak{c}\\}$

is the congruence space of $A$, denoted by $\operatorname{Cong}A$. For a
monoid scheme $X=\bigcup\;\operatorname{MSpec}A_{i}$, the congruence space of
$X$ is constructed by gluing the affine pieces $\operatorname{Cong}A_{i}$,
such that the result $X^{\textup{cong}}$ is independent of the choice of
affine covering.

We introduce the term domain for a pointed monoid that can be embedded as a
multiplicative submonoid into a field. Section 4.2 gives an intrinsic
characterization of domains, without mention to fields. A prime congruence
$\mathfrak{c}$ on a pointed monoid $A$ is strong if $A/\mathfrak{c}$ is a
domain. If $A$ is a $k$-algebra, we say that $\mathfrak{c}$ is a
$k$-congruence if it is strong and the composition $k\rightarrow A\rightarrow
A/\mathfrak{c}$ is injective. The set of $k$-congruences of $A$ with the
topology induced from $\operatorname{Cong}A$ is denoted by
$\operatorname{SCong}_{k}A$. Analogous to the case of prime congruences, we
globalize this construction to the case of a $k$-scheme $X$ to define
$\operatorname{SCong}_{k}X$, the strong $k$-congruence space of $X$.

An extension of pointed groups
${\mathbb{F}}\hookrightarrow{\mathbb{F}}^{\prime}$ is algebraic if for every
$\myalpha\in{\mathbb{F}}^{\prime}$ there exists $n\geq 1$ such that
$\myalpha^{n}\in{\mathbb{F}}$. A pointed group ${\mathbb{F}}$ that is also a
domain is called algebraically closed if there is no non-trivial algebraic
extension ${\mathbb{F}}\hookrightarrow{\mathbb{F}}^{\prime}$ with
${\mathbb{F}}^{\prime}$ being a pointed group and a domain. The following
result, which is Theorem 5.2, shows that the strong congruence space of a
toric monoid scheme over an algebraically closed basis reflects its expected
dimension.

######  Theorem A.

Let ${\mathbb{F}}$ be an algebraically closed pointed group and $\Sigma$ a fan
in a finite dimensional ${\mathbb{R}}$-vector space. Let $X(\Sigma)$ be the
toric monoid scheme associated to $\Sigma$ and $Y$ the strong
${\mathbb{F}}$-congruence space of
$X(\Sigma)\times_{{\mathbb{F}}_{1}}\operatorname{MSpec}{\mathbb{F}}$. Then $Y$
is a catenary space whose Krull dimension is the same as that of the complex
toric variety $X(\Sigma)_{\mathbb{C}}$.

### Relation to classical scheme theory

Let $R\mapsto R^{\bullet}$ be the forgetful functor from rings to pointed
monoids. It has a left adjoint, given by $A\mapsto
A_{\mathbb{Z}}:={\mathbb{Z}}[A]/\langle 1_{\mathbb{Z}}0_{A}\rangle$ on
objects. The inclusion

$\displaystyle A$ $\displaystyle\longrightarrow A_{\mathbb{Z}}^{\bullet}$
$\displaystyle a$ $\displaystyle\longmapsto 1_{\mathbb{Z}}a$

induces a continuous map
$p:\operatorname{Spec}(A_{\mathbb{Z}})\rightarrow\operatorname{MSpec}A$ that
sends a prime ideal $\mathfrak{p}$ of ${\mathbb{A}}_{\mathbb{Z}}$ to the prime
ideal $\mathfrak{p}\cap A$ of $A$. The inclusion also induces a map
$\mygamma:\operatorname{Spec}(A_{\mathbb{Z}})\rightarrow\operatorname{SCong}A$,
such that $p$ factorizes through it, given by

$\mathfrak{p}\mapsto\mygamma(\mathfrak{p}):=\\{(a,b)\in A\times
A\mid\overline{a}=\overline{b}\in A_{\mathbb{Z}}/\mathfrak{p}\\}.$

The construction above is globalized as follows: let $Y\mapsto Y^{\bullet}$ be
the forgetful functor from Grothendieck schemes to monoidal spaces. Let $X$ be
a monoid scheme and $\\{\operatorname{MSpec}A_{i}\\}_{i\in I}$ an affine open
covering of $X$. Let
$X_{\mathbb{Z}}=\textup{colim}\operatorname{Spec}(A_{i,{\mathbb{Z}}})$ be the
base extension of $X$ to ${\mathbb{Z}}$, which is a Grothendieck scheme that
does not depend on the choice of the covering
$\\{\operatorname{MSpec}A_{i}\\}_{i\in I}$. There exists a morphism of
monoidal spaces $X_{\mathbb{Z}}^{\bullet}\rightarrow X$ generalizing the map
$p$ above (see Section 1.3).

######  Theorem B.

The underlying continuous map of the morphism of monoidal spaces
$X_{\mathbb{Z}}^{\bullet}\rightarrow X$ factorizes through a surjective
continuous map $\mygamma:X_{\mathbb{Z}}\rightarrow\operatorname{SCong}X$
characterized by the property that if $U$ is an affine open of $X$, then
$\mygamma(U_{\mathbb{Z}})\subseteq\operatorname{SCong}U$ and
$\mygamma|_{U_{\mathbb{Z}}}:U_{\mathbb{Z}}\rightarrow\operatorname{SCong}U$ is
induced by the inclusion $\Gamma U\rightarrow(\Gamma
U_{\mathbb{Z}})^{\bullet}$ (see Theorem 4.1 and Section 4.2).

We have a description of the strong $k$-congruence space of a $k$-scheme in
terms of a map coming from a Grothendieck scheme.

######  Proposition .

Let $k$ be a domain, $X$ a $k$-scheme and $F$ a field containing $k$ as a
multiplicative submonoid. Let $\mygamma_{F}$ be the composition
$(X_{\mathbb{Z}}\times\operatorname{Spec}F)^{\bullet}\rightarrow
X_{\mathbb{Z}}^{\bullet}\rightarrow\operatorname{SCong}X$. Then there exists a
closed subscheme $Z$ of $X_{\mathbb{Z}}\times\operatorname{Spec}F$ whose image
by $\mygamma_{F}$ is $\operatorname{SCong}_{k}X$ (see Theorem 4.2).

Even without a good sheaf on the space of (strong) congruences of a monoid
scheme $X$, every point $x\in\operatorname{SCong}X$ has a meaningful “residue
field” $\mykappa(x)$ with the following property:

######  Proposition .

The factorization $\mygamma$ presented in Theorem B induces an injective
morphism of pointed groups
$\mygamma_{x}:\mykappa(\mygamma(x))\rightarrow\mykappa(x)^{\bullet}$ for every
$x\in X_{\mathbb{Z}}$ (see Section 4.1).

In analogy with classical scheme theory, we say that a morphism of monoid
schemes $\myvarphi:X\rightarrow Y$ is locally of finite type if every $x\in X$
has an affine open neighborhood $U$ and $\myvarphi(x)$ has an affine open
neighborhood $V$ containing $\myvarphi(U)$ such that $\Gamma U$ is a finitely
generated $\Gamma V$-algebra.

######  Theorem C.

Let $k$ be a domain and $X$ a $k$-scheme locally of finite type. Then
$\operatorname{SCong}_{k}X$ is a sober space (see Theorem 4.4).

Follow-up work. In [Smi93] Smirnov explains that a conjectural Hurwitz
inequality in ${\mathbb{F}}_{1}$-geometry implies the ABC conjecture. For this
he finds ad hoc definitions for the “compactification”
$\overline{\operatorname{Spec}{\mathbb{Z}}}$, for the projective line over
algebraic extensions of ${\mathbb{F}}_{1}$ and for maps
$\overline{\operatorname{Spec}{\mathbb{Z}}}\rightarrow{\mathbb{P}}_{{\mathbb{F}}_{1}}^{1}$
induced by rational numbers. We show in [Jar23] how these spaces and maps fit
into the framework of strong congruence spaces.

### Acknowledgements

The author thanks Oliver Lorscheid for several conversations and for his help
with preparing this text. The author also thanks Leo Herr, Alejandro Martínez
Méndez and Eduardo Vital for useful conversations. The present work was
carried out with the support of CNPq, National Council for Scientific and
Technological Development - Brazil.

##  1\. Preliminaries

In this section we recall the theory of monoid schemes. For more details, see
[Dei05], [CC10], [CLS12] and [CHWW15].

###  1.1. Basic definitions

A pointed monoid is a set $A$ equipped with an associative binary operation

$\displaystyle A\times A$ $\displaystyle\rightarrow A$ $\displaystyle(a,b)$
$\displaystyle\mapsto a\cdot b,$

called multiplication or product, such that $A$ has an element $0$, called
absorbing element or zero, and an element $1$, called one, satisfying $0\cdot
a=a\cdot 0=0$ and $1\cdot a=a\cdot 1=a$ for all $a$ in $A$. We sometimes use
$ab$ to denote $a\cdot b$. A pointed monoid is commutative if $ab=ba$ for all
$a,b$ in $A$. Throughout this text all monoids are commutative. A pointed
monoid is integral (or cancellative) if $ab=ac$ implies $a=0$ or $b=c$. A
pointed monoid is without zero divisors if $ab=0$ implies $a=0$ or $b=0$. Note
that if a pointed monoid is integral, then it is without zero divisors.

An ideal of $A$ is a subset $I\subseteq A$ such that $0\in I$ and $ax\in I$
for all $a\in A$ and $x\in I$. Every ideal $I$ induces an equivalence relation
$\sim$ on $A$ generated by $\\{(0,x)\in A\times A\mid x\in I\\}$. The quotient
set $A/I:=A/\sim$ is a pointed monoid with operation given by
$[a]\cdot[b]:=[ab]$, absorbing element $[0]$ and one $[1]$. The ideal $I$ is
prime if $A/I$ is without zero divisors.

A morphism of pointed monoids is a multiplicative map preserving zero and one.
We use ${\mathbb{F}}_{1}$ to denote the initial object $\\{0,1\\}$ of the
category of pointed monoids.

An element $u\in A$ is invertible if there exists $y\in A$ such that $uy=1$.
We denote the set of invertible elements of $A$ by $A^{\times}$. The set
$A\backslash A^{\times}$ is a prime ideal and contains every proper ideal of
$A$. A morphism of pointed monoids $f:A\rightarrow B$ is local if
$f^{-1}(B^{\times})=A^{\times}$. A pointed group is a pointed monoid $A$ such
that $A^{\times}=A\backslash\\{0\\}$.

If $f:A\rightarrow B$ is a morphism and $I$ is an ideal of $B$, then the set

$f^{*}(I):=\\{a\in A\mid f(a)\in I\\}$

is an ideal of $A$, called pullback of $I$ along $f$. If $I$ is prime, then
$f^{*}(I)$ is also prime.

A subset $S\subseteq A$ is multiplicative if it is multiplicatively closed and
contains $1$. In this case, we define the localization $S^{-1}A$ as follows:
as a set, $S^{-1}A:=(A\times S)/\sim$, where $\sim$ is the equivalence
relation given by $(a,s)\sim(a^{\prime},s^{\prime})$ if there exists $t\in S$
such that $tsa^{\prime}=ts^{\prime}a$. We denote the class of $(a,s)$ in
$S^{-1}A$ by $\frac{a}{b}$. The multiplication on $S^{-1}A$ is given by

$\dfrac{a}{s}.\dfrac{b}{t}:=\dfrac{ab}{st},$

with zero $\frac{0}{1}$ and one $\frac{1}{1}$. If $I$ is a prime ideal, then
$S:=A\backslash I$ is a multiplicative subset of $A$ and we denote the
localization $S^{-1}A$ by $A_{I}$. For $h\in A$, we use $A[h^{-1}]$ to denote
the monoid $\\{h^{n}\mid n\in{\mathbb{N}}\\}^{-1}A$.

If $A$ is without zero divisors, the group of fractions $A$ is the
localization $(A\backslash\\{0\\})^{-1}A$, denoted by $\textup{Frac}(A)$. It
is a pointed group and the natural map $A\rightarrow\textup{Frac}(A)$ is
injective.

Let $k$ be a pointed monoid. A $k$-algebra is a pointed monoid $A$ with a
morphism $k\rightarrow A$. We denote the image of an element $\mylambda\in k$
via the map $k\rightarrow A$ also by $\mylambda$. We say that $A$ is a
finitely generated $k$-algebra if there exists $a_{1},\dotsc,a_{n}$ in $A$
such that for every $a\in A$ there exist $\mylambda\in k$ and
$m_{1},\dotsc,m_{n}\in{\mathbb{N}}$ satisfying $a=\mylambda
a_{1}^{m_{1}}\cdots a_{n}^{m_{n}}$. A pointed monoid $A$ is finitely generated
if it is a finitely generated ${\mathbb{F}}_{1}$-algebra.

Let $M$ be a (not necessarily pointed) monoid with operation $+$. We define
the $k$-algebra $k[M]$ as the set of formal monomials

$\big{\\{}\mylambda\mychi^{m}\mid\mylambda\in k\backslash\\{0\\}\text{ and
}m\in M\big{\\}}\cup\big{\\{}0\big{\\}}$

with operation
$(\mylambda\mychi^{m})\cdot(\mymu\mychi^{n}):=(\mylambda\mymu)\mychi^{m+n}$
and $0\cdot(\mylambda\mychi^{m})=0$. For $M={\mathbb{N}}_{\geq 1}^{n}$ and
variables $T_{1},\dotsc,T_{n}$, we sometimes use $k[T_{1},\dotsc,T_{m}]$ to
denote $k[M]$, and $T_{1}^{m_{1}}\dotsc T_{n}^{m_{n}}$ for
$\mychi^{(m_{1},\dotsc,m_{n})}$.

###  1.2. The geometry of monoids

We use $\operatorname{MSpec}A$ to denote the set of prime ideals of the
pointed monoid $A$. For $h\in A$, we denote the set
$\\{I\in\operatorname{MSpec}A\mid h\notin I\\}$ by $U_{h}$. We endow
$\operatorname{MSpec}A$ with the topology generated by the basis $\\{U_{a}\mid
a\in A\\}$, and with the sheaf of pointed monoids
$\mathcal{O}_{\operatorname{MSpec}A}$ characterized by
$\Gamma(\mathcal{O}_{\operatorname{MSpec}A},U_{h})=A[h^{-1}]$ for $h\in A$,
with restriction maps

$\begin{array}[]{ccc}\Gamma(\mathcal{O}_{\operatorname{MSpec}A},U_{h})&\longrightarrow&\Gamma(\mathcal{O}_{\operatorname{MSpec}A},U_{gh})\\\
\dfrac{a}{h}&\longmapsto&\dfrac{ga}{gh}.\end{array}$

A subset of the form $U_{h}$ is called principal open.

A monoidal space is a pair $(X,\mathcal{O}_{X})$ where $X$ is a topological
space and $\mathcal{O}_{X}$ is a sheaf of pointed monoids on $X$. A morphism
of monoidal spaces $(X,\mathcal{O}_{X})\rightarrow(Y,\mathcal{O}_{Y})$ is a
pair $(\myvarphi,\myvarphi^{\\#})$ where $\myvarphi:X\rightarrow Y$ is a
continuous map and
$\myvarphi^{\\#}:\mathcal{O}_{Y}\rightarrow\myvarphi_{*}\mathcal{O}_{X}$ is a
morphism of sheaves of pointed monoids such that the induced map between
stalks
$\myvarphi^{\\#}_{x}:\mathcal{O}_{Y,\myvarphi(x)}\rightarrow\mathcal{O}_{X,x}$
is local for every $x\in X$.

An affine monoid scheme is a monoidal space isomorphic to
$(\operatorname{MSpec}A,\mathcal{O}_{\operatorname{MSpec}A})$ for some pointed
monoid $A$. A monoid scheme is a monoidal space that has an open covering by
affine monoid schemes. As a full subcategory of monoidal spaces, monoid
schemes form a category, denoted by $\operatorname{MSch}$.

Let $Y$ be a monoid scheme. An $Y$-scheme is a monoid scheme $X$ with a
morphism $X\rightarrow Y$. A morphism of $Y$-schemes $f:X\rightarrow Z$ is a
morphism of monoid schemes such that the diagram

${X}$${Z}$${Y}$$\scriptstyle{f}$

commutes. We denote the category of $Y$-schemes by $\operatorname{MSch}/Y$.
For the affine scheme $\operatorname{MSpec}k$, we write “$k$-scheme” and
“$\operatorname{MSch}/k$” instead of “$\operatorname{MSpec}k$-scheme” and
“$\operatorname{MSch}/\operatorname{MSpec}k$”, respectively.

###  1.3. Base extension to rings

Throughout this text all rings are commutative with unit. Let $R$ be a ring
and $A$ a pointed monoid.

