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# The topological shadow of ${{\mathbb{F}}_{1}}$-geometry: congruence spaces

Oliver Lorscheid  Oliver Lorscheid, University of Groningen, the Netherlands,
and IMPA, Rio de Janeiro, Brazil<EMAIL_ADDRESS>and  Samarpita Ray  Samarpita
Ray, Indian Statistical Institute, Bangalore, India<EMAIL_ADDRESS>
###### Abstract.

In this paper we introduce congruence spaces, which are topological spaces
that are canonically attached to monoid schemes and that reflect closed
topological properties. This leads to satisfactory topological
characterizations of closed morphisms and closed immersions as well as
separated and proper morphisms. We study congruence spaces thoroughly and
extend standard results from usual scheme theory to monoid schemes: a closed
immersion is the same as an affine morphism for which the pullback of sections
is surjective; a morphism is separated if and only if the image of the
diagonal is a closed subset of the congruence space; a valuative criterion for
separated and proper morphisms.

###### Contents

  1. 1 Preliminaries
  2. 2 Congruence spaces
  3. 3 Strongly reduced monoid schemes
  4. 4 Closed immersions and vanishing sets
  5. 5 Valuation monoids
  6. 6 Universally closed morphisms
  7. 7 Separated morphisms
  8. 8 Proper morphisms
  9. A Alternatives, ramifications, and why not

## Introduction

Monoid schemes are sometimes termed the _minimalist approach_ to
${{\mathbb{F}}_{1}}$-geometry and appear, in fact, as a special case of nearly
every other approach to ${{\mathbb{F}}_{1}}$-geometry (cf. [20]). They have
been utilized in and connected with various other areas of geometry: most
profoundly, monoid schemes are a generalization of toric geometry ([11]). They
are used to study the $K$-theory of toric varieties in positive characteristic
([6, 8]), cdh descent for toric varieties ([7]) and sheaf cohomology of toric
vector bundles ([14]). Stable homotopy groups of spheres and other spaces
resurface as the $K$-theory of monoid schemes ([3, 10, 13]). They enter as an
integral component into the theory of tropical schemes ([15, 16, 23, 26, 27,
28]). Other works on monoid schemes include [1, 2, 4, 17, 25, 29, 31, 32, 33,
34, 35, 36, 37].

Monoid schemes are understood from two equivalent perspectives: as topological
spaces together with a structure sheaf in monoids ([3, 5, 7, 9]) and as
sheaves on the category of monoids ([30, 38, 39]). The topological perspective
gives monoid schemes a concrete shape as spaces of prime ideals. The topos
theoretic interpretation justifies that prime ideals capture _open conditions_
on monoids. More explicitly, a family $\\{U_{i}\to X\\}_{i\in I}$ of open
immersions is a covering family in the canonical topology for monoid schemes
if and only if the $U_{i}$ cover $X$ as a topological space. As a consequence,
every monoid is isomorphic to the global sections of its spectrum and the
Yoneda functor of monoids into sheaves is fully faithful.

However, in contrast to usual scheme theory, prime ideals do not reflect
_closed conditions_. A prototypical example is the diagonal embedding
$\Delta:{\mathbb{A}}^{1}_{{\mathbb{F}}_{1}}\to{\mathbb{A}}^{2}_{{\mathbb{F}}_{1}}$,
which is not a closed topological embedding. This is illustrated in Figure 1:
the image of $\Delta$ (shaded in orange) is not among the closed subsets of
${\mathbb{A}}^{2}_{{\mathbb{F}}_{1}}$ (shaded in green).

$\Delta$ Figure 1. The diagonal embedding
$\Delta:{\mathbb{A}}^{1}_{{\mathbb{F}}_{1}}\to{\mathbb{A}}^{2}_{{\mathbb{F}}_{1}}$

This example shows that the underlying topological space of a monoid scheme
does not lead to a satisfactory notion of closed morphisms, which is a
prominent concept in usual scheme theory. Several subclasses of closed
morphisms allow for equivalent characterizations generalize to monoid schemes.
For example:

  * $\bullet$

_Closed immersions_ are defined as affine morphisms $\myvarphi:Y\to X$ for
which ${\mathcal{O}}_{X}\to\myvarphi_{\ast}{\mathcal{O}}_{Y}$ is surjective
([7, Definition 2.5]).

  * $\bullet$

_Separated morphisms_ are defined as morphisms $\myvarphi:Y\to X$ whose
diagonal $\Delta_{\myvarphi}:Y\to Y\times_{X}Y$ is a closed immersion ([7,
Definition 3.3]).

  * $\bullet$

_Proper morphisms_ are only defined in the finite type case in terms of the
valuative criterion for properness ([7, Definition 8.4]).

  * $\bullet$

There is no notion of _(universally) closed morphisms_ yet.

In this paper, we introduce a secondary topological space associated with
monoid schemes, which we call the _congruence space_ and which captures closed
conditions for monoid schemes. This leads to a satisfactory notion of closed
morphisms for monoid schemes and a topological characterization of the above
mentioned classes of morphisms.

### Earlier works on congruences

The idea to use congruences in the context of monoid schemes is not novel to
this paper. Berkovich develops in the unpublished text [1] a scheme theory for
monoids that is based on congruences, or quotients, rather than on ideals, or
submodules. In fact, this text features many concepts that appear in our text.

Berkovich’s approach leads, however, to a quite different notion of
${{\mathbb{F}}_{1}}$-scheme since the space of prime congruences does not
reflect the same notion of open coverings as monoid schemes (cf. section
A.2.3). In particular, there are topological coverings in the congruence
setting, which are not canonical. This leads to technical ramifications in
Berkovich’s theory in contrast to monoid schemes in the sense of our text.

Deitmar develops in [12] a variant of congruence schemes for sesquiads, which
are algebraic objects that generalize monoids. Also Deitmar’s theory deviates
from monoid schemes in the sense that it features non-canonical open
coverings.

### Follow-up work

Jarra builds up on the theory of congruence spaces as developed in this text:
he introduces in [19] an intrinsic subspace of the congruence space that
exhibits the expected dimension of monoid schemes over algebraically closed
monoids. In a second text [18], Jarra shows that this subspace coincide with
the spaces in Smirnov’s approach to the ABC-conjecture via Hurwitz formulas in
${{\mathbb{F}}_{1}}$-geometry.

### Congruence spaces

We refer the reader to section 1 for background on pointed monoids and monoid
schemes. Here we restrict ourselves to introduce the novel concepts of this
paper.