Define $A_{R}$ as the $R$-algebra $R[A]/\langle 1_{R}0_{A}\rangle$, where
$R[A]$ is the monoid algebra of $A$ over $R$ and $\langle 1_{R}0_{A}\rangle$
is the ideal generated by $1_{R}0_{A}$111The subscripts at right indicate if
the corresponding elements are in $R$ or $A$. In particular, $1_{R}$ is the
unit of $R$ and $0_{A}$ is the absorbing element of $A$. The assignment
$A\mapsto A_{R}$ extends to a functor from the category of pointed monoids to
$\textup{Alg}_{R}$, which is left-adjoint to the forgetful functor $B\mapsto
B^{\bullet}$ from $R$-algebras to pointed monoids. For an affine monoid scheme
$U=\operatorname{MSpec}A$, we use $U_{R}$ to denote the affine $R$-scheme
$\operatorname{Spec}(A_{R})$. We also define a functor $Y\mapsto Y^{\bullet}$
from schemes to monoidal spaces by simply forgetting the additive structure of
$\mathcal{O}_{Y}$.

Let $X$ be a monoid scheme. Let $\mathcal{U}_{X}$ be the category of all
affine open subschemes of $X$ with inclusions as morphisms. The base extension
of $X$ to $R$ (or $R$-realization of $X$) is the $R$-scheme

$X_{R}:=\underset{U\in\mathcal{U}_{X}}{\textup{colim}}\;U_{R}.$

If $\\{U_{i}\\}_{i\in I}$ is an affine covering of a monoid scheme $X$, then
$\\{U_{i,R}\\}_{i\in I}$ is an affine covering of $X_{R}$. If $U,V$ are affine
open subsets of $X$, then $U_{R}\cap V_{R}=(U\cap V)_{R}$ in $X_{R}$ (cf.
[CLS12, Cor. 3.3] and [CHWW15, Sec. 5]).

######  Proposition .

The scheme $X_{R}$ comes with a morphism $p_{R}:X_{R}^{\bullet}\rightarrow X$
of monoidal spaces that is functorial in $X$.

###### Proof.

By [CLS12, Thm. 3.2], one has a continuous map
$\mybeta:X_{\mathbb{Z}}\rightarrow X$, functorial in $X$, characterized by
$\mybeta^{-1}(U)=U_{\mathbb{Z}}$ for $U=\operatorname{MSpec}A$ affine open of
$X$ and $\mybeta|_{U_{\mathbb{Z}}}:U_{\mathbb{Z}}\rightarrow U$ sending a
prime ideal $\mathfrak{p}\in\operatorname{Spec}(A_{\mathbb{Z}})$ to
$\mathfrak{p}\cap A\in\operatorname{MSpec}A$. As
$X_{R}=X_{\mathbb{Z}}\times_{\operatorname{Spec}{\mathbb{Z}}}\operatorname{Spec}R$,
we define the underlying continuous map of $p_{R}$ as the composition

$X_{R}^{\bullet}=X_{R}\longrightarrow
X_{\mathbb{Z}}\xrightarrow{\;\;\mybeta\;\,}X.$

Note that $p_{R}^{-1}(U)=U_{R}$ for $U$ affine open of $X$. One defines the
morphism of sheaves $p_{R}^{\\#}:\mathcal{O}_{X}\rightarrow
p_{R,*}(\mathcal{O}_{X_{R}^{\bullet}})$ as the inclusion, characterized by

$p_{R}^{\\#}(U):A=\Gamma(\mathcal{O}_{X},U)\hookrightarrow
A_{R}^{\bullet}=\Gamma(\mathcal{O}_{X_{R}^{\bullet}},U_{R})$

for $U=\operatorname{MSpec}A$ affine open of $X$. ∎

##  2\. Congruence spaces

In this section we explain the construction of the congruence space associated
to a monoid scheme as introduced in [LR23, Section 2].

######  Definition .

Let $A$ be a pointed monoid. A congruence on $A$ is an equivalence relation
$\mathfrak{c}\subseteq A\times A$ such that $(ab,ac)\in\mathfrak{c}$ whenever
$(b,c)\in\mathfrak{c}$ and $a\in A$. The null ideal of a congruence
$\mathfrak{c}$ is the set

$I_{\mathfrak{c}}:=\\{a\in A\mid(a,0)\in\mathfrak{c}\\}.$

The congruence generated by a subset $E=\\{(a_{i},b_{i})\\}_{i\in I}\subseteq
A\times A$ is

$\langle E\rangle:=\bigcap\\{\mathfrak{c}\mid\mathfrak{c}\text{ is a
congruence containing }E\\},$

which is the smallest congruence containing $E$. The trivial congruence is
$\langle\emptyset\rangle$, which is the minimal congruence $\\{(a,b)\in
A\times A\mid a=b\\}$.

The quotient set $A/\mathfrak{c}$ is a pointed monoid with product
$[a]\cdot[b]:=[ab]$, absorbing element $[0]$ and one $[1]$. The congruence
$\mathfrak{c}$ is called prime if $A/\mathfrak{c}$ is integral. If
$\mathfrak{c}$ is prime, then $I_{\mathfrak{c}}$ is a prime ideal. The
congruence space of $A$ is the set

$\operatorname{Cong}A:=\\{\mathfrak{c}\subseteq A\times
A\mid\mathfrak{c}\text{ is a prime congruence of }A\\}$

with the topology generated by the sets

$U_{a,b}:=\\{\mathfrak{c}\in\operatorname{Cong}A\mid(a,b)\notin\mathfrak{c}\\},$

for $a,b\in A$, i.e., the coarsest topology such that $U_{a,b}$ is open for
all $a,b\in A$.

######  Remark .

The set $\\{U_{a,b}\mid a,b\in A\\}$ is not a basis of the topology on
$\operatorname{Cong}A$ in general (cf. [LR23, Example 2.10]).

######  Notation.

_For a congruence $\mathfrak{c}$, we sometimes use $a\sim_{\mathfrak{c}}b$ and
$a\sim b$ instead of $(a,b)\in\mathfrak{c}$. We also use $\langle a_{i}\sim
b_{i}\mid i\in I\rangle$ to denote the congruence
$\langle\\{(a_{i},b_{i})\\}_{i\in I}\rangle$._

######  Definition .

Let $f:A\rightarrow B$ be a morphism of pointed monoids. The congruence kernel
of $f$ is the set

$\textup{congker}(f):=\\{(a,b)\in A\times A\mid f(a)=f(b)\\},$

which is indeed a congruence on $A$. Note that it is prime if $B$ is integral.

Let $\mathfrak{d}$ be a congruence on $B$. The pullback of $\mathfrak{d}$
along $f$ is the set

$f^{*}(\mathfrak{d}):=\\{(x,y)\in A\times A\mid(f(x),f(y))\in\mathfrak{d}\\},$

which is a congruence on $A$. Note that $f^{*}(\mathfrak{d})$ is prime if
$\mathfrak{d}$ is prime. The association $A\mapsto\operatorname{Cong}A$ for
objects and $f\mapsto f^{*}$ for maps defines a contravariant functor from
pointed monoids to topological spaces.

Let $\mathfrak{c}$ be a congruence on $A$. The push-forward of $\mathfrak{c}$
along $f$ is the congruence

$f_{*}(\mathfrak{c}):=\langle f(a)\sim f(b)\mid(a,b)\in\mathfrak{c}\rangle.$

######  Proposition .

[LR23, Prop. 2.8] Let $A$ be a pointed monoid, $S$ a multiplicative subset of
$A$ and $\myiota_{S}:A\rightarrow S^{-1}A$ the localization map. Then the maps

${\operatorname{Cong}S^{-1}A}$${\\{\mathfrak{c}\in\operatorname{Cong}A\mid
S\cap
I_{\mathfrak{c}}=\emptyset\\}}$$\scriptstyle{\myiota_{S}^{\ast}}$$\scriptstyle{\myiota_{S,\ast}}$

are mutually inverse bijections.

######  Notation.

_If $\mathfrak{c}$ is a congruence on $A$, we use $S^{-1}\mathfrak{c}$ for the
congruence $\myiota_{S,*}(\mathfrak{c})$ on $S^{-1}A$._

Let $X$ be a monoid scheme and define
$U^{\textup{cong}}:=\operatorname{Cong}\Gamma U$ for $U\in\mathcal{U}_{X}$.
Given a morphism $U\rightarrow V$ in $\mathcal{U}_{X}$, one has an induced
continuous map $U^{\textup{cong}}\rightarrow V^{\textup{cong}}$.

######  Definition .

The congruence space of $X$ is the topological space

$X^{\textup{cong}}:=\underset{U\in\mathcal{U}_{X}}{\textup{colim}}\,U^{\textup{cong}}.$

There is a functor from monoid schemes to topological spaces

$(-)^{\textup{cong}}:\operatorname{MSch}\rightarrow\textup{Top}$

extending $X\mapsto X^{\textup{cong}}$ such that for every morphism
$\myvarphi:\operatorname{MSpec}B\rightarrow\operatorname{MSpec}A$ of affine
monoid schemes one has
$\myvarphi^{\textup{cong}}=\big{(}\myvarphi^{\\#}(\operatorname{MSpec}A)\big{)}^{*}:\operatorname{Cong}B\rightarrow\operatorname{Cong}A$.

The space $X^{\textup{cong}}$ comes equipped with a continuous projection
$\mypi_{X}:X^{\textup{cong}}\rightarrow X$ given by
$\mypi_{X}(\tilde{x})=I_{\mathfrak{c}}$ if $U\subseteq X$ is an affine open
and $\mathfrak{c}$ is a prime congruence on $\Gamma U$ such that
$\tilde{x}=\mathfrak{c}\in\operatorname{Cong}\Gamma U\subseteq
X^{\textup{cong}}$.

######  Proposition .

Let $X$ be a monoid scheme and $\\{U_{i}\\}_{i\in I}$ an open covering of $X$.
Then $\\{U_{i}^{\textup{cong}}\\}$ is an open covering of $X^{\textup{cong}}$.

###### Proof.

See [LR23, Prop. 2.13]. ∎

Let $X$ be a monoid scheme, $\tilde{x}\in X^{\textup{cong}}$ and
$x:=\mypi_{X}(\tilde{x})\in X$. Let $\mathcal{U}_{X,x}$ be the full
subcategory of $\mathcal{U}_{X}$ whose objects are the affine open
neighborhoods of $x$. For each $U\in\mathcal{U}_{X,x}$, let $i_{U}:\Gamma
U\rightarrow\mathcal{O}_{X,x}$ be the natural map into the colimit and
$\mathfrak{c}_{\tilde{x},U}\in\operatorname{Cong}\Gamma U$ the prime
congruence corresponding to $\tilde{x}$. We define the congruence

$\mathfrak{c}_{\tilde{x}}:=\Bigg{\langle}\underset{U\in\mathcal{U}_{x}}{\bigcup}i_{U,*}(\mathfrak{c}_{\tilde{x},U})\Bigg{\rangle}$

on $\mathcal{O}_{X,x}$, and the pointed group

$\mykappa(\tilde{x}):=\mathcal{O}_{X,x}/\mathfrak{c}_{\tilde{x}}.$

By [LR23, Lemma 2.17], $\mathfrak{c}_{\tilde{x}}$ is prime and for any
$U\in\mathcal{U}_{X,x}$ one has
$\mathfrak{c}_{\tilde{x}}=i_{U,*}(\mathfrak{c}_{\tilde{x},U})$ and
$\mykappa(\tilde{x})\simeq\textup{Frac}(\Gamma U/\mathfrak{c}_{\tilde{x},U})$.

A morphism of monoid schemes $\myvarphi:X\rightarrow Y$ induces an injective
morphism of pointed groups

$\mykappa\big{(}\myvarphi^{\textup{cong}}(\tilde{x})\big{)}\hookrightarrow\mykappa(\tilde{x})$

for every $\tilde{x}\in X^{\textup{cong}}$.

##  3\. The strong congruence space

######  Definition .

A domain is a pointed monoid $A$ that is isomorphic to a pointed submonoid of
$K^{\bullet}$ for some field $K$. A strong prime congruence on $A$ is a
congruence $\mathfrak{c}$ such that $A/\mathfrak{c}$ is a domain. We denote
the set of strong prime congruences on A by $\operatorname{SCong}A$.

Let $k$ be a pointed monoid and $A$ a $k$-algebra. A $k$-congruence on $A$ is
a strong prime congruence $\mathfrak{c}\in\operatorname{SCong}A$ such that the
natural map $k\rightarrow A/\mathfrak{c}$ is injective. The set of
$k$-congruences of $A$ is denoted by $\operatorname{SCong}_{k}A$. Note that
$\operatorname{SCong}A=\operatorname{SCong}_{{\mathbb{F}}_{1}}A$ and
$\operatorname{SCong}_{k}A$ is empty if $k$ is not a domain or if the map
$k\rightarrow A$ is not injective.

A $k$-prime of $A$ is a prime ideal $P\in\operatorname{MSpec}A$ such that the
natural map $k\rightarrow A/P$ is injective. The set of $k$-primes of $A$ is
denoted by $\operatorname{MSpec}_{k}A$. Note that if
$\mathfrak{c}\in\operatorname{SCong}_{k}A$, then
$I_{\mathfrak{c}}\in\operatorname{MSpec}_{k}A$.

######  Remark .

If $A$ is a domain, then $A$ is integral and for every $\myalpha$ in $A$ and
$n\geq 1$ one has $\\#\\{x\in A|\,x^{n}=\myalpha\\}\leq n$. Section 4.2 shows
that these two properties characterize domains.

######  Remark .

If $A$ is a domain, then any pointed submonoid of $\textup{Frac}(A)$ is a
domain.

######  Remark .

Throughout this text the term “domain” only refers to pointed monoids as
presented in Section 3, and not to nonzero commutative rings without zero
divisors.

######  Lemma .

Let $A$ be a $k$-algebra and $S$ a multiplicative subset of $A$. Then the map
$\mathfrak{c}\mapsto S^{-1}\mathfrak{c}$ gives a bijection

$\\{\mathfrak{c}\in\operatorname{SCong}_{k}A\mid I_{\mathfrak{c}}\cap
S=\emptyset\\}\xrightarrow{\sim}\operatorname{SCong}_{k}S^{-1}A.$

###### Proof.

Let $\mathfrak{c}$ be a $k$-congruence on $A$ with $I_{\mathfrak{c}}\cap
S=\emptyset$. Then there exists a field $K$ and an inclusion
$A/\mathfrak{c}\hookrightarrow K^{\bullet}$. As $K^{\bullet}$ is a pointed
group, the composition $A\rightarrow A/\mathfrak{c}\rightarrow K^{\bullet}$
induces a map $S^{-1}A\rightarrow K^{\bullet}$ whose kernel is
$S^{-1}\mathfrak{c}$. Thus $S^{-1}\mathfrak{c}\in\operatorname{SCong}S^{-1}A$.
As the diagram

${k}$${A}$${A/\mathfrak{c}}$${K^{\bullet}}$${S^{-1}A}$${S^{-1}A/S^{-1}\mathfrak{c}}$

commutes and the map $k\rightarrow K^{\bullet}$ given by the upper row is
injective, we conclude that $S^{-1}\mathfrak{c}\in\operatorname{SCong}_{k}A$.

By Section 2, the map $\mathfrak{c}\mapsto S^{-1}\mathfrak{c}$ is injective
and given $\mathfrak{d}$ in $\operatorname{SCong}_{k}S^{-1}A$, there exists
$\mathfrak{r}\in\operatorname{Cong}A$ such that
$\mathfrak{d}=S^{-1}\mathfrak{r}$. It only remains to show that $\mathfrak{r}$
is a $k$-congruence. The composition

$k\rightarrow A\rightarrow A/\mathfrak{r}\rightarrow
S^{-1}\big{(}A/\mathfrak{r}\big{)}\simeq S^{-1}A/S^{-1}\mathfrak{r}$

is injective, thus $k\rightarrow A/\mathfrak{r}$ is injective. As
$A/\mathfrak{r}$ is integral and $S^{-1}A/S^{-1}\mathfrak{r}$ is a domain,
$A/\mathfrak{r}$ is a domain. ∎

######  Lemma .

Let $A$ be a $k$-algebra, $\mathfrak{d}$ a congruence on $A$ and
$\mypi:A\twoheadrightarrow A/\mathfrak{d}$ the natural projection. Then the
map $\mathfrak{c}\mapsto\mypi_{*}(\mathfrak{c})$ gives a bijection

$\\{\mathfrak{c}\in\operatorname{SCong}_{k}A\mid\mathfrak{d}\subseteq\mathfrak{c}\\}\xrightarrow{\sim}\operatorname{SCong}_{k}(A/\mathfrak{d}).$

###### Proof.

By [LR23, Prop. 4.13], the map $\mathfrak{c}\mapsto\mypi_{*}(\mathfrak{c})$
gives a bijection

$\\{\mathfrak{c}\in\operatorname{Cong}A\mid\mathfrak{d}\subseteq\mathfrak{c}\\}\xrightarrow{\sim}\operatorname{Cong}(A/\mathfrak{d}).$

As $A/\mathfrak{c}\simeq(A/\mathfrak{d})\big{/}\mypi_{*}(\mathfrak{c})$ for
any congruence $\mathfrak{c}$ with $\mathfrak{d}\subseteq\mathfrak{c}$, the
result follows. ∎

######  Lemma .

Let $k$ be a domain, $f:A\rightarrow B$ a morphism of $k$-algebras and
$\mathfrak{q}$ a $k$-congruence on $B$. Then $f^{*}(\mathfrak{q})$ is a
$k$-congruence of $A$. This shows that the map $f^{*}$ restricts to a
continuous map
$f^{*}:\operatorname{SCong}_{k}B\rightarrow\operatorname{SCong}_{k}A$.