A _congruence on $A$_ is an equivalence relation ${\mathfrak{c}}$ on $A$ such
that $(a,b)\in{\mathfrak{c}}$ implies $(ac,bc)\in{\mathfrak{c}}$. The quotient
$A/{\mathfrak{c}}$ is naturally a monoid, which establishes a bijective
correspondence between the congruences of $A$ and the monoid quotients of $A$.
A congruence ${\mathfrak{p}}$ is _prime_ if $A/{\mathfrak{p}}$ is _integral_ ,
or multiplicatively cancellative.

The _congruence space of $A$_ is the set $\operatorname{Cong}(A)$ of all prime
congruences ${\mathfrak{p}}$ on $A$ together with the topology generated by
open subsets of the form

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

for $a,b\in A$. It comes with a continuous surjection
$\mypi_{A}:\operatorname{Cong}(A)\to\operatorname{MSpec}(A)$, which sends a
prime congruence ${\mathfrak{p}}$ to its nullideal $I_{\mathfrak{p}}=\\{a\in
A\mid(a,0)\in{\mathfrak{p}}\\}$.

The following result endows every monoid scheme with a congruence space; cf.
Propositions 2.4 and 2.5. Let $\operatorname{MSch}$ be the category of monoid
schemes and $\operatorname{Top}$ the category of topological spaces.

######  Theorem A.

There is a functor $\operatorname{MSch}\to\operatorname{Top}$ that associates
with every monoid scheme $X$ a topological space ${\widetilde{X}}$, called its
_congruence space_ , together with a continuous surjection
$\mypi_{X}:{\widetilde{X}}\to X$, which is characterized by the following
properties:

  1. (1)

If $X=\operatorname{MSpec}(A)$ is affine, then
${\widetilde{X}}=\operatorname{Cong}(A)$ and $\mypi_{X}=\mypi_{A}$.

  2. (2)

For every morphism $\myvarphi:Y\to X$ of monoid schemes, the diagram

${{\widetilde{Y}}}$${{\widetilde{X}}}$${Y}$${X}$$\scriptstyle{\tilde{\myvarphi}}$$\scriptstyle{\mypi_{Y}}$$\scriptstyle{\mypi_{X}}$$\scriptstyle{\myvarphi}$

commutes.

  3. (3)

Open immersions $\myvarphi:Y\to X$ induce open topological embeddings
$\tilde{\myvarphi}:{\widetilde{Y}}\to{\widetilde{X}}$.

  4. (4)

If $X=\bigcup_{i\in I}U_{i}$ is covered by open subschemes $U_{i}$, then
${\widetilde{X}}=\bigcup_{i\in I}{\widetilde{U}}_{i}$.

The notion of a structure sheaf for congruence spaces is problematic (cf.
section A.2); still every point comes with a well-defined notion of a _stalk_
and a _residue field_.

### Closed properties

The congruence space turns out to be a suitable space to capture closed
properties of (morphisms between) monoid schemes.

We study several morphism classes in this paper and explain our results in the
following. The systematic approach is to replace closed morphisms in usual
scheme theory by morphisms $\myvarphi:Y\to X$ of monoid schemes for which
$\tilde{\myvarphi}:{\widetilde{Y}}\to{\widetilde{X}}$ is closed. Note that
this interpolation between the monoid scheme and its congruence space makes
proofs significantly more subtle and difficult than in usual scheme theory.

### Closed points

As a first insight (Proposition 2.7), we find canonical bijections

$X({{\mathbb{F}}_{1}})\
\stackrel{{\scriptstyle\mychi_{X}}}{{\longrightarrow}}\ \big{\\{}\text{closed
points of ${\widetilde{X}}$}\big{\\}}\
\stackrel{{\scriptstyle\tilde{\mypi}_{X}}}{{\longrightarrow}}\ X$

where
$X({{\mathbb{F}}_{1}})=\operatorname{Hom}(\operatorname{MSpec}{{\mathbb{F}}_{1}},\,X)$
and where $\tilde{\mypi}_{X}$ is the restriction of the surjection
$\mypi_{X}:{\widetilde{X}}\to X$ to the subset of closed points of
${\widetilde{X}}$.

### Closed subschemes

A _closed subscheme_ of a monoid scheme $X$ is the isomorphism class of a
closed immersion $\myvarphi:Y\to X$. Closed subschemes can be characterized
similarly as for usual schemes if concepts are generalized appropriately.

A monoid $A$ is _strongly reduced_ if the trivial congruence
${\mathfrak{c}}_{\textup{triv}}=\\{(a,a)\in A\times A\mid a\in A\\}$ is a
prime congruence. A monoid scheme $X$ is strongly reduced if
${\mathcal{O}}_{X}(U)$ is strongly reduced for every open subset $U$. Every
monoid scheme has a strong reduction.

A _vanishing set_ of ${\widetilde{X}}$ is the image
$\tilde{\myvarphi}({\widetilde{Y}})$ for a closed immersion $\myvarphi:Y\to
X$. For affine monoid schemes $X=\operatorname{MSpec}(A)$, a vanishing set is
the intersection of closed subsets of the form
$V_{a,b}=\\{{\mathfrak{p}}\in{\widetilde{X}}\mid(a,b)\in{\mathfrak{p}}\\}$;
note that subsets of this form are not closed under finite unions, i.e.
vanishing sets are closed, but not every closed subset of ${\widetilde{X}}$ is
a vanishing set.

A _quasi-coherent congruence sheaf_ on $X$ is a quasi-coherent subsheaf of
${\mathcal{O}}_{X}\times{\mathcal{O}}_{X}$ that is a congruence on
${\mathcal{O}}_{X}(U)$ for every open subset $U$ of $X$. The congruence kernel
of a closed immersion $\myvarphi:Y\to X$ is the quasi-coherent congruence
sheaf defined by
$\operatorname{Congker}(\myvarphi)(U)=\\{(a,b)\in{\mathcal{O}}_{X}(U)\times{\mathcal{O}}_{X}(U)\mid\myvarphi^{\\#}(U)(a)=\myvarphi^{\\#}(U)(b)\\}$.

These relations yield a diagram

${\big{\\{}\text{vanishing sets of
$X$}\big{\\}}}$${\Big{\\{}\begin{array}[]{c}\text{strongly reduced}\\\
\text{closed subschemes of
$X$}\end{array}\Big{\\}}}$${\Big{\\{}\begin{array}[]{c}\text{quasi-
coherent}\\\ \text{congruence sheaves on
$X$}\end{array}\Big{\\}}}$${\big{\\{}\text{closed subschemes of
$X$}\big{\\}}}$bijectivebijective

which commutes for all paths that either start or end in the top row; cf.
Remark 4.4.

### Closed immersions

In [7], a _closed immersion_ of monoid schemes is defined as an affine
morphism $\myvarphi:Y\to X$ for which
${\mathcal{O}}_{X}\to\myvarphi_{\ast}{\mathcal{O}}_{Y}$ is surjective. _A
posteriori_ we gain the following topological characterization, which reflects
the usual definition of closed immersions of usual schemes; cf. Theorem 4.3.

######  Theorem B.

Let $\myvarphi:Y\to X$ be a morphism of monoid schemes. The following are
equivalent:

  1. (1)

$\myvarphi:Y\to X$ is a closed immersion.

  2. (2)

$\myvarphi:Y\to X$ is a quasi-compact topological embedding,
$\myvarphi^{\\#}:{\mathcal{O}}_{X}\to\myvarphi_{\ast}{\mathcal{O}}_{Y}$ is
surjective and the image of
$\tilde{\myvarphi}:{\widetilde{Y}}\to{\widetilde{X}}$ is a vanishing set.

### Separated morphisms

In [7], a morphism $\myvarphi:Y\to X$ is defined to be _separated_ if the
diagonal $\Delta_{\myvarphi}:Y\to Y\times_{X}Y$ is a closed immersion (see
[7]). _A posteriori_ the congruence space provides a topological
characterization. We say that $\myvarphi:Y\to X$ is _quasi-separated_ if
$\Delta_{\myvarphi}:Y\to Y\times_{X}Y$ is quasi-compact. The following is
Theorem 7.1.

######  Theorem C.

Let $\myvarphi:Y\to X$ be a morphism of monoid schemes. Then the following are
equivalent:

  1. (1)

$\myvarphi$ is separated;

  2. (2)

$\myvarphi$ is quasi-separated and the image of $\tilde{\Delta}_{\myvarphi}$
is a closed subset of the congruence space of $Y\times_{X}Y$.

### Closed and proper morphisms

We say that a morphism $\myvarphi:Y\to X$ of monoid schemes is

  * $\bullet$

_closed_ if the map $\tilde{\myvarphi}:{\widetilde{Y}}\to{\widetilde{X}}$
between congruence spaces is closed;

  * $\bullet$

_universally closed_ if for every morphism $\mypsi:X^{\prime}\to X$ the
canonical projection $X^{\prime}\times_{X}Y\to X^{\prime}$ is closed;

  * $\bullet$

_proper_ if it is of finite type, separated and universally closed.

### Valuative criteria

A _pointed group_ is a pointed monoid $G$ for which every nonzero element is
invertible and $0\neq 1$. A _valuation monoid_ is a submonoid $A$ of a pointed
group $G$ such that for every nonzero $a\in G$ either $a\in A$ or $a^{-1}\in
A$.

Let $\myvarphi:Y\to X$ be a morphism of monoid schemes. A _test diagram_ for
$\myvarphi$ is a commutative diagram of the form

${U=\operatorname{MSpec}(G)}$${Y}$${T=\operatorname{MSpec}(A)}$${X}$$\scriptstyle{\myiota}$$\scriptstyle{\myeta}$$\scriptstyle{\myvarphi}$$\scriptstyle{\mynu}$

where $A$ is a valuation monoid in the pointed group $G$ and $\myiota$ is
induced by the inclusion map $A\to G$. A _lift for the test diagram_ is a
morphism $\hat{\mynu}:T\to Y$ such that

${U=\operatorname{MSpec}(G)}$${Y}$${T=\operatorname{MSpec}(A)}$${X}$$\scriptstyle{\myiota}$$\scriptstyle{\myeta}$$\scriptstyle{\myvarphi}$$\scriptstyle{\mynu}$$\scriptstyle{\hat{\mynu}}$

commutes. The following summarizes Theorem 6.1, Theorem 7.2 and Theorem 8.1.

######  Theorem D.

A quasi-compact / quasi-separated / finite type morphism $\myvarphi:Y\to X$ of
monoid schemes is universally closed / separated / proper if and only if every
test diagram for $\myvarphi$ has at least / at most / exactly one lift.

### Content overview

Section 1 contains a short reminder on monoid schemes and introduces the
conventions for this paper. In section 2, we introduce congruence spaces and
study their properties. In section 3, we introduce the strong reduction of
monoids and monoid schemes and compare it with the usual reduction. In section
4, we exhibit a topological characterization of closed immersions and study
the relation between closed subschemes, vanishing sets and congruence sheaves.
In section 5, we introduce valuation monoids and characterize them as maximal
objects with respect to dominant morphisms. In section 6, we introduce
universally closed morphisms and establish the corresponding valuative
criterion. In section 7, we exhibit a topological characterization of
separated morphisms, which is based on a study of locally closed immersions,
and we establish the corresponding valuative criterion. In section 8, we
introduce proper morphisms and establish the corresponding valuative
criterion.

### Acknowledgements

The first author was supported by the Marie Curie Grant No. 101022339 of the
European Commission. The second author’s work was partially supported by the
SERB NPDF PDF/2020/000670 grant of Govt. of India, partially by the
IBS-R003-D1 grant of the IBS-CGP, POSTECH, South Korea and partially by the
DST/INSPIRE/04/2021/002904 grant of Govt. of India.

##  1\. Preliminaries

In this section we recall some basic facts about commutative pointed monoids
and monoid schemes in brevity. For further details and a more comprehensive
introduction into these concepts, we refer the reader to [3] and [7].

###  1.1. Pointed monoids

A _(commutative) pointed monoid_ is a commutative semigroup $A$ with a zero
(absorbing element) $0$ and an identity $1$, satisfying $a.0=0.a=0$ and
$a.1=1.a=a$ for all $a\in A$. A _morphism of pointed monoids_ is a semigroup
homomorphism that preserves the zero and identity.

An _ideal_ of a pointed monoid $A$ is a subset $I$ of $A$ such that $0\in I$
and $IA\subseteq A$. The _quotient monoid_ is the set $A/I=(A-I)\cup\\{0\\}$
together with the multiplication given by $a\cdot b=ab$ if $a,b\in A-I$ and
$a\cdot b=0$ otherwise. An ideal $P$ of $A$ is _prime_ if $ab\in P$ implies
$a\in P$ or $b\in P$ or, in other words, $A/P$ is without zero divisors. The
complement of the set of units $A^{\times}$ of a pointed monoid $A$ forms an
ideal and is therefore the unique maximal ideal of $A$ which we denote by
$M_{A}$.