###### Proof.

Note that $f$ induces an injection $A/f^{*}(\mathfrak{q})\hookrightarrow
B/\mathfrak{q}$. As $B/\mathfrak{q}$ is a domain, one has that
$A/f^{*}(\mathfrak{q})$ is also a domain. As the diagram

${A}$${B}$${k}$${A/f^{*}(\mathfrak{q})}$${B/\mathfrak{q}}$

commutates and the map $k\rightarrow B/\mathfrak{q}$ given by the bottom row
is injective, one has that $k\rightarrow A/f^{*}(\mathfrak{q})$ is injective.
Thus $f^{*}(\mathfrak{q})\in\operatorname{SCong}_{k}A$. ∎

Let $k$ be a pointed monoid and $X$ a $k$-scheme. For $U,V\subseteq X$ affine
open with $U\subseteq V$, let $\myiota_{UV}:U\hookrightarrow V$ be the
inclusion. Let $\mathcal{U}_{X,k}^{\textup{scong}}$ be the category whose
objects are the topological spaces
$U_{k}^{\textup{scong}}:=\operatorname{SCong}_{k}\Gamma U$ for affine open
subschemes $U$ of $X$, with

$\textup{Hom}_{\mathcal{U}_{X,k}^{\textup{cong}}}(U_{k}^{\textup{scong}},V_{k}^{\textup{scong}})=\smash{\left\\{\begin{array}[]{cl}\\{(\Gamma\myiota_{UV})^{*}\\}&\text{if
}U\subseteq V\\\ {}\emptyset&\text{if }U\not\subseteq V.\end{array}\right.}$

######  Definition .

The strong $k$-congruence space of $X$ is the topological space

$\operatorname{SCong}_{k}X:=\textup{colim}\,\mathcal{U}_{X,k}^{\textup{scong}}.$

Note that $\operatorname{SCong}_{k}X$ comes with a natural inclusion to
$X^{\textup{cong}}$. We denote the restriction of the projection
$\mypi_{X}:X^{\textup{cong}}\rightarrow X$ to $\operatorname{SCong}_{k}X$
again by $\mypi_{X}$.

######  Lemma .

Let $U$ and $V$ be affine open subschemes of a monoid scheme $X$ and $x\in
U\cap V$. Then there exists an affine open neighborhood $W\subseteq U\cap V$
of $x$ that is a principal open subset of both $U$ and $V$.

###### Proof.

Let $W\subseteq U\cap V$ be an affine open neighborhood of $x$. Let $A:=\Gamma
U$. By [CHWW15, Lemma 2.4], there exists
$\mathfrak{p}\in\operatorname{MSpec}A$ such that
$W=\operatorname{MSpec}A_{\mathfrak{p}}$. By [CHWW15, Lemma 1.3], $W$ is a
principal open of $U$.

Analogously, $W$ is a principal open of $V$. ∎

######  Lemma .

Let $X$ be a $k$-scheme and $\\{U_{i}\\}_{i\in I}$ a collection of open
subschemes that covers $X$. Then $\\{\operatorname{SCong}_{k}U_{i}\\}_{i\in
I}$ is an open covering of $\operatorname{SCong}_{k}X$.

###### Proof.

As $\\{U_{i}^{\textup{cong}}\\}_{i\in I}$ is an open covering of
$X_{k}^{\textup{cong}}$, it is enough to show that if $U$ is an open subscheme
of $X$, then
$\operatorname{SCong}_{k}U=U^{\textup{cong}}\cap\operatorname{SCong}_{k}X$.

Let $x\in U^{\textup{cong}}\cap\operatorname{SCong}_{k}X$. Then there exists
$V=\operatorname{MSpec}M\subseteq X$ affine open such that $x\in
V_{k}^{\textup{scong}}$. Let $W=\operatorname{MSpec}N\subseteq U$ affine open
such that $x\in W^{\textup{cong}}$. By Section 3, there exists
$Y=\operatorname{MSpec}A\subseteq W\cap V$ affine open containing
$\mypi_{X}(x)$ that is a principal open of both $W$ and $V$. Thus $x\in
Y^{\textup{cong}}\subseteq(W\cap V)^{\textup{cong}}$. Let $s\in M$ such that
$M[s^{-1}]\simeq A$. One has
$x\in\operatorname{SCong}_{k}M\cap\operatorname{Cong}M[s^{-1}]$. By Section 3,
this implies

$x\in\operatorname{SCong}_{k}M[s^{-1}]=Y_{k}^{\textup{scong}}\subseteq\operatorname{SCong}_{k}U.$

∎

######  Proposition .

There is an extension of the association $X\mapsto\operatorname{SCong}_{k}X$
to a subfunctor of
$(-)^{\textup{cong}}:\operatorname{MSch}/k\rightarrow\textup{Top}$.

###### Proof.

One knows that $\operatorname{SCong}_{k}X$ is a subspace of
$X^{\textup{cong}}$ for any $X\in\operatorname{MSch}/k$.

Next we define the functor $\operatorname{SCong}_{k}(-)$ on morphisms. Let
$\myvarphi:X\rightarrow Y$ be a morphism in $\operatorname{MSch}/k$. Let
$\\{U_{\myalpha}\\}_{\myalpha\in\Lambda}$ be an affine open covering of $X$
and $\\{V_{\myalpha}\\}_{\myalpha\in\Lambda}$ an affine open covering of $Y$
such that $\myvarphi(U_{\myalpha})\subseteq V_{\myalpha}$ for any
$\myalpha\in\Lambda$. Let $\myvarphi_{\myalpha}:U_{\myalpha}\rightarrow
V_{\myalpha}$ be the restriction of $\myvarphi$. Note that
$\myvarphi_{\myalpha}$ is induced by a morphism of pointed monoids
$\mygamma_{\myalpha}:\Gamma V_{\myalpha}\rightarrow\Gamma U_{\myalpha}$. By
Section 3,
$\mygamma_{\myalpha}^{\textup{cong}}(U_{\myalpha,k}^{\textup{scong}})\subseteq
V_{\myalpha,k}^{\textup{scong}}$. By Section 3,
$\\{U_{\myalpha,k}^{\textup{scong}}\\}_{\myalpha\in\Lambda}$ is an open
covering of $\operatorname{SCong}_{k}X$ and
$\\{V_{\myalpha,k}^{\textup{scong}}\\}_{\myalpha\in\Lambda}$ is an open
covering of $\operatorname{SCong}_{k}Y$. Thus

$\myvarphi^{\textup{cong}}(\operatorname{SCong}_{k}X)=\underset{\myalpha\in\Lambda}{\bigcup}\mygamma_{\myalpha}^{\textup{cong}}(U_{\myalpha,k}^{\textup{scong}})\subseteq\operatorname{SCong}_{k}Y.$

Define
$\operatorname{SCong}_{k}\myvarphi:\operatorname{SCong}_{k}X\rightarrow\operatorname{SCong}_{k}Y$
as the restriction of $\myvarphi^{\textup{cong}}$.

The fact that $\operatorname{SCong}_{k}(-)$ is a subfunctor of
$(-)^{\textup{cong}}$ follows from the construction above. ∎

##  4\. Relation to classical scheme theory

###  4.1. The map $\mygamma:X_{R}\rightarrow\operatorname{SCong}X$

We construct a continuous map $\mygamma$ such that the underlying continuous
map of $p_{R}:X_{R}^{\bullet}\rightarrow X$ factorizes as

${X_{R}}$${\operatorname{SCong}X}$${X.}$$\scriptstyle{\mygamma}$$\scriptstyle{p_{R}}$$\scriptstyle{\mypi_{X}}$

Let $A$ be a pointed monoid and $R$ a ring. A morphism of pointed monoids
$\myvarphi:A\rightarrow R^{\bullet}$ induces a continuous map
$\myvarphi^{*}:\operatorname{Spec}R\rightarrow\operatorname{SCong}A$ that
sends a prime ideal $\mathfrak{p}$ to the strong prime congruence

$\myvarphi^{*}(\mathfrak{p}):=\operatorname{congker}(\mypi_{\mathfrak{p}}^{\bullet}\circ\myvarphi)=\big{\\{}(a,b)\in
A\times A\;\big{|}\;\overline{\myvarphi(a)}=\overline{\myvarphi(b)}\text{ in
}R/\mathfrak{p}\big{\\}},$

where $\mypi_{\mathfrak{p}}:R\rightarrow R/\mathfrak{p}$ is the natural
projection.

######  Theorem 4.1.

Let $X$ be a monoid scheme and $R$ a ring. The underlying continuous map of
the morphism of monoidal spaces $p_{R}:X_{R}^{\bullet}\rightarrow X$
factorizes through a continuous map
$\mygamma:X_{R}\rightarrow\operatorname{SCong}X$ characterized by the property
that if $U$ is an affine open of $X$, then
$\mygamma(U_{R})\subseteq\operatorname{SCong}U$ and
$\mygamma|_{U_{R}}:U_{R}\rightarrow\operatorname{SCong}U$ is induced by the
inclusion $\Gamma U\rightarrow(\Gamma U_{R})^{\bullet}$.

###### Proof.

As $X_{R}=X_{\mathbb{Z}}\times_{\mathbb{Z}}\operatorname{Spec}R$, it is enough
to show the result for $R={\mathbb{Z}}$. The result follows for $X$ affine by
Section 1.3 and the discussion above. We just need to verify that if $U$ and
$V$ are affine open subsets of $X$, then the diagram

${U_{\mathbb{Z}}\cap
V_{\mathbb{Z}}}$${U_{\mathbb{Z}}}$${V_{\mathbb{Z}}}$${\operatorname{SCong}X}$

commutes. By [CLS12, Cor. 3.3], $U_{\mathbb{Z}}\cap V_{\mathbb{Z}}=(U\cap
V)_{\mathbb{Z}}$ and, by Section 3, the intersection $U\cap V$ has a covering
by principal open subschemes of both $U$ and $V$. As the diagram

${U_{\mathbb{Z}}}$${W_{\mathbb{Z}}}$${V_{\mathbb{Z}}}$${\operatorname{SCong}U}$${\operatorname{SCong}W}$${\operatorname{SCong}V}$${\operatorname{SCong}X}$

commutes for every $W$ that is a principal open subscheme of both $U$ and $V$,
the result follows. ∎

######  Proposition .

The factorization of $p_{R}$ through $\mygamma$, as in Theorem 4.1, induces an
injective morphism of pointed groups
$\mygamma_{x}:\mykappa(\mygamma(x))\rightarrow\mykappa(x)^{\bullet}$ for every
$x\in X_{R}$.

###### Proof.

Let $x\in X_{R}$ and $U=\operatorname{MSpec}A\subseteq X$ an affine open
subscheme such that $x\in U_{R}$. Let $\mathfrak{p}_{x}$ be the prime ideal of
$A_{\mathbb{Z}}$ corresponding to $x$. As $\mygamma(x)$ is defined as the
point corresponding to the congruence kernel $\mathfrak{c}$ of the map

$A\rightarrow
A_{\mathbb{Z}}^{\bullet}\twoheadrightarrow(A_{\mathbb{Z}}/\mathfrak{p}_{x})^{\bullet},$

one has an injective morphism of pointed groups

$\mykappa(\mygamma(x))=\textup{Frac}(A/\mathfrak{c})\rightarrow\big{(}\textup{Frac}(A_{\mathbb{Z}}/\mathfrak{p}_{x})\big{)}^{\bullet}=\mykappa(x)^{\bullet}.$

∎

###  4.2. Characterization of $\operatorname{SCong}_{k}X$

We give a characterization of $\operatorname{SCong}_{k}X$ in terms of a
continuous map coming from the base extension of $X$ to a field that contains
$k$ as a multiplicative submonoid.

######  Lemma .

Let $A$ be an integral pointed monoid such that $\\#\\{x\in A\mid
x^{n}=\myalpha\\}\leq n$ for all $\myalpha\in A$ and $n\geq 1$. Then
$\\#\\{y\in\textup{Frac}(A)\mid y^{n}=1\\}\leq n$ for all $n\geq 1$.

###### Proof.

Let $n\geq 1$ and $a_{1},\dotsc,a_{n+1},b_{1},\dotsc,b_{n+1}\in
A\backslash\\{0\\}$ such that $\big{(}\frac{a_{i}}{b_{i}}\big{)}^{n}=1$ for
all $i=1,\dotsc,n+1$. Define $\myalpha:=(b_{1}\cdots b_{n+1})^{n}$. As
$a_{i}^{n}=b_{i}^{n}$, one has

$\myalpha=\Bigg{(}\underset{i\neq j}{\prod}b_{i}\cdot a_{j}\Bigg{)}^{n}$

for all $1\leq j\leq n+1$. Thus there exists $\ell\neq m$ such that

$\underset{i\neq\ell}{\prod}b_{i}\cdot a_{\ell}=\underset{i\neq
m}{\prod}b_{i}\cdot a_{m}.$

As $A$ is integral, one has $b_{m}a_{\ell}=b_{\ell}a_{m}$, which implies
$\frac{a_{\ell}}{b_{\ell}}=\frac{a_{m}}{b_{m}}$. ∎

######  Lemma .

Let $K$ be a field, $A$ a pointed submonoid of $K^{\bullet}$ and $S$ a (not
necessarily pointed) monoid. Let $\mathfrak{c}$ be a prime congruence on
$A[S]$222See Section 1.1 for the definition of $A[S]$. such that:

  1. (i)

The natural map $A\rightarrow A[S]/\mathfrak{c}$ is injective;

  2. (ii)

$\\#\\{x\in A[S]/\mathfrak{c}\;|\;\,x^{n}=\myalpha\\}\leq n$ for every
$\myalpha$ in $A[S]/\mathfrak{c}$ and $n\geq 1$.

Then there exists $\mathfrak{p}\in\operatorname{Spec}K[S]$ such that
$\mathfrak{c}=i^{*}(\mathfrak{p})$, where $i:A[S]\rightarrow(K[S])^{\bullet}$
is the natural inclusion.

###### Proof.

As $\mathfrak{c}$ is prime, $M:=(A[S]/\mathfrak{c})\backslash\\{0\\}$ is a
monoid. Let $G$ be the Grothendieck group of $M$. Note that
$A\backslash\\{0\\}\rightarrow G$ is injective. Let $I$ be the ideal of $K[G]$
generated by

$\big{\\{}1_{K}\mylambda_{G}-\mylambda_{K}1_{G}\in K[G]\;\big{|}\;\mylambda\in
A\big{\\}},$

where, for $a$ in $A$, the subscript in $a_{K}$ (resp. $a_{G}$) indicates $a$
as an element of $K$ (resp. $G$). Define the following maps:

  1. (i)

The natural inclusion $i:A[S]\hookrightarrow K[S]^{\bullet}$;

  2. (ii)

$f:A[S]\rightarrow K[G]^{\bullet}$, where $f(x)=1\overline{x}$ for $x\neq 0$,
and $f(0)=0$;

  3. (iii)

The morphism of $K$-algebras $g:K[S]\rightarrow K[G]/I$ induced by $s\mapsto
1\overline{s}+I$ for $s\in S$, where $\overline{s}$ is the class of $s$ in
$G$;

  4. (iv)

The natural projection $\mypi:K[G]\twoheadrightarrow K[G]/I$.

We have a commutative diagram of pointed monoids

${A[S]}$${(K[S])^{\bullet}}$${(K[G])^{\bullet}}$${(K[G]/I)^{\bullet}.}$$\scriptstyle{i}$$\scriptstyle{f}$$\scriptstyle{g^{\bullet}}$$\scriptstyle{\mypi^{\bullet}}$

We show that there exists a field $E$ and a morphism of rings
$\mypsi:K[G]\rightarrow E$ satisfying $I\subseteq\ker(\mypsi)$ such that the
induced map $z\mapsto\mypsi(1z)$ from $G$ to $E$ is injective. Thus
$\mathfrak{p}:=g^{-1}\big{(}\mypi(\ker(\mypsi)\big{)}$ is a prime ideal of
$K[S]$ satisfying $i^{*}(\mathfrak{p})=\mathfrak{c}$.

Let $\Gamma$ be the set $2^{K[G]}$ and $E$ the algebraic closure of the field
$K(T_{\mygamma}|\,\mygamma\in\Gamma)$. Let $\mathcal{H}$ be the set of
$K$-algebra morphisms $\myvarphi:K[H_{\myvarphi}]\rightarrow E$ satisfying:

  1. (i)

$H_{\myvarphi}$ is a subgroup of $G$ containing $A\backslash\\{0\\}$;

  2. (ii)

The induced map $z\mapsto\myvarphi(1z)$ from $H_{\myvarphi}$ to $E$ is
injective;

  3. (iii)

$\\{1_{K}\mylambda_{G}-\mylambda_{K}1_{G}\in K[H_{\myvarphi}]\mid\mylambda\in
A\\}\subseteq\ker(\myvarphi)$.