Let $S$ be a _multiplicative subset_ of a pointed monoid $A$, i.e. $1\in S$
and $ab\in S$ for all $a,b\in S$. The _localization of $A$ at $S$_ is the
monoid $S^{-1}A=S\times A/\sim$ where $(s,a)\sim(s^{\prime},a^{\prime})$ if
and only if $tsa^{\prime}=ts^{\prime}a$ for some $t\in S$. We denote the
equivalence class of $(s,a)$ by $\tfrac{a}{s}$. The association
$a\mapsto\tfrac{a}{1}$ defines the _localization map_ $\myiota_{S}:A\to
S^{-1}A$. Particular examples of localizations are $A[h^{-1}]=S_{h}^{-1}A$
with $S_{h}=\\{h^{i}\\}_{i\geqslant 0}$ for $h\in A$ and $A_{P}=S_{P}^{-1}A$
with $S_{P}=A-P$ for a prime ideal $P$ of $A$.

The category of pointed monoids is complete and cocomplete. In particular, it
comes with a tensor product $A\otimes_{C}B$ of any pair of morphisms $C\to A$
and $C\to B$. The pointed monoid ${{\mathbb{F}}_{1}}=\\{0,1\\}$ is its initial
object.

Let $A$ be a pointed monoid. The _free monoid in $t_{1},\dotsc,t_{n}$ over
$A$_ is the pointed monoid

$A[t_{1},\dotsc,t_{n}]\ =\ \big{\\{}at_{1}^{e_{1}}\cdots
t_{n}^{e_{n}}\,\big{|}\,e_{1},\dotsc,e_{n}\in{\mathbb{N}}\big{\\}}$

with product

$(at_{1}^{e_{1}}\cdots t_{n}^{e_{n}})\ \cdot\ (bt_{1}^{f_{1}}\cdots
t_{n}^{f_{n}})\ =\ (ab)t_{1}^{e_{1}+f_{1}}\cdots t_{n}^{e_{n}+f_{n}}$

and identification $0\cdot t_{1}^{e_{1}}\cdots t_{n}^{e_{n}}=0\cdot
t_{1}^{f_{1}}\cdots t_{n}^{f_{n}}$ for all $e_{i},f_{i}\in{\mathbb{N}}$.

###  1.2. Integral and zero divisor free monoids

An _integral monoid_ is a pointed monoid $A$ such that $ab=ac$ implies $b=c$
for all $a,b,c\in A$ with $a\neq 0$. A _zero divisor free monoid_ is a pointed
monoid $A$ for which $S=A-\\{0\\}$ is multiplicative subset.

Let $A$ be zero divisor free, $S=A-\\{0\\}$ and $i_{S}:A\to S^{-1}A$ be the
localization map. The _group completion of $A$_ is
$\operatorname{Frac}A=S^{-1}A$. The _integral quotient of $A$_ is
$A^{{\overline{\textup{int}}}}=i_{S}(A)$.

Note that $\operatorname{Frac}A$ is a pointed group and
$A^{\overline{\textup{int}}}$ is an integral monoid. The localization map
defines canonical maps $A\to\operatorname{Frac}A$ and $A\to
A^{\overline{\textup{int}}}$. We mention the following universal properties of
$\operatorname{Frac}A$ and $A^{\overline{\textup{int}}}$ without proof.

######  Lemma .

Let $A$ be a pointed monoid without zero divisors. Then every morphism $f:A\to
B$ into a pointed group $B$ factors uniquely through $\operatorname{Frac}A$
and every morphism $f:A\to B$ into an integral monoid $B$ factors uniquely
through $A^{{\overline{\textup{int}}}}$.

###  1.3. Monoid schemes

Let $A$ be a pointed monoid. The _prime spectrum of $A$_ is the set

$\operatorname{MSpec}(A)\ =\ \\{P\subseteq A\mid P\text{\leavevmode\nobreak\
is a prime ideal of }A\\}$

that carries the topology generated by the open subsets $U_{h}=\\{P\mid
h\notin P\\}$ for $h\in A$, together with a sheaf ${\mathcal{O}}_{X}$ of
monoids on $X=\operatorname{MSpec}(A)$ that is characterized by its values
${\mathcal{O}}_{X}(U_{h})=A[h^{-1}]$ and its stalks
${\mathcal{O}}_{X,P}=A_{P}$.

A _monoid scheme_ is a topological space $X$ together with a sheaf
${\mathcal{O}}_{X}$ of monoids that is locally isomorphic to spectra of
pointed monoids. A monoid scheme is _affine_ if it isomorphic the spectrum of
a pointed monoid.

A _morphism of monoid schemes $X$ and $Y$_ is a continuous map $\myvarphi:X\to
Y$ together with a morphism of sheaves
$\myvarphi^{\\#}:{\mathcal{O}}_{X}\to\myvarphi_{\ast}{\mathcal{O}}_{Y}$ of
pointed monoids for which the morphism of stalks
$\myvarphi_{x}:{\mathcal{O}}_{X,x}\to{\mathcal{O}}_{Y,y}$ are _local_ for
every $x\in X$ and $y=\myvarphi(x)$ in the sense that
$\myvarphi_{x}^{-1}({\mathcal{O}}_{Y,y}^{\times})={\mathcal{O}}_{X,x}^{\times}$.
We denote the category of monoid schemes by $\operatorname{MSch}$.

Note that $\operatorname{MSpec}$ extends to an anti-equivalence of
$\operatorname{Mon}$ with the category of affine monoid schemes: a morphism
$f:A\to B$ of pointed monoids defines the morphism
$f^{\ast}:\operatorname{MSpec}(B)\to\operatorname{MSpec}(A)$ with
$f^{\ast}(P)=f^{-1}(P)$. It has the global section functor
$\Gamma:\operatorname{MSch}\to\operatorname{Mon}$, which sends a monoid scheme
$X$ to $\Gamma X={\mathcal{O}}_{X}(X)$, as an adjoint:

$\operatorname{Hom}(X,\operatorname{MSpec}(A))\ =\ \operatorname{Hom}(A,\Gamma
X).$

###  1.4. Base extensions

The tensor product of pointed monoids extends to a fibre product
$X\times_{Z}Y$ of monoid schemes, which defines the base extension
$X_{B}=X\otimes_{A}B$ of a monoid scheme $X$ over $A$ to a $A$-algebra $B$.

Given a pointed monoid $A$, we denote its _associated ring_ by
$A^{+}={\mathbb{Z}}[A]/\langle 0\rangle$, which is equal to
${\mathbb{Z}}[A]-\\{0\\}$ as an abelian group. Given a monoid scheme $X$ over
$A$ and a morphism $f:A\to k$ from a pointed monoid $A$ to (the underlying
monoid of) a ring $k$, we obtain a functorial base extension
$X_{k}^{+}=X\otimes_{A}k$ that is characterized by the properties that it
commutes with colimits of open immersions and that for affine
$X=\operatorname{MSpec}(B)$,

$X_{k}\ =\ \operatorname{Spec}\big{(}B^{+}\otimes_{A^{+}}k\big{)}.$

###  1.5. Finite type morphisms

A morphism $f:A\to B$ of pointed monoids is _of finite type_ if there are
elements $b_{1},\ldots,b_{n}\in B$ such that every element of $B$ can be
expressed as a monomial in the $b_{i}$ and elements of $f(A)$.

A morphism $\myvarphi:X\to Y$ of monoid schemes is _of finite type_ if there
are affine open coverings $Y=\bigcup_{i\in I}V_{i}$ and
$\myvarphi^{-1}(V_{i})=\bigcup_{j\in J_{i}}U_{i,j}$ for all $i\in I$ such that
$J_{i}$ is finite for all $i\in I$ and such that
$\Gamma{\myvarphi_{i,j}}:\Gamma{V_{i}}\to\Gamma{U_{i,j}}$ is of finite type
for all $i\in I$ and $j\in J_{i}$.

##  2\. Congruence spaces

In this section, we introduce congruence spaces for monoid schemes, which are
secondary and subordinate topological spaces that translate topological
property from usual scheme theory appropriately to the world of monoid
schemes.

###  2.1. Congruences

The algebraic backbone for congruence spaces are congruences. We review their
definition and first properties. For more details, see [22, Chapter 3].

######  Definition .

A _congruence_ of a pointed monoid $A$ is an equivalence relation
$\mathfrak{c}$ on $A$ that is closed under multiplication by $A$, i.e.
$(a,b)\in\mathfrak{c}$ implies $(ac,bc)\in\mathfrak{c}$ for all $a,b,c\in A$.
The _nullideal of ${\mathfrak{c}}$_ is

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

The _quotient of $A$ by a congruence ${\mathfrak{c}}$_ is the set
$A/{\mathfrak{c}}$ of equivalence classes with multiplication given by
$[a]\cdot[b]=[ab]$.

Note that the product of elements of $A/{\mathfrak{c}}$ is well-defined on
representatives and turns $A/{\mathfrak{c}}$ into a pointed monoid with zero
$[0]$ and identity $[1]$. The quotient map $\mypi:A\to A/{\mathfrak{c}}$ is a
pointed monoid morphism.

Every subset $S$ of $A\times A$ is contained in a smallest congruence

$\langle S\rangle\ =\ \bigcap_{S\subset{\mathfrak{c}}}\ {\mathfrak{c}}$

that contains $S$, which we call the _congruence generated by $S$_.

######  Definition .

Let $f:A\to B$ be a morphism of pointed monoids. The _congruence kernel of
$f$_ is

$\operatorname{congker}(f):=\\{(a,a^{\prime})\in A\times A\mid
f(a)=f(a^{\prime})\\}.$

The congruence kernel of a morphism is indeed a congruence and that every
congruence ${\mathfrak{c}}$ of $A$ is the congruence kernel of the quotient
map $A\to A/{\mathfrak{c}}$. This establishes a bijective correspondence
between the quotients of $A$ and the congruences on $A$.

######  Definition .

A _prime congruence_ of $A$ is a congruence $\mathfrak{c}$ such that
$A/\mathfrak{c}$ is integral, i.e., $(ab,ac)\in\mathfrak{c}$ implies
$(a,0)\in\mathfrak{c}$ or $(b,c)\in\mathfrak{c}$ for all $a,b,c\in A$.

If $\mathfrak{c}$ is a prime congruence, then $I_{\mathfrak{c}}$ is a prime
ideal. The congruence kernel $\operatorname{congker}(f)$ of a morphism $f:A\to
B$ into an integral monoid $B$ is a prime congruence.

######  Definition .

Let $f:A\to B$ be a morphism of pointed monoids, ${\mathfrak{c}}$ a congruence
on $A$ and ${\mathfrak{d}}$ a congruence on $B$. The _push-forward of
${\mathfrak{c}}$ along $f$_ is the congruence

$f_{\ast}({\mathfrak{c}})\ =\ \langle(f(a),f(b))\in B\times
B\mid(a,b)\in{\mathfrak{c}}\\}\rangle$

of $B$ and the _pullback of ${\mathfrak{d}}$ along $f$_ is the congruence

$f^{\ast}({\mathfrak{d}})\ =\ \\{(a,b)\in A\times
A\mid(f(a),f(b))\in{\mathfrak{d}}\\}$

on $A$.

######  Proposition .

Let $A$ be a pointed monoid, ${\mathfrak{c}}$ a congruence on $A$ and
$\mypi:A\to A/{\mathfrak{c}}$ the quotient map. Then the pushforward and
pullback define mutually inverse bijections

${\big{\\{}\text{congruences on $A$ that contain
${\mathfrak{c}}$}\big{\\}}}$${\big{\\{}\text{congruences on
$A/{\mathfrak{c}}$}\big{\\}}}$$\scriptstyle{\mypi_{\ast}}$$\scriptstyle{\mypi^{\ast}}$

that restrict to bijections between the respective subsets of prime
congruences.

###### Proof.

This result follows immediately from the identification of congruences with
quotients of $A$ and the fact that the quotients of $A/{\mathfrak{c}}$
correspond to the quotients of $A$ that factor through $\mypi:A\to
A/{\mathfrak{c}}$. The additional claim about prime congruences follows from
the characterization of prime congruences ${\mathfrak{p}}$ in terms of the
quotient $A/{\mathfrak{p}}$, which agrees on both sides of the bijection. ∎

###  2.2. Compatibility with localizations

######  Definition .

Let $A$ be a pointed monoid, $S$ a multiplicative subset and $\mathfrak{c}$ a
congruence of $A$. The _localization of $\mathfrak{c}$ at $S$_ is

$S^{-1}\mathfrak{c}\ =\
\big{\\{}(\tfrac{a}{s},\tfrac{a^{\prime}}{s^{\prime}})\in S^{-1}A\times
S^{-1}A\leavevmode\nobreak\ \big{|}\leavevmode\nobreak\
(ts^{\prime}a,tsa^{\prime})\in\mathfrak{c}\text{\leavevmode\nobreak\ for
some\leavevmode\nobreak\ }t\in S\big{\\}}.$

Note that $S^{-1}\mathfrak{c}$ is a congruence of $S^{-1}A$. Localizations of
monoids are exact in the following sense.