Let $\leq$ be the partial order on $\mathcal{H}$ given by
$\myvarphi\leq\mypsi$ if $H_{\myvarphi}\subseteq H_{\mypsi}$ and
$\mypsi|_{K[H_{\myvarphi}]}=\myvarphi$. Note that $\mathcal{H}$ is non-empty,
because there exists a morphism of $K$-algebras $\mytau:K[\langle
A\backslash\\{0\\}\rangle]\rightarrow E$ such that
$\mytau(1_{K}a_{G})=a_{K}1_{G}$, where $\langle A\backslash\\{0\\}\rangle$ is
the subgroup of $G$ generated by $A\backslash\\{0\\}$. Let $\mathcal{C}$ be a
chain in $\mathcal{H}$. Define
$H_{\mybeta}:=\underset{\myvarphi\in\mathcal{C}}{\bigcup}H_{\myvarphi}$ and
note that $H_{\mybeta}$ is a group. For $\myvarphi,\mypsi\in\mathcal{C}$, if
$y\in K[H_{\myvarphi}]\cap K[H_{\mypsi}]$ one has $\myvarphi(y)=\mypsi(y)$.
Thus the map

$\begin{array}[]{cccl}H_{\mybeta}&\longrightarrow&E&\\\
x&\longmapsto&\myvarphi(x)&\text{if }x\in K[H_{\myvarphi}],\end{array}$

induces an element $\mybeta\in\mathcal{H}$. Note that $\mybeta$ is an upper
bound for $\mathcal{C}$. By Zorn’s lemma, $\mathcal{H}$ has a maximal element
$\Phi$. Next we show that $H_{\Phi}=G$.

Suppose that there is an $h\in G\backslash H_{\Phi}$. We have three cases to
analyze:

Case 1: $\\{\ell>0\mid h^{\ell}\in H_{\Phi}\\}=\emptyset$, i.e., $h$ is not a
torsion element and $\langle h\rangle\cap H_{\Phi}=\\{1_{G}\\}$. In this case,
$H_{\Omega}:=\langle H_{\Phi},h\rangle$ is isomorphic to
$H_{\Phi}\times{\mathbb{Z}}$. As $|\Gamma|>\big{|}K[G]\big{|}$, there exists
$\myalpha\in\Gamma$ such that $T_{\myalpha}\notin\operatorname{im}\Phi$. Let
$\Omega:K[H_{\Omega}]\rightarrow E$ be the morphism of $K$-algebras such that
$\Omega|_{K[H_{\Phi}]}=\Phi$ and $\Omega(1h)=T_{\myalpha}$. Therefore,
$\Omega\in\mathcal{H}$ and $\Phi<\Omega$, which contradicts the maximality of
$\Phi$.

Case 2: $\\{\ell>0\mid h^{\ell}\in H_{\Phi}\\}\neq\emptyset$ and $h$ is a
torsion element. Then there exist

$w:=\min\\{\ell>0\mid h^{\ell}\in H_{\Phi}\\}\quad\text{and}\quad
s:=\textup{ord}(h^{w})$

such that $sw=\textup{ord}(h)$. Note that $\Phi(1h^{w})$ is a root of $1$ in
$E$ and let $\myzeta\in E$ be a primitive $w^{\textup{th}}$-root of
$\Phi(1h^{w})$. Note that $\textup{ord}(h^{w})=\textup{ord}(\Phi(1h^{w}))$.
Let $H_{\Omega}$ be the subgroup $\langle H_{\Phi},h\rangle$. There exists a
morphism of groups $\myomega:H_{\Omega}\rightarrow E^{*}$ such that
$\myomega(x)=\Phi(1x)$ for $x\in H_{\Phi}$, and $\myomega(h)=\myzeta$. Next we
show that $\myomega$ is injective.

If $\myomega$ is not injective, then there exists $y\in H_{\Phi}$ and
$m\in\\{1,\dotsc,w-1\\}$ such that $\myomega(yh^{-m})=1$, which implies
$\Phi(y)=\myomega(y)=\myomega(h^{m})=\myzeta^{m}$. Then

$\Phi(1y^{sw})=\myzeta^{msw}=\myomega((h^{sw})^{m})=\myomega(1)=\Phi(1),$

which implies $y^{sw}=1$. As $\textup{ord}(h)=sw$, by Section 4.2, one has
$y\in\langle h\rangle$. As the intersection $\langle h\rangle\cap H_{\Phi}$ is
equal to $\langle h^{w}\rangle$, there exists $t>0$ such that $y=h^{wt}$. This
implies $\myzeta^{wt-m}=\myomega(yh^{-m})=1$ and, consequently, that
$sw=\textup{ord}(\myzeta)$ divides $wt-m$, which is a contradiction because
$w$ does not divide $m$.

Let $\Omega:K[H_{\Omega}]\rightarrow E$ be the morphism of $K$-algebras
induced by $\myomega$. Note that $\Omega\in\mathcal{H}$ and $\Phi<\Omega$,
which contradicts the maximality of $\Phi$.

Case 3: $\\{\ell>0\mid h^{\ell}\in H_{\Phi}\\}\neq\emptyset$ and $h$ is not a
torsion element. Let

$w:=\min\\{\ell>0\mid h^{\ell}\in H_{\Phi}\\}$

and fix $\myxi\in E$ such that $\myxi^{w}=\Phi(1h^{w})$. Suppose that
$\langle\myxi\rangle\cap\Phi(1H_{\Phi})\neq\langle\myxi^{w}\rangle$. Define
$m:=\min\\{\ell>0\mid\myxi^{\ell}\in\Phi(1H_{\Phi})\\}$. Then $w>m$ and exists
$y\in H_{\Phi}$ such that $\Phi(1y)=\myxi^{m}$. Write $w=qm+r$ with $q,r$
integers satisfying $0\leq r<m$. Note that $h^{w}y^{-q}\in H_{\Phi}$ and
$\Phi(1h^{w}y^{-q})=\myxi^{r}$. As $r<m$ and $\Phi|_{1H_{\Phi}}$ is injective,
$r=0$ and $h^{w}=y^{q}$. Let $\tilde{h}:=h^{m}y^{-1}$. Note that
$\tilde{h}\notin H_{\Phi}$ and $\tilde{h}^{q}=1$. Analogously to Case 2, we
get a contradiction. Thus
$\langle\myxi\rangle\cap\Phi(1H_{\Phi})=\langle\myxi^{w}\rangle$.

Let $H_{\Omega}$ be the subgroup $\langle H_{\Phi},h\rangle$. There exists an
injective morphism of groups $\myomega:H_{\Omega}\rightarrow E^{*}$ such that
$\myomega(x)=\Phi(1x)$ for $x\in H_{\Phi}$, and $\myomega(h)=\myxi$. Let
$\Omega:K[H_{\Omega}]\rightarrow E$ be the morphism of $K$-algebras induced by
$\myomega$. Note that $\Omega\in\mathcal{H}$ and $\Phi<\Omega$, which
contradicts the maximality of $\Phi$. ∎

######  Proposition .

Let $A$ be an integral pointed monoid such that $\\#\\{x\in
A|\,x^{n}=\myalpha\\}\leq n$ for every $\myalpha$ in $A$ and $n\geq 1$. Then
there exists a field $K$ of characteristic zero and an injective morphism of
pointed monoids $A\hookrightarrow K^{\bullet}$.

###### Proof.

Note that $A$ is isomorphic to ${\mathbb{F}}_{1}[A]/\mathfrak{c}$, where
$\mathfrak{c}:=\langle 0_{{\mathbb{F}}_{1}}1_{A}\sim
1_{{\mathbb{F}}_{1}}0_{A}\rangle$. As ${\mathbb{F}}_{1}$ is a pointed
submonoid of ${\mathbb{C}}^{\bullet}$, by Section 4.2, there exists
$\mathfrak{p}\in\operatorname{Spec}\,{\mathbb{C}}[A]$ such that
$\mathfrak{c}=i^{*}(\mathfrak{p})$, where
$i:{\mathbb{F}}_{1}[A]\rightarrow{\mathbb{C}}[A]^{\bullet}$ is the inclusion.

Let $K$ be the field $\textup{Frac}({\mathbb{C}}[A]/\mathfrak{p})$. Then the
composition

$A\simeq{\mathbb{F}}_{1}[A]/\mathfrak{c}\rightarrow({\mathbb{C}}[A]/\mathfrak{p})^{\bullet}\rightarrow
K^{\bullet}$

is an injective morphism $A\hookrightarrow K^{\bullet}$ of pointed monoids. ∎

The next result, which follows from Section 4.2, gives an intrisic (i.e.,
without mention to fields) description of domains.

######  Corollary .

Let $A$ be a pointed monoid. Then $A$ is a domain if, and only if, $A$ is
integral and satisfies $\\#\\{x\in A|\,x^{n}=\myalpha\\}\leq n$ for every
$\myalpha$ in $A$ and $n\geq 1$.

######  Remark .

By Section 4.2, every domain is isomorphic to a pointed submonoid of
$K^{\bullet}$ for some field $K$ of characteristic zero.

######  Proposition .

Let $B$ be a pointed monoid, $F$ a field and $k$ a domain that is a pointed
submonoid of both $B$ and $F^{\bullet}$. Let $R:=F[B]/I$, where $I$ is the
ideal generated by $\\{\mylambda_{k}1_{B}-1_{k}\mylambda_{B}\mid\mylambda\in
k\\}$. Then the map $\operatorname{Spec}R\rightarrow\operatorname{SCong}B$,
induced by $b\mapsto\overline{1b}$, which sends
$\mathfrak{p}\in\operatorname{Spec}R$ to the congruence kernel of the
composition $B\rightarrow R^{\bullet}\rightarrow(R/\mathfrak{p})^{\bullet}$,
has image $\operatorname{SCong}_{k}B$.

###### Proof.

Let $\myvarphi:B\rightarrow R^{\bullet}$ given by
$\myvarphi(b):=\overline{1b}$ and $\mathfrak{p}\in\operatorname{Spec}R$. As
$F$ is a field, the composition $F\rightarrow R\rightarrow R/\mathfrak{p}$ is
injective. As

${k}$${B}$${F^{\bullet}}$${R^{\bullet}.}$$\scriptstyle{\myvarphi}$

commutes, one has $\myvarphi^{*}(\mathfrak{p})\in\operatorname{SCong}_{k}B$.
Therefore $\operatorname{im}\myvarphi^{*}\subseteq\operatorname{SCong}_{k}B$.
Next we prove that
$\operatorname{SCong}_{k}B\subseteq\operatorname{im}\myvarphi^{*}$.

Let $\mathfrak{d}$ be a $k$-congruence on $B$. Define the congruence

$\mathfrak{a}:=\langle\\{\mylambda_{k}1_{B}\sim
1_{k}\mylambda_{B}\mid\mylambda\in k\\}\cup\\{1a\sim
1b\mid(a,b)\in\mathfrak{d}\rangle$

on $k[B]$. The map $\overline{b}\mapsto\overline{1b}$ defines an isomorphism
$B/\mathfrak{d}\xrightarrow{\sim}k[B]/\mathfrak{a}$. As $B/\mathfrak{d}$ is a
domain, one has $\mathfrak{a}\in\operatorname{SCong}k[B]$. By Section 4.2,
there exists $\mathfrak{q}\in\operatorname{Spec}F[B]$ such that
$\mathfrak{a}=i^{*}(\mathfrak{q})$, where $i:k[B]\rightarrow(F[B])^{\bullet}$
is the inclusion.

Let $\mypi:F[B]\rightarrow R$ be the natural projection. As
$\mylambda_{k}1_{B}\sim_{\mathfrak{a}}1_{k}\mylambda_{B}$, one has

$\mylambda_{k}1_{B}-1_{k}\mylambda_{B}\in\mathfrak{q}\quad\text{for
all}\quad\mylambda\in k.$

This implies $I\subseteq\mathfrak{q}$ and, consequently,
$\mypi(\mathfrak{q})\in\operatorname{Spec}R$. As the diagram

${B}$${R^{\bullet}}$${B/\mathfrak{d}}$${\big{(}R/\mypi(\mathfrak{q})\big{)}^{\bullet}}$${k[B]/\mathfrak{a}}$${(F[B]/\mathfrak{q})^{\bullet}}$$\scriptstyle{\myvarphi}$

$\scriptstyle\sim$

$\scriptstyle\sim$

commutes,
$B/\mathfrak{d}\rightarrow\big{(}R/\mypi(\mathfrak{q})\big{)}^{\bullet}$ is
injective and $\myvarphi^{*}\big{(}\mypi(\mathfrak{q})\big{)}=\mathfrak{d}$. ∎

######  Lemma .

Let $A$ be a pointed monoid, $B$ an $A$-algebra and $R$ a ring such that
$R^{\bullet}$ is an $A$-algebra. Let $I$ be the ideal of $R[B]$ generated by
$\\{\mylambda_{R}1_{B}-1_{R}\mylambda_{B}\mid\mylambda\in A\\}$. Then the
rings $R[B]/I$ and $B_{\mathbb{Z}}\otimes_{A_{\mathbb{Z}}}R$ are isomorphic.

###### Proof.

The map $b\mapsto(1b)\otimes 1$ from $B$ to
$B_{\mathbb{Z}}\otimes_{A_{\mathbb{Z}}}R$ induces a morphism of $R$-algebras
$f:R[B]\rightarrow B_{\mathbb{Z}}\otimes_{A_{\mathbb{Z}}}R$. As
$I\subseteq\ker f$, one has a morphism of $R$-algebras

$\begin{array}[]{cccc}\overline{f}:&R[B]/I&\longrightarrow&B_{\mathbb{Z}}\otimes_{A_{\mathbb{Z}}}R\\\
&\overline{1b}&\longmapsto&(1b)\otimes 1.\end{array}$

As the map $(\sum n_{i}b_{i},r)\mapsto\overline{\sum(n_{i}r)b_{i}}$ from
$B_{\mathbb{Z}}\times R$ to $R[B]/I$ is $A_{\mathbb{Z}}$-bilinear, it induces
a morphism of $A_{\mathbb{Z}}$-algebras

$\begin{array}[]{cccc}g:&B_{\mathbb{Z}}\otimes_{A_{\mathbb{Z}}}R&\longrightarrow&R[B]/I\\\
&(\sum n_{i}b_{i})\otimes
r&\longmapsto&\overline{\sum(n_{i}r)b_{i}}.\end{array}$

As $\overline{f}$ and $g$ are mutually inverse, the result follows. ∎

######  Theorem 4.2.

Let $k$ be a domain, $X$ a $k$-scheme and $F$ a field such that $k$ is a
pointed submonoid of $F^{\bullet}$. Then
$X_{\mathbb{Z}}\times_{k_{\mathbb{Z}}}\operatorname{Spec}F$ is a closed
subscheme of $X_{F}$ whose image by $\mygamma$ is $\operatorname{SCong}_{k}X$.

###### Proof.

For each $U=\operatorname{MSpec}B$ affine open of $X$, define

$I_{U}:=\langle\mylambda_{k}1_{B}-1_{k}\mylambda_{B}\mid\mylambda\in
k\rangle\triangleleft B_{F}.$

Note that $B_{F}/I_{U}\simeq F[B]/I_{U}^{\prime}$, where
$I_{U}^{\prime}:=\langle\mylambda_{k}1_{B}-1_{k}\mylambda_{B}\mid\mylambda\in
k\rangle\triangleleft F[B]$. Let $\mathcal{I}_{U}$ be the sheaf of ideals on
$U_{F}$ induced by $I_{U}$. If $U$ and $V$ are affine open subsets of $X$ and
$W\subseteq U\cap V$ is a principal open of both $U$ and $V$, then
$\mathcal{I}_{U}|_{W}=\mathcal{I}_{W}=\mathcal{I}_{V}|_{W}$. Thus there exists
a sheaf of ideals $\mathcal{I}$ on $X_{F}$ such that
$\mathcal{I}|_{U_{F}}=\mathcal{I}_{U}$ for every affine open $U\subseteq X$.

Let $Z$ be the closed subscheme of $X_{F}$ corresponding to $\mathcal{I}$. If
$U\subseteq X$ is an affine open, one has
$Z\times_{X_{F}}U_{F}=\operatorname{Spec}(B_{F}/I_{U})$ and, by Section 4.2,

$\mygamma(Z\times_{X_{F}}U_{F})=\operatorname{SCong}_{k}U\subseteq\operatorname{SCong}X.$

By Section 4.2, $Z=X_{\mathbb{Z}}\times_{k_{\mathbb{Z}}}\operatorname{Spec}F$
and, by Section 3, $\mygamma(Z)=\operatorname{SCong}_{k}X$. ∎

######  Corollary .

The map $\mygamma:X_{\mathbb{Z}}\rightarrow\operatorname{SCong}X$ is
surjective.

###### Proof.

As ${\mathbb{F}}_{1}\subseteq{\mathbb{Q}}$ and
$\operatorname{SCong}X=\operatorname{SCong}_{{\mathbb{F}}_{1}}X$, by Theorem
4.2, the map $X_{\mathbb{Q}}\rightarrow\operatorname{SCong}X$ is surjective.
As the diagram

${X_{\mathbb{Q}}}$${X_{\mathbb{Z}}}$${\operatorname{SCong}X}$

commutes, the result follows. ∎

###  4.3. The topology of $\operatorname{SCong}_{k}X$

We prove that if $X$ is a $k$-scheme locally of finite type, then
$\operatorname{SCong}_{k}X$ is a sober space, i.e., every irreducible closed
subset of $\operatorname{SCong}_{k}X$ has a unique generic point.

######  Theorem 4.3.

Let $k$ be a domain and $B$ a finitely generated $k$-algebra. Then
$\operatorname{SCong}_{k}B$ is sober.

###### Proof.

If $k\rightarrow B$ is not injective, then
$\operatorname{SCong}_{k}B=\emptyset$ has no irreducible closed subsets.

Suppose that $k\rightarrow B$ is injective. Let $F$ be a field with an
inclusion $k\hookrightarrow F^{\bullet}$ and define

$R:=F[B]/\langle 1_{F}\mylambda_{B}-\mylambda_{F}1_{B}\mid\mylambda\in
k\rangle.$

Let $\myvarphi:B\rightarrow R^{\bullet}$ be given by $b\mapsto\overline{1b}$.
By Section 4.2,
$\myvarphi^{*}:\operatorname{Spec}R\rightarrow\operatorname{SCong}B$ has image
$\operatorname{SCong}_{k}B$.

Let $Z\subseteq\operatorname{SCong}_{k}B$ be an irreducible closed subset.
Then $(\myvarphi^{*})^{-1}(Z)$ is a closed subset of $\operatorname{Spec}R$.
As $R$ is a Noetherian ring, there exists
$W_{1},\dotsc,W_{n}\subseteq\operatorname{Spec}R$ irreducible closed subsets
such that $(\myvarphi^{*})^{-1}(Z)=W_{1}\cup\cdots\cup W_{n}$. For each $i$,
let $\myeta_{i}\in\operatorname{Spec}R$ be the generic point of $W_{i}$. In
$\operatorname{SCong}_{k}B$, one has

$Z=\overline{\myvarphi^{*}\big{(}(\myvarphi^{*})^{-1}(Z)\big{)}}=\underset{i=1}{\overset{n}{\bigcup}}\;\overline{\myvarphi^{*}(W_{i})}.$

As $Z$ is irreducible, there exists $\ell$ such that
$Z=\overline{\myvarphi^{*}(W_{\ell})}$. Note that
$(\myvarphi^{*})^{-1}\big{(}\overline{\\{\myvarphi^{*}(\myeta_{\ell})\\}}\big{)}$
is a closed subset of $\operatorname{Spec}R$ that contains $\myeta_{\ell}$.
Thus
$W_{\ell}\subseteq(\myvarphi^{*})^{-1}\big{(}\overline{\\{\myvarphi^{*}(\myeta_{\ell})\\}}\big{)}$,
which implies

$\myvarphi^{*}(W_{\ell})\subseteq\myvarphi^{*}\bigg{(}(\myvarphi^{*})^{-1}\big{(}\overline{\\{\myvarphi^{*}(\myeta_{\ell})\\}}\big{)}\bigg{)}=\overline{\\{\myvarphi^{*}(\myeta_{\ell})\\}}$

and, consequently,
$Z=\overline{\myvarphi^{*}(W_{\ell})}\subseteq\overline{\\{\myvarphi^{*}(\myeta_{\ell})\\}}$.
Therefore $\myvarphi^{*}(\myeta_{\ell})$ is a generic point of $Z$.

As $\operatorname{Cong}B$ is a $T_{0}$ space and $\operatorname{SCong}_{k}B$
is a subspace of $\operatorname{Cong}B$, one has uniqueness. ∎

######  Definition .

A morphism of monoid schemes $\myvarphi:X\rightarrow Y$ is locally of finite
type if $Y$ has an affine open covering $\\{\operatorname{MSpec}B_{i}\\}_{i\in
I}$ such that, for each $i\in I$, the subscheme
$\myvarphi^{-1}(\operatorname{MSpec}B_{i})$ has an affine open covering
$\\{\operatorname{MSpec}A_{ij}\\}_{j\in J_{i}}$ with $A_{ij}$ finitely
generated $B_{i}$-algebra for all $j\in J_{i}$.

An $Y$-scheme $X$ is locally of finite type if the map $X\rightarrow Y$ is
locally of finite type.

######  Theorem 4.4.

Let $k$ be a domain and $X$ a $k$-scheme locally of finite type. Then
$\operatorname{SCong}_{k}X$ is sober.

###### Proof.

Let $\myvarphi:X\rightarrow\operatorname{MSpec}k$ be the morphism that makes
$X$ a $k$-scheme and $Z$ an irreducible closed subset of
$\operatorname{SCong}_{k}X$. Let $U=\operatorname{MSpec}B$ be an affine open
of $X$ such that $Z\cap\operatorname{SCong}_{k}U\neq\emptyset$. Then there
exists $\\{f_{i}\\}_{i\in I}\subseteq B$ such that
$\\{\operatorname{MSpec}\;B[f_{i}^{-1}]\\}_{i\in I}$ covers $U$ and, for each
$i$, there exists $\\{g_{ij}\\}_{j\in J_{i}}\subseteq k$, such that
$\\{\operatorname{MSpec}\;k[g_{ij}^{-1}]\\}_{i\in I}$ covers
$\myvarphi^{-1}(\operatorname{MSpec}B[f_{i}^{-1}])$ and with $B[f_{i}]$ a
finitely generated $k[g_{ij}]$-algebra for all $j\in J_{i}$. Note that
$B[f_{i}]$ is a finitely generated $k$-algebra for all $i\in I$.

Let $\ell\in I$ such that
$Z_{\ell}:=Z\cap\operatorname{SCong}_{k}B[f_{\ell}^{-1}]$ is not empty. As
$Z_{\ell}$ is an irreducible closed subset of
$\operatorname{SCong}_{k}B[f_{\ell}^{-1}]$, by Theorem 4.3, $Z_{\ell}$ has a
unique generic point $\myeta$. As $Z_{\ell}$ is a non-empty open subset of
$Z$, one has that $\myeta$ is the generic point of $Z$. ∎

##  5\. Strong congruence space of toric monoid schemes

###  5.1. Algebraically closed pointed groups

We introduce algebraically closed pointed groups and gather some results about
them (cf. [Dei08, pp. 519-520]).

######  Definition .

An extension of pointed groups is an injective morphism
$f:{\mathbb{G}}_{1}\hookrightarrow{\mathbb{G}}_{2}$. In this case we identify
${\mathbb{G}}_{1}$ with its image via $f$. The extension is trivial if it is
an isomorphism. An element $\mylambda\in{\mathbb{G}}_{2}$ is algebraic over
${\mathbb{G}}_{1}$ if there exists $n\geq 1$ such that
$\mylambda^{n}\in{\mathbb{G}}_{1}$. We say that ${\mathbb{G}}_{2}$ is
algebraic over ${\mathbb{G}}_{1}$ if every $x\in{\mathbb{G}}_{2}$ is algebraic
over ${\mathbb{G}}_{1}$.

######  Definition .

A pointed group ${\mathbb{G}}$ is algebraically closed if for every $\myalpha$
in ${\mathbb{G}}^{\times}$ and $n\geq 1$ one has
$\\#\\{x\in{\mathbb{G}}|\,x^{n}=\myalpha\\}$ equal to $n$.

######  Remark .

By Section 4.2, every algebraically closed pointed group is a domain.

######  Proposition .

Let ${\mathbb{G}}$ be a pointed group that is a domain. Then there exists an
algebraically closed pointed group $\overline{{\mathbb{G}}}$ and an algebraic
extension $\myiota:{\mathbb{G}}\hookrightarrow\overline{{\mathbb{G}}}$ such
that every extension of pointed groups
$f:{\mathbb{G}}\rightarrow{\mathbb{G}}^{\prime}$ with ${\mathbb{G}}^{\prime}$
algebraically closed factorizes as

${{\mathbb{G}}}$${{\mathbb{G}}^{\prime}}$${\overline{{\mathbb{G}}}}$$\scriptstyle{\myiota}$$\scriptstyle{f}$$\scriptstyle{j}$

with $j:\overline{{\mathbb{G}}}\rightarrow{\mathbb{G}}^{\prime}$ injective. We
call $\overline{{\mathbb{G}}}$ an algebraic closure of ${\mathbb{G}}$.

###### Proof.

By Section 4.2, there exists a field $K$ of characteristic zero and an
injective morphism $\myvarphi:{\mathbb{G}}\hookrightarrow K^{\bullet}$ (see
Section 4.2). Let $L$ be an algebraic closure of $K$ (in the usual field-
theoretic sense) and define

$\overline{{\mathbb{G}}}:=\\{\mylambda\in L\mid\exists n\geq 1\text{ such that
}\mylambda^{n}\in{\mathbb{G}}\\}.$

Note that $\overline{{\mathbb{G}}}$ is an algebraically closed pointed group
and $\operatorname{im}\myvarphi\subseteq\overline{{\mathbb{G}}}$. The map

$\begin{array}[]{cccc}\myiota:&{\mathbb{G}}&\longhookrightarrow&\overline{{\mathbb{G}}}\\\
&\myalpha&\longmapsto&\myvarphi(\myalpha)\end{array}$

makes $\overline{{\mathbb{G}}}$ algebraic over ${\mathbb{G}}$.

Let $f:{\mathbb{G}}\rightarrow{\mathbb{G}}^{\prime}$ be an extension of
pointed groups with ${\mathbb{G}}^{\prime}$ algebraically closed. We say that
a morphism $g:{\mathbb{G}}_{g}\rightarrow{\mathbb{G}}^{\prime}$ is an
extension of $f$ if it is injective, ${\mathbb{G}}_{g}$ is a pointed subgroup
of $\overline{{\mathbb{G}}}$ containing ${\mathbb{G}}$ and $f$ factorizes
through $g$. Let $\mathcal{E}$ be the set of all extensions of $f$, with the
partial order given by $g\leq h$ if
${\mathbb{G}}_{g}\subseteq{\mathbb{G}}_{h}$ and $h|_{{\mathbb{G}}_{g}}=g$. By
Zorn’s lemma, $\mathcal{E}$ has a maximal element $z$. Next we show that
${\mathbb{G}}_{z}=\overline{{\mathbb{G}}}$.

Suppose that there exists
$\mylambda\in\overline{{\mathbb{G}}}\backslash{\mathbb{G}}_{z}$. Let
$m:=\textup{min}\\{n\geq 1\mid\mylambda^{n}\in{\mathbb{G}}_{z}\\}$ and
$x:=\mylambda^{m}$. Let $\widetilde{{\mathbb{G}}}$ be the pointed submonoid of
$\overline{{\mathbb{G}}}$ generated by ${\mathbb{G}}_{z}$ and $\mylambda$. As
${\mathbb{G}}^{\prime}$ is algebraically closed and
$\widetilde{{\mathbb{G}}}\simeq{\mathbb{G}}_{z}[T]/\langle T^{m}\sim
x\rangle$, there exists $y\in{\mathbb{G}}^{\prime}\backslash\textup{im}(z)$
with $y^{m}=x$ and an extension
$r:\widetilde{{\mathbb{G}}}\rightarrow{\mathbb{G}}^{\prime}$ of $z$ sending
$\mylambda$ to $y$. Then $r\in\mathcal{E}$ and $r>z$, which is a
contradiction. Therefore ${\mathbb{G}}_{z}=\overline{{\mathbb{G}}}$. ∎

######  Corollary .

Let ${\mathbb{G}}$ be a pointed group that is also a domain. If
${\mathbb{G}}_{1}$ and ${\mathbb{G}}_{2}$ are two algebraic closures of
${\mathbb{G}}$, then ${\mathbb{G}}_{1}\simeq{\mathbb{G}}_{2}$.

###### Proof.

By Section 5.1, there exists an injective morphism of ${\mathbb{G}}$-algebras
$\myvarphi:{\mathbb{G}}_{1}\rightarrow{\mathbb{G}}_{2}$. Let
$\mylambda\in{\mathbb{G}}_{2}$. As $\mylambda$ is algebraic over
${\mathbb{G}}$, there exists $m:=\textup{min}\\{n\geq
1\mid\mylambda^{n}\in{\mathbb{G}}\\}$. Let $L_{i}:=\\{x\in{\mathbb{G}}_{i}\mid
x^{m}=\mylambda^{m}\\}$, for $i=1,2$. As $\myvarphi(L_{1})\subseteq L_{2}$ and
$\\#L_{1}=m=\\#L_{2}$, one has $\myvarphi(L_{1})=L_{2}$. As $\mylambda\in
L_{2}$, we conclude that $\mylambda\in\textup{im}\,\myvarphi$. ∎

######  Example .

By the proof of Section 5.1, an algebraic closure of ${\mathbb{F}}_{1}$ is

${\mathbb{F}}_{1^{\infty}}=\\{z\in{\mathbb{C}}\mid\exists n\geq 1\text{ such
that }z^{n}=1\\}\cup\\{0\\}.$

######  Proposition .

Let ${\mathbb{G}}$ be a pointed group that is also a domain. Then $\
{\mathbb{G}}$ is algebraically closed if, and only if, there is no non-trivial
algebraic extension ${\mathbb{G}}\hookrightarrow{\mathbb{G}}^{\prime}$ with
${\mathbb{G}}^{\prime}$ a pointed group that is also a domain.

###### Proof.

Assume that ${\mathbb{G}}$ is algebraically closed. Let
${\mathbb{G}}\hookrightarrow{\mathbb{G}}^{\prime}$ be an algebraic extension
of pointed groups with ${\mathbb{G}}^{\prime}$ a domain. If
$\mylambda\in({\mathbb{G}}^{\prime})^{\times}$, then there exists $n\geq 1$
such that $\myalpha:=\mylambda^{n}\in{\mathbb{G}}$. Note that
$I:=\\{x\in{\mathbb{G}}\mid x^{n}=\myalpha\\}$ has $n$ elements. As $I$ is
contained in $J:=\\{y\in{\mathbb{G}}^{\prime}\mid y^{n}=\myalpha\\}$, which
has at most $n$ elements, one has $I=J$. As $\mylambda\in J$, we conclude that
$\mylambda\in{\mathbb{G}}$. Thus ${\mathbb{G}}={\mathbb{G}}^{\prime}$, i.e.,
the extension ${\mathbb{G}}\hookrightarrow{\mathbb{G}}^{\prime}$ is trivial.

The opposite direction follows from Section 5.1. ∎

###  5.2. Main theorem

We prove that the strong congruence space of a toric monoid scheme over an
algebraically closed pointed group reflects its expected dimension.

We use the following notations from [Bra01] and [Eur15] for toric geometry.
For a torsion-free abelian group $L$ of finite rank, we use $L_{\mathbb{R}}$
to denote the vector space $L\otimes_{\mathbb{Z}}{\mathbb{R}}$. Throughout
this subsection, fix $M$ and $N$ dual torsion-free abelian groups of finite
rank $n$ with perfect pairing $\langle\;,\;\rangle:M\times
N\rightarrow{\mathbb{Z}}$, which extends to an ${\mathbb{R}}$-linear pairing
$\langle\;,\;\rangle:M_{\mathbb{R}}\times
N_{\mathbb{R}}\rightarrow{\mathbb{R}}$. A rational polyhedral cone in
$N_{\mathbb{R}}$ is a set of the form

$\mysigma=\text{Cone}(z_{1},\dotsc,z_{\ell}):=\\{\mylambda_{1}z_{1}+\cdots+\mylambda_{\ell}z_{\ell}\in
N_{\mathbb{R}}\mid\mylambda_{i}\in{\mathbb{R}}_{\geq 0},\text{ for all }i\\}$

for some $z_{1},\dotsc,z_{\ell}\in N$, where we identify $N$ with its image in
$N_{\mathbb{R}}$ via $x\mapsto x\otimes 1$. The dimension of $\mysigma$ is the
dimension of the linear subspace generated by $\mysigma$. The dual cone of
$\mysigma$ is the set

$\mysigma^{\vee}:=\\{x\in M_{\mathbb{R}}\mid\langle x,y\rangle\geq 0,\text{
for all }y\in\mysigma\\}.$

It is a fact that $\mysigma^{\vee}$ is a rational polyhedral cone in
$M_{\mathbb{R}}$ and $(\mysigma^{\vee})^{\vee}=\mysigma$. A face of $\mysigma$
is a set of the form

$\mybeta=\mysigma\cap H_{m},\quad\text{where}\quad H_{m}:=\\{u\in
N_{\mathbb{R}}\mid\langle m,u\rangle=0\\},$

for some $m\in\mysigma^{\vee}$. Any face of $\mysigma$ is a rational
polyhedral cone. We say that $\mysigma$ is strongly convex if
$\dim\mysigma^{\vee}=\dim M_{\mathbb{R}}=n$. Equivalently, $\mysigma$ is
strongly convex if it does not contain any non-zero linear subspace.

A fan is a finite collection $\Sigma$ of rational polyhedral cones in
$N_{\mathbb{R}}$ satisfying:

  1. (i)

Every $\mysigma\in\Sigma$ is strongly convex;

  2. (ii)

Every face of a cone in $\Sigma$ is also in $\Sigma$;

  3. (iii)

If $\mysigma_{1},\mysigma_{2}\in\Sigma$, then
$\mybeta:=\mysigma_{1}\cap\mysigma_{2}$ is a face of both $\mysigma_{1}$ and
$\mysigma_{2}$.

We define the dimension of $\Sigma$ as $\dim\Sigma:=\dim N_{\mathbb{R}}=n$.