######  Lemma .

Let $f:A\to B$ be a morphism of pointed monoids. Let $S$ be a multiplicative
subset of $A$ and $f_{S}:S^{-1}A\to f(S)^{-1}B$ the induced morphism. Then
$\ker(f_{S})=S^{-1}\ker(f)$ and
$\operatorname{congker}(f_{S})=S^{-1}\operatorname{congker}(f)$. In
particular, $f_{S}$ is injective if $f$ is so.

###### Proof.

An element $\frac{a}{s}\in S^{-1}A$ is in the kernel of $f_{S}$ if and only if
$\frac{f(a)}{f(s)}=\frac{0}{1}$, which means that
$f(t)f(1)f(a)=f(t)f(s)f(0)=0$ for some $t\in S$. Equivalently $ta\in\ker(f)$
or, in other words, $\frac{a}{s}=\frac{at}{st}\in S^{-1}(\ker(f))$. This shows
$\ker(f_{S})=S^{-1}(\ker(f))$.

A pair $(\frac{a}{s},\frac{a^{\prime}}{s^{\prime}})$ is in the congruence
kernel of $f_{S}$ if and only if
$f_{S}(\frac{a}{s})=f_{S}(\frac{a^{\prime}}{s^{\prime}})$, which means that
$f(tsa^{\prime})=f(t)f(s)f(a^{\prime})=f(t)f(s^{\prime})f(a)=f(ts^{\prime}a)$
for some $t\in S$. Equivalently
$(tsa^{\prime},ts^{\prime}a)\in\operatorname{congker}(f)$ or, in other words,
$\frac{a}{s}=\frac{at}{st}\in S^{-1}(\ker(f))$. This shows
$\operatorname{congker}(f_{S})=S^{-1}(\operatorname{congker}(f))$.

If $f$ is injective, then $\operatorname{congker}(f)$ is the trivial
congruence ${\mathfrak{c}}_{\textup{triv}}=\\{(a,a)\in\mid a\in A\\}$. Thus
$\operatorname{congker}(f_{S})=S^{-1}{\mathfrak{c}}_{\textup{triv}}$ is
trivial and consequently $f_{S}$ is injective. ∎

######  Proposition .

Let $\myiota_{S}:A\to S^{-1}A$ be a localization of the pointed monoid $A$ at
$S$. Then the maps

${\big{\\{}\begin{array}[]{c}\text{prime congruences on
}S^{-1}A\end{array}\big{\\}}}$${\Big{\\{}\begin{array}[]{c}\text{prime
congruences ${\mathfrak{c}}$}\\\ \text{on $A$ with $I_{\mathfrak{c}}\cap
S=\emptyset$}\end{array}\Big{\\}}}$$\scriptstyle{\myiota_{S}^{\ast}}$$\scriptstyle{\myiota_{S,\ast}}$

are mutually inverse bijections.

###### Proof.

We first check that the maps are well defined. Let ${\mathfrak{d}}$ be a prime
congruence on $S^{-1}A$. Then
$(ac,bc)\in\myiota_{S}^{\ast}({\mathfrak{d}})=\\{(a,a^{\prime})\in A\times
A\leavevmode\nobreak\ |\leavevmode\nobreak\
(\frac{a}{1},\frac{a^{\prime}}{1})\in{\mathfrak{d}}\\}$ implies
$(\frac{ac}{1},\frac{bc}{1})\in{\mathfrak{d}}$ which implies
$(\frac{c}{1},\frac{0}{1})\in{\mathfrak{d}}$ or
$(\frac{ac}{1},\frac{bc}{1})\in{\mathfrak{d}}$. Thus we have
$(c,0)\in\myiota_{S}^{\ast}({\mathfrak{d}})$ or
$(a,b)\in\myiota_{S}^{\ast}({\mathfrak{d}})$ which shows that
$\myiota_{S}^{\ast}({\mathfrak{d}})$ is a prime congruence on $A$. Also, if
$(\frac{a}{1},\frac{0}{1})\in{\mathfrak{d}}$ then clearly
$\frac{a}{1}\notin(S^{-1}A)^{\times}$ which implies $a\notin S$. Thus
$I_{\myiota_{S}^{\ast}({\mathfrak{d}})}\cap S=\emptyset$ and this shows that
$\myiota_{S}^{\ast}$ is well defined.

Let ${\mathfrak{c}}$ be a prime congruence on $A$ such that
$I_{\mathfrak{c}}\cap S=\emptyset$. Consider
$(\frac{a}{s}\frac{c}{r},\frac{b}{t}\frac{c}{r})\in\myiota_{S,\ast}({\mathfrak{c}})=\langle\\{(\frac{a}{1},\frac{b}{1})\in
S^{-1}A\times S^{-1}A\leavevmode\nobreak\ |\leavevmode\nobreak\
(a,b)\in{\mathfrak{c}}\\}\rangle$. Since
$\frac{a}{s}\frac{c}{r}=\frac{act}{srt}$ and
$\frac{b}{t}\frac{c}{r}=\frac{bcs}{trs}$ in $S^{-1}A$, we have
$(\frac{act}{srt},\frac{bcs}{trs})\in\myiota_{S,\ast}({\mathfrak{c}})$.
Therefore $(\frac{act}{1},\frac{bcs}{1})\in\myiota_{S,\ast}({\mathfrak{c}})$
which implies $(actr^{\prime},bcsr^{\prime})\in{\mathfrak{c}}$ for some
$r^{\prime}\in S$. Thus either $(cr^{\prime},0)\in{\mathfrak{c}}$ or
$(at,bs)\in{\mathfrak{c}}$. This implies either
$(\frac{cr^{\prime}}{1},\frac{0}{1})\in\myiota_{S,\ast}({\mathfrak{c}})$ or
$(\frac{at}{1},\frac{bs}{1})\in\myiota_{S,\ast}({\mathfrak{c}})$. Since
$(\frac{cr^{\prime}}{1},\frac{0}{1})\in\myiota_{S,\ast}({\mathfrak{c}})$
implies
$(\frac{cr^{\prime}}{r^{\prime}},\frac{0r^{\prime}}{r^{\prime}})=(\frac{c}{1},\frac{0}{1})\in\myiota_{S,\ast}({\mathfrak{c}})$
and $(\frac{at}{1},\frac{bs}{1})\in\myiota_{S,\ast}({\mathfrak{c}})$ implies
$(\frac{at}{ts},\frac{bs}{ts})=(\frac{a}{s},\frac{b}{t})\in\myiota_{S,\ast}({\mathfrak{c}})$
it follows that $\myiota_{S,\ast}({\mathfrak{c}})$ is a prime congruence on
$S^{-1}A$. Therefore $\myiota_{S,\ast}$ is well defined.