Given a rational polyhedral cone $\mysigma\subseteq N_{\mathbb{R}}$, we define
the affine semigroups

$S_{\mysigma}:=\mysigma^{\vee}\cap M\quad\text{and}\quad
S_{\mysigma}^{\vee}:=\mysigma\cap N,$

which are finitely generated monoids. Note that
$S_{(\mysigma^{\vee})}=S_{\mysigma}^{\vee}$.

######  Remark .

The notations $\dim\Sigma$ and $S_{\mysigma}^{\vee}$ are introduced in the
present text.

Given a fan $\Sigma$, let $\mathcal{U}$ be the category whose objects are the
elements of $\Sigma$ and the morphisms are the inclusions of faces. Given two
cones $\mytau\subseteq\mysigma$ in $\Sigma$, the induced map

$\operatorname{MSpec}{\mathbb{F}}_{1}[S_{\mytau}]\rightarrow\operatorname{MSpec}{\mathbb{F}}_{1}[S_{\mysigma}]$

is an open embedding. Define $X(\Sigma)$ as the monoid scheme

$\underset{\mysigma\in\mathcal{U}}{\textup{colim}}\operatorname{MSpec}{\mathbb{F}}_{1}[S_{\mysigma}].$

Let $A$ be a pointed monoid. The monoid scheme

$X(\Sigma)\times_{{\mathbb{F}}_{1}}\operatorname{MSpec}A$

has an open covering given by
$\\{\operatorname{MSpec}A[S_{\mysigma}]\\}_{\mysigma\in\Sigma}$ (cf. [Dei05,
Prop 3.1]).

Let ${\mathbb{F}}$ be a pointed group that is also a domain, $\mysigma$ a
rational polyhedral cone and $\mytau$ a face of $\mysigma^{\vee}$. Note that
the set
$\mathfrak{p}_{\mytau}:=({\mathbb{F}}[S_{\mysigma}]\backslash{\mathbb{F}}[S_{\mytau}^{\vee}])\cup\\{0\\}$
is an ${\mathbb{F}}$-prime of ${\mathbb{F}}[S_{\mysigma}]$. The next result
shows that every prime in
$\operatorname{MSpec}_{\mathbb{F}}{\mathbb{F}}[S_{\mysigma}]$ has this form.

######  Lemma .

Let ${\mathbb{F}}$ be a pointed group that is also a domain and $\mysigma$ a
rational polyhedral cone. Then the map $\mytau\mapsto\mathfrak{p}_{\mytau}$ is
a bijection between the set of faces of $\mysigma^{\vee}$ and
$\operatorname{MSpec}_{\mathbb{F}}{\mathbb{F}}[S_{\mysigma}]$ that reverses
inclusions.

###### Proof.

It is enough to show that the map $\\{\textup{faces of
}\mysigma^{\vee}\\}\rightarrow\operatorname{MSpec}_{\mathbb{F}}{\mathbb{F}}[S_{\mysigma}]$
above is surjective.

Let
$\mathfrak{p}\in\operatorname{MSpec}_{\mathbb{F}}{\mathbb{F}}[S_{\mysigma}]$.
We use $S_{\mysigma}$ to denote the semigroup
${\mathbb{F}}_{1}[S_{\mysigma}]\backslash\\{0\\}\subseteq{\mathbb{F}}[S_{\mysigma}]$
and define $\mathfrak{q}:=\mathfrak{p}\cap S_{\mysigma}$. As
$S_{\mysigma}\backslash\mathfrak{q}$ is a multiplicative subset, by [Ogu18,
Prop. 2.2.1], there exists a morphism of semigroups
$h:S_{\mysigma}\rightarrow{\mathbb{N}}$ such that
$h^{-1}(0)=S_{\mysigma}\backslash\mathfrak{q}$. We have a unique extension of
$h$ to a morphism of groups $\tilde{h}:M\rightarrow{\mathbb{Z}}$ and to a map
of ${\mathbb{R}}$-algebras $f:M_{\mathbb{R}}\rightarrow{\mathbb{R}}$. One
knows that there exists a unique $u\in N_{\mathbb{R}}$ such that $f(m)=\langle
m,u\rangle$ for all $m\in M_{\mathbb{R}}$. Note that $u\in\mysigma$ and define
$\mytau:=\\{m\in\mysigma^{\vee}\mid\langle m,u\rangle=0\\}$, which is a face
of $\mysigma^{\vee}$. Next we show that $\mathfrak{p}=\mathfrak{p}_{\mytau}$.

If $\mylambda.\mychi^{m}\in\mathfrak{p}\backslash\\{0\\}$, one has
$m\in\mathfrak{q}$, which implies $h(m)\neq 0$ and, consequently,
$\mylambda.\mychi^{m}$ not in ${\mathbb{F}}[S_{\mytau}^{\vee}]$. Thus
$\mathfrak{p}\subseteq\mathfrak{p}_{\mytau}$. If
$\myalpha.\mychi^{s}\in\mathfrak{p}_{\mytau}\backslash\\{0\\}$, then $s\notin
S_{\mytau}^{\vee}=S_{\mysigma}\cap\mytau$, which implies $h(s)\neq 0$ and,
consequently, $s\in\mathfrak{q}$. Therefore
$\mathfrak{p}_{\mytau}\subseteq\mathfrak{p}$. ∎

######  Lemma .

Let $\mysigma$ be a rational polyhedral cone and $\mytau\subseteq\myalpha$
faces of $\mysigma^{\vee}$ such that $\dim\,\mytau=\dim\,\myalpha$. Then
$\mytau=\myalpha$.

###### Proof.

Let $t\in(\mysigma^{\vee})^{\vee}$ such that $\mytau=\mysigma^{\vee}\cap
H_{t}$. As $\dim\,\mytau=\dim\,\myalpha$, one has
$\textup{span}\,\mytau=\textup{span}\,\myalpha$. As $H_{t}$ is a linear
subspace, one has $\myalpha\subseteq H_{t}$ if, and only if,
$\textup{span}\,\myalpha\subseteq H_{t}$. Note that
$\mytau\subseteq\myalpha\subseteq\textup{span}\,\myalpha\cap\mysigma^{\vee}=\textup{span}\,\mytau\cap\mysigma^{\vee}\subseteq
H_{t}\cap\mysigma^{\vee}=\mytau$. Thus $\mytau=\myalpha$. ∎

######  Lemma .

Let $\mysigma$ be a rational polyhedral cone and $\mytau\subseteq\myalpha$
faces of $\mysigma^{\vee}$. Let $\ell$ be an integer such that
$\dim\,\mytau\leq\ell\leq\dim\,\myalpha$. Then there exists a face $\mydelta$
of $\mysigma^{\vee}$ of dimension $\ell$ with
$\mytau\subseteq\mydelta\subseteq\myalpha$.

###### Proof.

If $\mybeta$ is a face of $\myalpha$ and $\mydelta$ is a face of $\mybeta$,
then $\mydelta$ is a face of $\myalpha$. Thus it is enough to prove the result
for $\ell=\dim\myalpha-1$, which is done in [Ful93, p. 10]. ∎

######  Lemma .

Let $G$ be a torsion-free abelian group of finite rank and $H_{1},H_{2}$
subgroups such that $H_{1}\cap H_{2}=\\{0\\}$ and $G/H_{i}$ is torsion-free
for $i=1,2$. Then $G/(H_{1}\oplus H_{2})$ is torsion free.

###### Proof.

Let $\mypi_{i}:G\rightarrow G/H_{i}$ be the natural projection for $i=1,2$. As
$G/H_{1}$ is torsion-free, it is also free, thus there exists a morphism
$\mytheta:G/H_{1}\rightarrow G$ such that
$\mypi_{1}\circ\mytheta=\textup{id}_{G/H_{1}}$. For each $g\in G$ there exists
$h_{g}\in H_{1}$ satisfying $\mytheta(g+H_{1})=g+h_{g}$, thus the kernel of
$\mypi_{2}\circ\mytheta$ is $(H_{1}\oplus H_{2})/H_{1}$.

As $G/H_{2}$ is torsion-free and one has the injection
$(G/H_{1})/\ker(\mypi_{2}\circ\mytheta)\hookrightarrow G/H_{2}$ and the
isomorphism

$(G/H_{1})/\big{(}(H_{1}\oplus H_{2})/H_{1}\big{)}\simeq G/(H_{1}\oplus
H_{2}),$

the result follows. ∎

######  Theorem 5.1.

Let ${\mathbb{F}}$ be an algebraically closed pointed group and
$\mysigma\subseteq N_{\mathbb{R}}$ a strongly convex rational polyhedral cone.
Then $\operatorname{SCong}_{\mathbb{F}}{\mathbb{F}}[S_{\mysigma}]$ is a
catenary space of Krull dimension $n$ (the rank of $N$).

###### Proof.

Firstly, we define two functions $\mathfrak{N}$ and $\mathfrak{T}$ from
$\operatorname{SCong}_{\mathbb{F}}{\mathbb{F}}[S_{\mysigma}]$ to
${\mathbb{N}}$ whose sum is analogous to the height of a prime ideal in a
ring. Let $\mathfrak{c}$ be an ${\mathbb{F}}$-congruence on
${\mathbb{F}}[S_{\mysigma}]$. By Section 5.2, there exists a unique face
$\mytau_{\mathfrak{c}}$ of $\mysigma^{\vee}$ such that
$\mathfrak{p}_{\mytau_{\mathfrak{c}}}=I_{\mathfrak{c}}$. Define
$\mathfrak{N}(\mathfrak{c})$ as $n-\dim\,\mytau_{\mathfrak{c}}$.

Let $\mathfrak{d}_{\mathfrak{c}}$ be the strong ${\mathbb{F}}$-congruence on
${\mathbb{F}}[S_{\mytau_{\mathfrak{c}}}^{\vee}]\simeq{\mathbb{F}}[S_{\mysigma}]/I_{\mathfrak{c}}$
induced by $\mathfrak{c}$. Let $\Lambda_{\mathfrak{c}}$ be the linear subspace
of $M_{\mathbb{R}}$ generated by $\mytau_{\mathfrak{c}}$. As
$I_{\mathfrak{d}_{\mathfrak{c}}}=\\{0\\}$ and
${\mathbb{F}}[S_{\Lambda_{\mathfrak{c}}}^{\vee}]=\textup{Frac}({\mathbb{F}}[S_{\mytau_{\mathfrak{c}}}^{\vee}])$,
one has $\mathfrak{d}_{\mathfrak{c}}=f^{*}(\mathfrak{a}_{\mathfrak{c}})$,
where
$f:{\mathbb{F}}[S_{\mytau_{\mathfrak{c}}}^{\vee}]\rightarrow{\mathbb{F}}[S_{\Lambda_{\mathfrak{c}}}^{\vee}]$
is the inclusion and
$\mathfrak{a}_{\mathfrak{c}}:=f_{*}(\mathfrak{d}_{\mathfrak{c}})\in\operatorname{SCong}{\mathbb{F}}[S_{\Lambda_{\mathfrak{c}}}^{\vee}]$.
Let $\equiv$ be the equivalence relation on
$S_{\Lambda_{\mathfrak{c}}}^{\vee}$ given by $a\equiv b$ if
$\mychi^{a}\sim_{\mathfrak{a}_{\mathfrak{c}}}\mylambda\mychi^{b}$ for some
$\mylambda\in{\mathbb{F}}^{\times}$ and define the subgroup
$H_{\mathfrak{c}}:=\\{a\in S_{\Lambda_{\mathfrak{c}}}^{\vee}\mid a\equiv 0\\}$
of $S_{\Lambda_{\mathfrak{c}}}^{\vee}$. Note that

$\mathfrak{a}_{\mathfrak{c}}=\langle\mychi^{a}\sim\mylambda\mid a\in
H_{\mathfrak{c}}\text{ and
}\mychi^{a}\sim_{\mathfrak{a}_{\mathfrak{c}}}\mylambda\rangle.$

Define $\mathfrak{T}(\mathfrak{c})$ as $\textup{rk}(H_{\mathfrak{c}})$ and
note that
$\mathfrak{t}(\mathfrak{c})\leq\textup{rk}(S_{\Lambda_{\mathfrak{c}}}^{\vee})=\textup{dim}\,\mytau_{\mathfrak{c}}$.

Next we show that $S_{\Lambda_{\mathfrak{c}}}^{\vee}/H_{\mathfrak{c}}$ is
torsion-free. Let $y\in S_{\Lambda_{\mathfrak{c}}}^{\vee}$ and $r>0$ such that
$ry\in H_{\mathfrak{c}}$. Then there exists
$\mylambda\in{\mathbb{F}}^{\times}$ such that
$(\mychi^{y})^{r}\sim_{\mathfrak{a}_{\mathfrak{c}}}\mylambda$. As
${\mathbb{F}}$ is algebraically closed and $\mathfrak{a}_{\mathfrak{c}}$ is
strong, there exists $\myzeta\in{\mathbb{F}}$ such that
$\myzeta^{r}=\mylambda$ and
$\mychi^{y}\sim_{\mathfrak{a}_{\mathfrak{c}}}\myzeta$, thus $y\in
H_{\mathfrak{c}}$.

Let $\mathfrak{c}_{1}$ and $\mathfrak{c}_{2}$ be ${\mathbb{F}}$-congruences on
${\mathbb{F}}[S_{\mysigma}]$ such that
$\mathfrak{c}_{1}\subseteq\mathfrak{c}_{2}$. Let $\mathcal{C}$ be a maximal
chain in $\operatorname{SCong}_{\mathbb{F}}{\mathbb{F}}[S_{\mysigma}]$ with
minimum $\mathfrak{c}_{1}$ and maximum $\mathfrak{c}_{2}$. We claim that
$\mathcal{C}$ is finite. Note that
$\mathcal{C}_{I}:=\\{I_{\mathfrak{c}}\mid\mathfrak{c}\in\mathcal{C}\\}$ is a
chain in $\operatorname{MSpec}_{\mathbb{F}}{\mathbb{F}}[S_{\mysigma}]$, thus
$\\#\mathcal{C}_{I}\leq n$, and given $J\in\mathcal{C}_{I}$, one has that
$\mathcal{H}_{J}:=\\{H_{\mathfrak{c}}\mid\mathfrak{c}\in\mathcal{C}\text{ with
}I_{\mathfrak{c}}=J\\}$ is a chain whose length is bounded by
$\textup{dim}\,\mytau$, where $\mytau$ is the unique face of $\mysigma^{\vee}$
satisfying $\mathfrak{p}_{\mytau}=J$. Next we show that the length of
$\mathcal{C}$ is
$\big{(}\mathfrak{N}(\mathfrak{c}_{2})+\mathfrak{T}(\mathfrak{c}_{2})\big{)}-\big{(}\mathfrak{N}(\mathfrak{c}_{1})+\mathfrak{T}(\mathfrak{c}_{1})\big{)}$.

It is enough to show that if $\mathfrak{r}\subsetneq\mathfrak{r}^{\prime}$ are
two consecutive congruences in $\mathcal{C}$, then

$\big{(}\mathfrak{T}(\mathfrak{r^{\prime}})-\mathfrak{T}(\mathfrak{r})\big{)}+\big{(}\mathfrak{N}(\mathfrak{r^{\prime}})-\mathfrak{N}(\mathfrak{r})\big{)}=1.$

Let
$\mypi:{\mathbb{F}}[S_{\mysigma}]\rightarrow{\mathbb{F}}[S_{\mytau_{\mathfrak{r}}}^{\vee}]\simeq{\mathbb{F}}[S_{\mysigma}]/I_{\mathfrak{r}}$
be the projection. If
$\mathfrak{c}\in\operatorname{SCong}_{\mathbb{F}}{\mathbb{F}}[S_{\mysigma}]$
satisfies $I_{\mathfrak{r}}\subseteq I_{\mathfrak{c}}$, then
$\mypi_{*}(\mathfrak{c})\in\operatorname{SCong}_{\mathbb{F}}{\mathbb{F}}[S_{\mytau_{\mathfrak{r}}}^{\vee}]$,
$\mathfrak{N}(\mypi_{*}(\mathfrak{c}))=\mathfrak{N}(\mathfrak{c})-\mathfrak{N}(\mathfrak{r})$
and $\mathfrak{T}(\mypi_{*}(\mathfrak{c}))=\mathfrak{T}(\mathfrak{c})$, thus
we can assume $I_{\mathfrak{r}}=\\{0\\}$. Therefore
$\Lambda_{\mathfrak{r}}=M_{\mathbb{R}}$ and we have two cases to analyze:

Case 1: $I_{\mathfrak{r^{\prime}}}=\\{0\\}$. Then $H_{\mathfrak{r}}\subseteq
H_{\mathfrak{r^{\prime}}}$, which implies
$\mathfrak{T}(\mathfrak{r})\leq\mathfrak{T}(\mathfrak{r^{\prime}})$. One has
the injection $H_{\mathfrak{r^{\prime}}}/H_{\mathfrak{r}}\hookrightarrow
S_{\Lambda_{\mathfrak{r}}}^{\vee}/H_{\mathfrak{r}}$, which shows that
$H_{\mathfrak{r^{\prime}}}/H_{\mathfrak{r}}$ is torsion-free. If
$\mathfrak{T}(\mathfrak{r})=\mathfrak{T}(\mathfrak{r^{\prime}})$, then
$H_{\mathfrak{r}}=H_{\mathfrak{r^{\prime}}}$ and we conclude that
$\mathfrak{r}=\mathfrak{r^{\prime}}$. Thus
$\mathfrak{T}(\mathfrak{r})<\mathfrak{T}(\mathfrak{r^{\prime}})$.

If $\mathfrak{T}(\mathfrak{r})\leq\mathfrak{T}(\mathfrak{r^{\prime}})-2$, let
$s:=\mathfrak{T}(\mathfrak{r^{\prime}})$, $\ell:=\mathfrak{T}(\mathfrak{r})$
and $\\{b_{1},\dotsc,b_{s}\\}$ a ${\mathbb{Z}}$-basis of
$H_{\mathfrak{r^{\prime}}}$ such that $\\{b_{1},\dotsc,b_{\ell}\\}$ is a
${\mathbb{Z}}$-basis of $H_{\mathfrak{r}}$. Then, for each $1\leq i\leq s$,
there exists a unique $\mylambda_{i}\in{\mathbb{F}}^{\times}$ such that
$b_{i}\sim_{\mathfrak{a}_{\mathfrak{r^{\prime}}}}\mylambda_{i}$. As
$\mathfrak{r}\subseteq\mathfrak{r^{\prime}}$, one has
$b_{j}\sim_{\mathfrak{a}_{\mathfrak{r}}}\mylambda_{j}$ for $1\leq j\leq\ell$.
Let
$\mathfrak{a}\in\operatorname{SCong}_{\mathbb{F}}{\mathbb{F}}[S_{\Lambda_{\mathfrak{r}}}^{\vee}]$
be the congruence $\langle b_{i}\sim\mylambda_{i}\mid 1\leq
i\leq\ell+1\rangle$ and define $\mathfrak{c}:=g^{*}(\mathfrak{a})$, where
$g:{\mathbb{F}}[S_{\mysigma}]\rightarrow{\mathbb{F}}[S_{\Lambda_{\mathfrak{r}}}^{\vee}]$
is the inclusion. Thus
$\mathfrak{r}\subsetneq\mathfrak{c}\subsetneq\mathfrak{r^{\prime}}$, which is
a contradiction. Therefore,
$\mathfrak{T}(\mathfrak{r})=\mathfrak{T}(\mathfrak{r^{\prime}})-1$.

Case 2: $I_{\mathfrak{r^{\prime}}}\neq\\{0\\}$. Suppose that
$H_{\mathfrak{r^{\prime}}}\not\subset H_{\mathfrak{r}}$, which is equivalent
to $H_{\mathfrak{r}}\cap H_{\mathfrak{r^{\prime}}}\subsetneq
H_{\mathfrak{r^{\prime}}}$. One has that
$H_{\mathfrak{r^{\prime}}}/H_{\mathfrak{r}}\cap H_{\mathfrak{r^{\prime}}}$ is
torsion free, because $S_{\Lambda_{\mathfrak{r^{\prime}}}}^{\vee}\subsetneq
S_{\Lambda_{\mathfrak{r}}}^{\vee}$ and
$H_{\mathfrak{r^{\prime}}}/H_{\mathfrak{r}}\cap
H_{\mathfrak{r^{\prime}}}\hookrightarrow
S_{\Lambda_{\mathfrak{r}}}^{\vee}/H_{\mathfrak{r}}$. Let
$\ell:=\textup{rk}(H_{\mathfrak{r}}\cap H_{\mathfrak{r^{\prime}}})$ and
$s:=\mathfrak{T}(\mathfrak{r^{\prime}})$. Then there exists a
${\mathbb{Z}}$-basis $\\{b_{1},\dotsc,b_{s}\\}$ of $H_{\mathfrak{r^{\prime}}}$
such that $\\{b_{1},\dotsc,b_{\ell}\\}$ is a ${\mathbb{Z}}$-basis of
$H_{\mathfrak{r}}\cap H_{\mathfrak{r^{\prime}}}$. For each $1\leq i\leq s$,
let $\mylambda_{i}\in{\mathbb{F}}^{\times}$ such that
$b_{i}\sim_{\mathfrak{a}_{\mathfrak{r^{\prime}}}}\mylambda_{i}$. Let
$\mathfrak{b}\in\operatorname{SCong}_{\mathbb{F}}{\mathbb{F}}[S_{\Lambda_{\mathfrak{r^{\prime}}}}^{\vee}]$
be the congruence $\langle b_{i}\sim\mylambda_{i}\mid 1\leq i\leq\ell\rangle$
and define $\mathfrak{c}:=p^{*}(\mathfrak{b})$, where $p$ is the composition

${\mathbb{F}}[S_{\mysigma}]\relbar\joinrel\twoheadrightarrow{\mathbb{F}}[S_{\mysigma}]/I_{\mathfrak{r^{\prime}}}\simeq{\mathbb{F}}[S_{\mytau_{\mathfrak{r^{\prime}}}}^{\vee}]\longhookrightarrow{\mathbb{F}}[S_{\Lambda_{\mathfrak{r^{\prime}}}}^{\vee}].$

Note that $I_{\mathfrak{c}}=I_{\mathfrak{r^{\prime}}}$ and
$H_{\mathfrak{c}}=H_{\mathfrak{r}}\cap H_{\mathfrak{r^{\prime}}}$, thus
$\mathfrak{r}\subsetneq\mathfrak{c}\subsetneq\mathfrak{r^{\prime}}$, which is
a contradiction. Therefore $H_{\mathfrak{r^{\prime}}}\subseteq
H_{\mathfrak{r}}$.

Let $s:=\mathfrak{T}(\mathfrak{r^{\prime}})$, $\\{x_{1},\dotsc,x_{n}\\}$ a
${\mathbb{Z}}$-basis of $S_{\Lambda_{\mathfrak{r}}}^{\vee}$ such that
$\\{x_{1},\dotsc,x_{s}\\}$ is a ${\mathbb{Z}}$-basis of
$G:=H_{\mathfrak{r^{\prime}}}$ and $\mylambda_{i}\in{\mathbb{F}}^{\times}$
such that $x_{i}\sim_{\mathfrak{a}_{\mathfrak{r^{\prime}}}}\mylambda_{i}$ for
$1\leq i\leq s$. The map of groups

$\begin{array}[]{ccl}S_{\Lambda_{\mathfrak{r}}}^{\vee}&\longrightarrow&({\mathbb{F}}[S_{\Lambda_{\mathfrak{r}}}^{\vee}/G])^{\times}\\\
&&\\\
x_{i}&\longmapsto&\smash{\left\\{\begin{array}[]{ll}\mylambda_{i}&\text{if
}1\leq i\leq s\\\ {}\mychi^{\overline{x_{i}}}&\text{if }s+1\leq i\leq
n\end{array}\right.}\end{array}$

induces a surjective morphism of ${\mathbb{F}}$-algebras
$\myvarphi:{\mathbb{F}}[S_{\Lambda_{\mathfrak{r}}}^{\vee}]\twoheadrightarrow{\mathbb{F}}[S_{\Lambda_{\mathfrak{r}}}^{\vee}/G]$
with congruence kernel
$\myiota_{*}(\mathfrak{a}_{\mathfrak{r^{\prime}}})\subseteq\mathfrak{a}_{\mathfrak{r}}$,
where
$\myiota:{\mathbb{F}}[S_{\Lambda_{\mathfrak{r^{\prime}}}}^{\vee}]\rightarrow{\mathbb{F}}[S_{\Lambda_{\mathfrak{r}}}^{\vee}]$
is the inclusion. As
$S_{\Lambda_{\mathfrak{r}}}^{\vee}/S_{\Lambda_{\mathfrak{r^{\prime}}}}^{\vee}$
and $S_{\Lambda_{\mathfrak{r^{\prime}}}}^{\vee}/G$ are free, one has that
$S_{\Lambda_{\mathfrak{r}}}^{\vee}/G$ is free, which implies that
$\overline{S_{\mysigma}}:=\big{\\{}\overline{x}\in
S_{\Lambda_{\mathfrak{r}}}^{\vee}/G\;\big{|}\;x\in S_{\mysigma}\big{\\}}$ is
an affine semigroup. Note that
$\myvarphi({\mathbb{F}}[S_{\mysigma}])={\mathbb{F}}[\overline{S_{\mysigma}}]$
and define

$\begin{array}[]{cccc}\mypsi:&{\mathbb{F}}[S_{\mysigma}]&\longrightarrow&{\mathbb{F}}[\overline{S_{\mysigma}}]\\\
&a&\longmapsto&\myvarphi(a),\end{array}$

$\mathfrak{s}:=\mypsi_{*}(\mathfrak{r})$ and
$\mathfrak{s^{\prime}}:=\mypsi_{*}(\mathfrak{r^{\prime}})$. Let
$\widetilde{M}$ be the group $S_{\Lambda_{\mathfrak{r}}}^{\vee}/G$ and
$\widetilde{N}:=\textup{Hom}(\widetilde{M},{\mathbb{Z}})$ its dual. Let
$y_{1},\dotsc,y_{w}\in M$ be generators of $S_{\mysigma}$ as a monoid and
$\myrho:=\textup{Cone}(\overline{y_{1}},\dotsc,\overline{y_{w}})^{\vee}\subseteq\widetilde{N}_{\mathbb{R}}$.
Note that $\overline{S_{\mysigma}}=S_{\myrho}$ and
$\textup{congker}(\mypsi)\subseteq\mathfrak{r}$, which implies
$\mathfrak{s},\mathfrak{s^{\prime}}\in\operatorname{SCong}_{\mathbb{F}}{\mathbb{F}}[S_{\myrho}]$.
As $H_{\mathfrak{s}}=H_{\mathfrak{r}}/G$ and
$H_{\mathfrak{s^{\prime}}}=H_{\mathfrak{r^{\prime}}}/G=\\{0\\}$, it is enough
to show that

$\mathfrak{N}(\mathfrak{s^{\prime}})-\mathfrak{T}(\mathfrak{s})=1.$

Define $m:=\textup{rk}(\widetilde{M})$, $d:=\mathfrak{T}(\mathfrak{s})$ and
$e:=m-\mathfrak{N}(\mathfrak{s^{\prime}})$. Let $\\{u_{1},\dotsc,u_{m}\\}$ be
a ${\mathbb{Z}}$-basis of $\widetilde{M}$ such that $\\{u_{1},\dotsc,u_{d}\\}$
is a ${\mathbb{Z}}$-basis of $H_{\mathfrak{s}}$, and
$\mybeta_{i}\in{\mathbb{F}}^{\times}$ with
$\mychi^{u_{i}}\sim_{\mathfrak{a}_{\mathfrak{s}}}\mybeta_{i}$ for $1\leq i\leq
d$. Analogous to what is done above, there exists a surjective morphism of
${\mathbb{F}}$-algebras

$\begin{array}[]{cccl}\widetilde{\myvarphi}:&{\mathbb{F}}[\widetilde{M}]&\longrightarrow&{\mathbb{F}}[\widetilde{M}/H_{\mathfrak{s}}]\\\
&&\\\
&\mychi^{u_{i}}&\longmapsto&\smash{\left\\{\begin{array}[]{ll}\mybeta_{i}&\text{if
}1\leq i\leq d\\\ {}\mychi^{\overline{u_{i}}}&\text{if }d+1\leq i\leq
m\end{array}\right.}\end{array}$

noindent such that
$\operatorname{congker}(\widetilde{\myvarphi})=\mathfrak{a}_{\mathfrak{s}}$
and
$\widetilde{\myvarphi}({\mathbb{F}}[S_{\myrho}])={\mathbb{F}}[\overline{S_{\myrho}}]$,
where $\overline{S_{\myrho}}$ is the affine semigroup
$\\{\overline{y}\in\widetilde{M}/H_{\mathfrak{s}}\mid y\in S_{\myrho}\\}$. As
the composition

${\mathbb{F}}[S_{\mytau_{\mathfrak{s^{\prime}}}}^{\vee}]\longhookrightarrow{\mathbb{F}}[S_{\myrho}]\relbar\joinrel\twoheadrightarrow{\mathbb{F}}[S_{\myrho}]/\mathfrak{s}\relbar\joinrel\twoheadrightarrow{\mathbb{F}}[S_{\myrho}]/\mathfrak{s^{\prime}}$

is an isomorphism, one has the commutative diagram

${{\mathbb{F}}[S_{\mytau_{\mathfrak{s^{\prime}}}}^{\vee}]}$${{\mathbb{F}}[S_{\myrho}]/\mathfrak{s}}$${{\mathbb{F}}[S_{\Lambda_{\mathfrak{s^{\prime}}}}^{\vee}]}$${{\mathbb{F}}[\widetilde{M}]/\mathfrak{a}_{\mathfrak{s}}\simeq{\mathbb{F}}[\widetilde{M}/H_{\mathfrak{s}}],}$

which shows that $S_{\Lambda_{\mathfrak{s^{\prime}}}}^{\vee}\cap
H_{\mathfrak{s}}=\\{0\\}$. By Section 5.2, there exists a ${\mathbb{Z}}$-basis
$\\{c_{1},\dotsc,c_{m}\\}$ of $\widetilde{M}$ such that $c_{i}=u_{i}$ for
$1\leq i\leq d$ and $\\{c_{d+1},\dotsc,c_{d+e}\\}$ is a ${\mathbb{Z}}$-basis
of $S_{\Lambda_{\mathfrak{s^{\prime}}}}^{\vee}$. Therefore $d+e\leq m$, i.e.,
$\mathfrak{T}(\mathfrak{s})\leq\mathfrak{N}(\mathfrak{s^{\prime}})$. Next we
show that $\mathfrak{T}(\mathfrak{s})=\mathfrak{N}(\mathfrak{s^{\prime}})-1$.

Suppose that $\mathfrak{T}(\mathfrak{s})=\mathfrak{N}(\mathfrak{s^{\prime}})$.
Then the map
${\mathbb{F}}[S_{\Lambda_{\mathfrak{s^{\prime}}}}^{\vee}]\rightarrow{\mathbb{F}}[\widetilde{M}/H_{\mathfrak{s}}]$
is surjective. Let $Q$ be the submonoid of $\widetilde{M}$ generated by
$S_{\myrho}$ and $S_{\Lambda_{\mathfrak{s^{\prime}}}}^{\vee}$. As
${\mathbb{F}}[Q]$ is a localization of ${\mathbb{F}}[S_{\myrho}]$ by elements
not in $I_{\mathfrak{s^{\prime}}}$ or $I_{\mathfrak{s}}=\\{0\\}$, one has that
$q_{*}(\mathfrak{s})$ and $q_{*}(\mathfrak{s^{\prime}})$ are
${\mathbb{F}}$-congruences on ${\mathbb{F}}[Q]$, where
$q:{\mathbb{F}}[S_{\myrho}]\rightarrow{\mathbb{F}}[Q]$ is the inclusion map.
Note that
$\textup{congker}(\widetilde{\myvarphi}|_{{\mathbb{F}}[Q]})=q_{*}(\mathfrak{s})$.
As the diagram

${{\mathbb{F}}[Q]}$${{\mathbb{F}}[Q]/q_{*}(\mathfrak{s})}$${{\mathbb{F}}[\widetilde{M}/H_{\mathfrak{s}}]}$${{\mathbb{F}}[Q]/q_{*}(\mathfrak{s^{\prime}})\simeq{\mathbb{F}}[S_{\Lambda_{\mathfrak{s^{\prime}}}}^{\vee}]}$${{\mathbb{F}}[Q]}$$\scriptstyle{\widetilde{\myvarphi}}$$\scriptstyle{\widetilde{\myvarphi}}$

commutes, one has
$q_{*}(\mathfrak{s^{\prime}})=\textup{congker}(\widetilde{\myvarphi}|_{{\mathbb{F}}[Q]})=q_{*}(\mathfrak{s})$,
which is a contradiction.

Suppose that
$\mathfrak{T}(\mathfrak{s})\leq\mathfrak{N}(\mathfrak{s^{\prime}})-2$. Let
$\widetilde{\mypsi}:=\widetilde{\myvarphi}|_{{\mathbb{F}}[S_{\myrho}]}$ and
$\mathfrak{u}:=\langle\overline{c_{m}}\sim 1\rangle$ on
${\mathbb{F}}[\widetilde{M}/H_{\mathfrak{s}}]$. As $\mathfrak{u}$ is an
${\mathbb{F}}$-congruence, one has that
$\mathfrak{v}:=\widetilde{\mypsi}^{*}(\mathfrak{u})$ is an
${\mathbb{F}}$-congruence on ${\mathbb{F}}[S_{\myrho}]$ satisfying
$\mathfrak{s}\subsetneq\mathfrak{v}\subsetneq\mathfrak{s^{\prime}}$, which is
a contradiction.