Clearly, we have
${\mathfrak{c}}\subseteq\myiota_{S}^{*}(\myiota_{S,\ast}({\mathfrak{c}}))$ and
${\mathfrak{d}}\subseteq\myiota_{S,\ast}(\myiota_{S}^{\ast}({\mathfrak{d}}))$.
Consider any
$(a,a^{\prime})\in\myiota_{S}^{*}(\myiota_{S,\ast}({\mathfrak{c}}))$. Then
$(\frac{a}{1},\frac{a^{\prime}}{1})\in\myiota_{S,\ast}({\mathfrak{c}})$ which
implies $(sa,sa^{\prime})\in{\mathfrak{c}}$ for some $s\in S$. Since
$I_{\mathfrak{c}}\cap S=\emptyset$, we have $(s,0)\notin{\mathfrak{c}}$ and
therefore we must have $(a,a^{\prime})\in{\mathfrak{c}}$ as desired. Consider
any
$(\frac{a}{s},\frac{b}{t})\in\myiota_{S,\ast}(\myiota_{S}^{\ast}({\mathfrak{d}}))$.
Then
$(\frac{ast}{s},\frac{bst}{t})=(\frac{at}{1},\frac{bs}{1})\in\myiota_{S,\ast}(\myiota_{S}^{\ast}({\mathfrak{d}}))$.
This implies $(atr,bsr)\in\myiota_{S}^{\ast}({\mathfrak{d}})$ for some $r\in
S$. But $(r,0)\notin\myiota_{S}^{\ast}({\mathfrak{d}})$ since
$I_{\myiota_{S}^{\ast}({\mathfrak{d}})}\cap S=\emptyset$. Therefore
$(at,bs)\in\myiota_{S}^{\ast}({\mathfrak{d}})$ which implies
$(\frac{at}{1},\frac{bs}{1})\in{\mathfrak{d}}$ or equivalently,
$(\frac{a}{s},\frac{b}{t})\in{\mathfrak{d}}$. This shows that
$\myiota_{S}^{\ast}$ and $\myiota_{S,\ast}$ are mutually inverse bijections. ∎

###  2.3. The congruence space of a pointed monoid

######  Definition .

Let $A$ be a pointed monoid. The _congruence space of $A$_ is the topological
space $\operatorname{Cong}(A)$ of all prime congruences on $A$ whose topology
is generated by the open subsets

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

where $a$ and $b$ vary through all pairs of elements of $A$. We denote the
complement $U_{a,b}$ by
$V_{a,b}=\\{{\mathfrak{p}}\in\operatorname{Cong}(A)\mid(a,b)\in{\mathfrak{p}}\\}$.

######  Example .

The collection $\\{U_{a,b}\mid a,b\in A\\}$ of open subsets is in general not
a basis for the topology on $\operatorname{Cong}(A)$, as in the example of the
pointed monoid $A=\mathbb{F}_{1}[t]$. We have

$\operatorname{Cong}(A)\ =\ \big{\\{}\mathfrak{c}_{\textup{triv}},\
\langle(t,0)\rangle,\ \langle(t,t^{k})\rangle\mid k\geqslant 0\big{\\}}.$

The topology of $\operatorname{Cong}(A)$ is generated by the open subsets of
the forms $U_{t^{n},0}$ and $U_{t^{n},t^{m}}$ for $n>m\geqslant 0$. The
intersection $U_{t,0}\cap U_{t,1}$ contains the prime congruence $\langle
t,t^{2}\rangle$, but it does not contain any non-empty open subset of the form
$U_{a,b}$. This shows that the family $\\{U_{a,b}\mid a,b\in A\\}$ does not
form a basis for the topology of $\operatorname{Cong}(A)$, but only a
subbasis.

The following result upgrades the congruence space to a functor
$\operatorname{Cong}:\operatorname{Mon}\to\operatorname{Top}$.

######  Lemma .

Let $f:A\to B$ be a morphism of pointed monoids and ${\mathfrak{q}}$ a prime
congruence of $B$. Then $f^{\ast}({\mathfrak{q}})$ is a prime congruence of
$A$, and the map $f^{\ast}:\operatorname{Cong}(B)\to\operatorname{Cong}(A)$ is
continuous.

###### Proof.

We begin with the verification that ${\mathfrak{p}}=f^{\ast}({\mathfrak{q}})$
is prime. Consider $(ca,cb)\in{\mathfrak{p}}$, i.e. $f(c)f(a)=f(ca)\sim
f(cb)=f(c)f(b)$ in ${\mathfrak{q}}$. Since ${\mathfrak{q}}$ is prime, this
means that either $f(c)\sim 0$ or $f(a)\sim f(b)$ in ${\mathfrak{q}}$. Thus
$c\sim 0$ or $a\sim b$ in ${\mathfrak{p}}$, which shows that ${\mathfrak{p}}$
is prime.

It suffices to verify the continuity of $f^{\ast}$ on a subbasis. For
$U_{a_{1},a_{2}}\subseteq\operatorname{Cong}(A)$, we have

$\displaystyle(f^{\ast})^{-1}(U_{a_{1},a_{2}})\ $ $\displaystyle=\
\\{\mathfrak{d}\in\operatorname{Cong}(B)\mid(a_{1},a_{2})\notin
f^{\ast}(\mathfrak{d})\\}$ $\displaystyle=\
\\{\mathfrak{d}\in\operatorname{Cong}(B)\mid(f(a_{1}),f(a_{2}))\notin\mathfrak{d})\\}=\
U_{f(a_{1}),f(a_{2})}$

which shows $f^{\ast}$ is continuous. ∎

###  2.4. The congruence space of a monoid scheme

The results from section 2.2 allow us to extend the definition of the
congruence space from pointed monoids, or affine monoid schemes, to arbitrary
monoid schemes.