The topological dimension of
$\operatorname{SCong}_{\mathbb{F}}{\mathbb{F}}[S_{\mysigma}]$ is $n$ because
for any maximal congruence $\mathfrak{c}$ in
$\operatorname{SCong}_{\mathbb{F}}{\mathbb{F}}[S_{\mysigma}]$, one has
${\mathbb{F}}[S_{\mysigma}]/\mathfrak{c}\simeq{\mathbb{F}}$ and

$\mathfrak{T}(\mathfrak{c})=\textup{rk}(H_{\mathfrak{c}})=\textup{rk}(S_{\Lambda_{\mathfrak{c}}}^{\vee})=\dim\mytau_{\mathfrak{c}}=n-\mathfrak{N}(\mathfrak{c}).$

∎

######  Definition .

Let $k$ be a pointed monoid and $X$ a $k$-scheme. We define the $k$-dimension
of $X$ as the Krull dimension of the topological space
$\operatorname{SCong}_{k}X$, and denote it by $\textup{dim}_{k}X$.

######  Theorem 5.2.

Let ${\mathbb{F}}$ be an algebraically closed pointed group, $\Sigma$ a fan
and

$X:=X(\Sigma)\times_{{\mathbb{F}}_{1}}\operatorname{MSpec}{\mathbb{F}}$

the toric ${\mathbb{F}}$-scheme associated to $\Sigma$. Then
$\operatorname{SCong}_{\mathbb{F}}X$ is a catenary space of dimension
$\dim_{\mathbb{F}}X=\dim\Sigma$.

###### Proof.

By Section 3,
$\\{\operatorname{SCong}_{\mathbb{F}}{\mathbb{F}}[S_{\mysigma}]\mid\mysigma\in\Sigma\\}$
is an open covering of $\operatorname{SCong}_{\mathbb{F}}X$. By Theorem 5.1,
each $\operatorname{SCong}_{\mathbb{F}}{\mathbb{F}}[S_{\mysigma}]$ is catenary
of dimension $n$, which implies $\dim_{\mathbb{F}}X=n$. By [Sta23, Lemma
5.11.5], $X$ is catenary. ∎

##  6\. Examples: The torus and affine space

Throughout this section, we fix an algebraically closed pointed group
${\mathbb{F}}$ and give an explicit description of the strong congruence space
of the torus and affine space over ${\mathbb{F}}$.

######  Notation.

For $y=(y_{1},\dotsc,y_{n})\in{\mathbb{Z}}^{n}$, we use $X^{y}$ to denote the
monomial $X_{1}^{y_{1}}\cdots X_{n}^{y_{n}}$.

###  6.1. The torus

The $n$-torus over ${\mathbb{F}}$ is the affine monoid scheme

${\mathbb{T}}_{\mathbb{F}}^{n}:=\operatorname{MSpec}{\mathbb{F}}[X_{1}^{\pm},\dotsc,X_{n}^{\pm}].$

Let $\\{z_{1},\dotsc,z_{c}\\}$ be part of a ${\mathbb{Z}}$-basis of
${\mathbb{Z}}^{n}$ and
$\mylambda_{1},\dotsc,\mylambda_{c}\in{\mathbb{F}}^{\times}$. Define the
congruence

$\mathfrak{c}:=\langle X^{z_{i}}\sim\mylambda_{i}\mid i=1,\dotsc,c\rangle$

on ${\mathbb{F}}[X_{1}^{\pm},\dotsc,X_{n}^{\pm}]$. Let
$z_{c+1},\dotsc,z_{n}\in{\mathbb{Z}}^{n}$ such that $\\{z_{1},\dotsc,z_{n}\\}$
is a ${\mathbb{Z}}$-basis. The map

$\begin{array}[]{cccc}\myvarphi:&{\mathbb{F}}[X_{1}^{\pm},\dotsc,X_{n}^{\pm}]&\longrightarrow&{\mathbb{F}}[X_{1}^{\pm},\dotsc,X_{n}^{\pm}]\\\
&X_{i}&\longmapsto&X^{z_{i}}\end{array}$

defines an isomorphism of ${\mathbb{F}}$-algebras. As

$\myvarphi^{*}(\mathfrak{c})=\langle X_{i}\sim\mylambda_{i}\mid
i=1,\dotsc,c\rangle,$

one has
${\mathbb{F}}[X_{1}^{\pm},\dotsc,X_{n}^{\pm}]/\mathfrak{c}\simeq{\mathbb{F}}[X_{1}^{\pm},\dotsc,X_{n}^{\pm}]/\myvarphi^{*}(\mathfrak{c})\simeq{\mathbb{F}}[X_{c+1}^{\pm},\dotsc,X_{n}^{\pm}]$.
Therefore $\mathfrak{c}$ is an ${\mathbb{F}}$-congruence.

The next result shows that every ${\mathbb{F}}$-congruence on
${\mathbb{F}}[X_{1}^{\pm},\dotsc,X_{n}^{\pm}]$ has this form.

######  Proposition .

Let
$\mathfrak{c}\in\operatorname{SCong}_{\mathbb{F}}{\mathbb{T}}_{\mathbb{F}}^{n}$.
Then there exists an integer $0\leq c\leq n$, constants
$\mylambda_{1},\dotsc,\mylambda_{c}\in{\mathbb{F}}^{\times}$ and
$z_{1},\dotsc,z_{c}\in{\mathbb{Z}}^{n}$ part of a ${\mathbb{Z}}$-basis such
that

$\mathfrak{c}=\langle X^{z_{i}}\sim\mylambda_{i}\mid i=1,\dotsc,c\rangle.$

###### Proof.

As ${\mathbb{F}}[X_{1}^{\pm},\dotsc,X_{n}^{\pm}]$ is a pointed group, one has
$I_{\mathfrak{c}}=\\{0\\}$. Let

$H:=\\{z\in{\mathbb{Z}}^{n}\mid X^{z}\sim_{\mathfrak{c}}\mylambda\text{ for
some }\mylambda\in{\mathbb{F}}^{\times}\\}.$

Note that $H$ is a subgroup of ${\mathbb{Z}}^{n}$. We show that
${\mathbb{Z}}^{n}/H$ is torsion-free. Let $z\in{\mathbb{Z}}^{n}$ and $m>0$
such that $mz\in H$. Then there exists $\mylambda\in{\mathbb{F}}^{\times}$
such that $(X^{z})^{m}\sim_{\mathfrak{c}}\mylambda$. Let
$\myalpha\in{\mathbb{F}}$ such that $\myalpha^{m}=\mylambda$ and
$\myzeta\in{\mathbb{F}}$ be a primitive $m^{\textup{th}}$-root of $1$, i.e., a
generator of the group $\mymu_{m}:=\\{w\in{\mathbb{F}}\mid w^{m}=1\\}$. As
$\mathfrak{c}$ is an ${\mathbb{F}}$-congruence, the sets

$\big{\\{}x\in{\mathbb{F}}[X_{1}^{\pm},\dotsc,X_{n}^{\pm}]/\mathfrak{c}\;\big{|}\;x^{m}=\mylambda\big{\\}}\quad\text{and}\quad\big{\\{}[\myzeta^{i}\myalpha]\;\big{|}\;i=0,\dotsc,m-1\big{\\}}$

coincide. As $[X^{z}]^{m}=\mylambda$, there exists $0\leq j\leq m-1$ such that
$X^{z}\sim_{\mathfrak{c}}\myzeta^{j}\myalpha$. Thus $z\in H$. Then there
exists a ${\mathbb{Z}}$-basis $\\{z_{1},\dotsc,z_{c}\\}$ of $H$ that can be
completed to a ${\mathbb{Z}}$-basis of ${\mathbb{Z}}^{n}$.

As $\mathfrak{c}$ is an ${\mathbb{F}}$-congruence and ${\mathbb{F}}$ is a
pointed group, there exist unique
$\mylambda_{1},\dotsc,\mylambda_{c}\in{\mathbb{F}}^{\times}$ such that
$X^{z_{i}}\sim_{\mathfrak{c}}\mylambda_{i}$ for $i=1,\dotsc,c$. Next we show
that $\mathfrak{c}$ is equal to

$\mathfrak{d}:=\langle X^{z_{i}}\sim\mylambda_{i}\mid i=1,\dotsc,c\rangle.$

Let $g,h\in({\mathbb{F}}[X_{1}^{\pm},\dotsc,X_{n}^{\pm}])^{\times}$ such that
$g\sim_{\mathfrak{c}}h$. Then $gh^{-1}\sim_{\mathfrak{c}}1$. Let
$\myomega\in{\mathbb{F}}^{\times}$ and $z\in{\mathbb{Z}}^{n}$ such that
$gh^{-1}=\myomega X^{z}$. As $X^{z}\sim_{\mathfrak{c}}\myomega^{-1}$, one has
$X^{z}\in H$. Thus there exist $m_{1},\dotsc,m_{c}\in{\mathbb{Z}}$ such that
$z=\sum m_{i}z_{i}$, which implies
$\myomega^{-1}\sim_{\mathfrak{c}}X^{z}\sim_{\mathfrak{c}}\prod\mylambda_{i}^{m_{i}}$.
As $\mathfrak{c}$ is an ${\mathbb{F}}$-congruence, one has
$\myomega^{-1}=\prod\mylambda_{i}^{m_{i}}$. Note that
$X^{z}\sim_{\mathfrak{d}}\prod\mylambda_{i}^{m_{i}}$, which implies

$g\sim_{\mathfrak{d}}gh^{-1}h\sim_{\mathfrak{d}}\myomega
X^{z}h\sim_{\mathfrak{d}}\myomega\myomega^{-1}h\sim_{\mathfrak{d}}h.$

Therefore $g\sim_{\mathfrak{d}}h$. ∎

###  6.2. Affine space

The affine $n$-space over ${\mathbb{F}}$ is the affine monoid scheme

${\mathbb{A}}_{\mathbb{F}}^{n}:=\operatorname{MSpec}{\mathbb{F}}[X_{1},\dotsc,X_{n}].$

We show the analogue of Section 6.1 in the case of the affine space. For a
subset $L\subseteq\\{1,\dotsc,n\\}$, let $\mathfrak{p}_{L}$ be the prime ideal
$\langle X_{i}\mid i\in L\rangle\in{\mathbb{A}}_{\mathbb{F}}^{n}$. Note that

${\mathbb{F}}[X_{1},\dotsc,X_{n}]/\mathfrak{p}_{L}\simeq{\mathbb{F}}[X_{i}\mid
i\notin L].$

Thus, for an ${\mathbb{F}}$-congruence $\mathfrak{d}$ on
${\mathbb{F}}[X_{i}^{\pm}\mid i\notin L]=\textup{Frac}({\mathbb{F}}[X_{i}\mid
i\notin L])$, one has that $\myvarphi^{*}(\mathfrak{d})$ is an
${\mathbb{F}}$-congruence on ${\mathbb{F}}[X_{1},\dotsc,X_{n}]$, where
$\myvarphi$ is the composition

${\mathbb{F}}[X_{1},\dotsc,X_{n}]\relbar\joinrel\twoheadrightarrow{\mathbb{F}}[X_{1},\dotsc,X_{n}]/\mathfrak{p}_{L}\simeq{\mathbb{F}}[X_{i}\mid
i\notin L]\longhookrightarrow{\mathbb{F}}[X_{i}^{\pm}\mid i\notin L].$

By Section 6.1, $\mathfrak{d}=\langle X^{z_{i}}\sim\mylambda_{i}\mid
i=1,\dotsc,c\rangle$, where
$\\{z_{1},\dotsc,z_{c}\\}\subset{\mathbb{Z}}^{n-\\#L}$ is part of a
${\mathbb{Z}}$-basis, which implies that

$\myvarphi^{*}(\mathfrak{d})=\langle\\{X_{i}\sim 0\mid i\in L\\}\cup\\{a\sim
b\mid a,b\in{\mathbb{F}}[X_{i}\mid i\notin L]\text{ with
}a\sim_{\mathfrak{d}}b\\}\rangle.$

The next result shows that every
$\mathfrak{c}\in\operatorname{SCong}_{\mathbb{F}}{\mathbb{A}}_{\mathbb{F}}^{n}$
has this form.

######  Proposition .

Let
$\mathfrak{c}\in\operatorname{SCong}_{\mathbb{F}}{\mathbb{A}}_{\mathbb{F}}^{n}$.
Then there exists a subset $L\subseteq\\{1,\dotsc,n\\}$, an integer $0\leq
c\leq n-\\#L$, constants
$\mylambda_{1},\dotsc,\mylambda_{c}\in{\mathbb{F}}^{\times}$ and
$\\{z_{1},\dotsc,z_{c}\\}\subseteq{\mathbb{Z}}^{n-\\#L}$ part of a
${\mathbb{Z}}$-basis such that

$\mathfrak{c}=\langle\\{X_{i}\sim 0\mid i\in L\\}\cup\\{a\sim b\mid
a,b\in{\mathbb{F}}[X_{i}\mid i\notin L]\text{ with
}a\sim_{\mathfrak{d}}b\\}\rangle,$

where $\mathfrak{d}$ is the congruence $\langle X^{z_{i}}\sim\mylambda_{i}\mid
i=1,\dotsc,c\rangle$ on ${\mathbb{F}}[X_{i}^{\pm}\mid i\notin L]$.

###### Proof.

As $I_{\mathfrak{c}}\in{\mathbb{A}}_{\mathbb{F}}^{n}$, there exists
$L\subseteq\\{1,\dotsc,n\\}$ such that $I_{\mathfrak{c}}=\mathfrak{p}_{L}$.
Let

$\mypi:{\mathbb{F}}[X_{1},\dotsc,X_{n}]\relbar\joinrel\twoheadrightarrow{\mathbb{F}}[X_{1},\dotsc,X_{n}]/\mathfrak{p}_{L}\simeq{\mathbb{F}}[X_{i}\mid
i\notin L]$

be the projection,

$\myiota:{\mathbb{F}}[X_{i}\mid i\notin
L]\longhookrightarrow{\mathbb{F}}[X_{i}^{\pm}\mid i\notin L]$

the localization map, $\mathfrak{r}:=\mypi_{*}(\mathfrak{c})$ and
$\mathfrak{d}:=\myiota_{*}(\mypi_{*}(\mathfrak{c}))$. Note that
$\mathfrak{r}\in\operatorname{SCong}_{\mathbb{F}}{\mathbb{A}}_{\mathbb{F}}^{n-\\#L}$
and $\mathfrak{c}=\mypi^{*}(\mathfrak{r})$. As $\mathfrak{r}$ satisfies
$I_{\mathfrak{r}}=\\{0\\}$, by Section 3, one has
$\mathfrak{d}\in\operatorname{SCong}_{\mathbb{F}}{\mathbb{T}}_{\mathbb{F}}^{n-\\#L}$
and $\mathfrak{r}=\myiota^{*}(\mathfrak{d})$. Thus the result follows from the
discussion above, because
$\mathfrak{c}=(\myiota\circ\mypi)^{*}(\mathfrak{d})$. ∎

## References

  * [Ber12] Vladimir G. Berkovich.  Algebraic geometry over the field with one element.  Unfinished project. Online available at https://www.wisdom.weizmann.ac.il/~vova/Algeom_2012.pdf, 2012.
  * [Bra01] Jean-Paul Brasselet.  Introduction to Toric Varieties.  Publicações Matemáticas. IMPA, Rio de Janeiro, 2001.
  * [CC10] Alain Connes and Caterina Consani.  From monoids to hyperstructures: in search of an absolute arithmetic.  In Casimir Force, Casimir Operators and the Riemann Hypothesis, page 147–198. De Gruyter, 2010.
  * [CHWW15] Guillermo Cortiñas, Christian Haesemeyer, Mark E. Walker, and Charles Weibel.  Toric varieties, monoid schemes and cdh descent.  Journal für die reine und angewandte Mathematik (Crelle’s Journal), 2015(698):1–54, 2015.
  * [CLS12] Chenghao Chu, Oliver Lorscheid, and Rekha Santhanam.  Sheaves and k-theory for ${\mathbb{F}}_{1}$-schemes.  Advances in Mathematics, 229(4):2239–2286, 2012.
  * [Dei05] Anton Deitmar.  Schemes over $\mathbb{F}_{1}$.  In Number Fields and Function Fields—Two Parallel Worlds, page 87–100. Birkhäuser Boston, 2005.
  * [Dei08] Anton Deitmar.  ${\mathbb{F}}_{1}$-schemes and toric varieties.  Contributions to Algebra and Geometry, 49(2):517–525, 2008.
  * [Dei13] Anton Deitmar.  Congruence schemes.  International Journal of Mathematics, 24(2):1350009, 2013.
  * [Eur15] Christopher Eur.  An Introduction to Toric Varieties.  Bachelor’s thesis, Harvard University, 2015.
  * [Ful93] William Fulton.  Introduction to Toric Varieties.  Annals of Mathematics Studies. Princeton University Press, 1993.
  * [Jar23] Manoel Jarra.  On Smirnov’s approach to the ABC conjecture.  Preprint, arXiv:2306.16637, 2023.
  * [LR23] Oliver Lorscheid and Samarpita Ray.  The topological shadow of $\mathbb{F}_{1}$-geometry: congruence spaces.  Preprint, arXiv:2305.12801, 2023.
  * [Ogu18] Arthur Ogus.  Lectures on Logarithmic Algebraic Geometry.  Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2018.
  * [Smi93] Alexander L. Smirnov.  Hurwitz inequalities for number fields.  St. Petersburg Math. J., 4(2):357–375, 1993.
  * [Sta23] The Stacks project authors.  The Stacks project.  https://stacks.math.columbia.edu, 2023.