######  Definition .

Let $X$ be a monoid scheme and $\mathcal{U}_{X}$ the category of all affine
open subschemes of $X$ together with all open immersions. Let
${\mathcal{U}}^{\textup{cong}}_{X}$ be the category of congruence spaces
$U^{\textup{cong}}=\operatorname{Cong}(\Gamma U)$ where $U\in\mathcal{U}_{X}$
together with the induced maps. The congruence space associated with $X$ is
the topological space

$X^{\textup{cong}}\ =\ \operatorname{colim\,}\mathcal{U}^{\textup{cong}}_{X}.$

It comes with canonical inclusions $\myiota_{U}:\operatorname{Cong}(\Gamma
U)\to X^{\textup{cong}}$ for every affine open $U$ of $X$.

Note that since every affine open covering of an affine monoid scheme
$X=\operatorname{MSpec}(A)$ must contain $X$ itself as its largest open
subset, the canonical inclusion
$\myiota_{\operatorname{MSpec}(A)}:\operatorname{Cong}(A)\to(\operatorname{MSpec}(A))^{\textup{cong}}$
is a homeomorphism. The term “canonical inclusions” is justified by the
following result.

Recall that a morphism $\myiota:Y\to X$ of monoid schemes is an _open
immersion_ if it is an open topological embedding and if $\mathcal{O}_{Y}$ is
the restriction of $\mathcal{O}_{X}$ to $Y$.

######  Proposition .

The assignment $X\mapsto X^{\textup{cong}}$ extends uniquely to a functor
$(-)^{\textup{cong}}:\operatorname{MSch}\to\operatorname{Top}$ such that for
every morphism $f:A\to B$ of pointed monoids and
$\myvarphi=f^{\ast}:\operatorname{MSpec}(B)\to\operatorname{MSpec}(A)$, the
diagram

${\operatorname{Cong}(B)}$${\operatorname{Cong}(A)}$${(\operatorname{MSpec}(B))^{\textup{cong}}}$${(\operatorname{MSpec}(A))^{\textup{cong}}}$$\scriptstyle{\myvarphi}$$\scriptstyle{\myiota_{\operatorname{MSpec}(B)}}$$\scriptstyle{\cong}$$\scriptstyle{\myiota_{\operatorname{MSpec}(A)}}$$\scriptstyle{\cong}$$\scriptstyle{\myvarphi^{\textup{cong}}}$

commutes. It satisfies that

  1. (1)

if $\myiota:Y\to X$ is an open immersion, then
$\myiota^{\textup{cong}}:Y^{\textup{cong}}\to X^{\textup{cong}}$ is an open
topological embedding;

  2. (2)

if $\\{U_{i}\\}$ is an open covering of $X$, then
$\\{U_{i}^{\textup{cong}}\\}$ is an open covering of $X^{\textup{cong}}$.

###### Proof.

The functor $(-)^{\textup{cong}}:\operatorname{MSch}\to\operatorname{Top}$ is
defined on objects, and the commutativity requirement determines
$\myvarphi^{\textup{cong}}$ for morphisms $\myvarphi:X\to Y$ of affine monoid
schemes. If we can extend $(-)^{\textup{cong}}$ to a functor on all monoid
schemes, then uniqueness follows from the functoriality of
$(-)^{\textup{cong}}$ and the local nature of morphisms; namely, given
$\myvarphi:Y\to X$, there are affine open coverings $\\{U_{i}\\}$ of $X$ and
$\\{V_{i}\\}$ of $Y$ with $\myvarphi(V_{i})\subseteq U_{i}$, and $\myvarphi$
is uniquely determined by its restrictions $\myvarphi_{i}:V_{i}\to U_{i}$ to
affine monoid schemes.

We are left with the construction of
$\myvarphi^{\textup{cong}}:Y^{\textup{cong}}\to X^{\textup{cong}}$ for an
arbitrary morphism $\myvarphi:Y\to X$. As a first step, we establish property
(1) for an open immersion $\myiota:Y\to X$ between affine monoid schemes
$Y=\operatorname{MSpec}(B)$ and $X=\operatorname{MSpec}(A)$. Since
$i:Y\hookrightarrow X$ is an open immersion, it follows from [7, Lemma 1.3]
and [7, Lemma 2.4] that $B\operatorname{Cong}(A)[s^{-1}]$ for some $s\in A$.
Next we verify that $i^{\textup{cong}}$ is injective. Let
$\mathfrak{c},\mathfrak{d}\in{Y}^{\textup{cong}}$ be such that
$i^{\textup{cong}}(\mathfrak{c})=i^{\textup{cong}}(\mathfrak{d})$, i.e.,
$\\{(a,a^{\prime})\in A\times
A\mid(\frac{a}{1},\frac{a^{\prime}}{1})\in\mathfrak{c}\\}=\\{(a,a^{\prime})\in
A\times A\mid(\frac{a}{1},\frac{a^{\prime}}{1})\in\mathfrak{d}\\}$. Then we
have

$(\frac{a_{1}}{s^{n}},\frac{a_{2}}{s^{m}})\in\mathfrak{c}\iff(\frac{a_{1}s^{m}}{1},\frac{a_{2}s^{n}}{1})\in\mathfrak{c}\iff(\frac{a_{1}s^{m}}{1},\frac{a_{2}s^{n}}{1})\in\mathfrak{d}\iff(\frac{a_{1}}{s^{n}},\frac{a_{2}}{s^{m}})\in\mathfrak{d}.$

That $i^{\textup{cong}}$ is an open embedding follows if we can show that any
open subset of $Y^{\textup{cong}}$ of the form
$U_{\frac{a_{1}}{s^{n}},\frac{a_{2}}{s^{m}}}$ maps to an open subset of
$X^{\textup{cong}}$ under $i^{\textup{cong}}$. Note that we have
$i^{\textup{cong}}(U_{\frac{a_{1}}{s^{n}},\frac{a_{2}}{s^{m}}})\subseteq
U_{a_{1}s^{m},a_{2}s^{n}}$. Consider any $\mathfrak{d}\in
U_{a_{1}s^{m},a_{2}s^{n}}$. We define

$\mathfrak{c}:=\\{(\frac{a}{s^{k}},\frac{a^{\prime}}{s^{r}})\in B\times
B\mid(as^{r+n_{0}},a^{\prime}s^{k+n_{0}})\in\mathfrak{d}\text{\leavevmode\nobreak\
for some\leavevmode\nobreak\ }n_{0}>0\\}.$

We leave it to the reader to check that
$\mathfrak{c}\in({\operatorname{MSpec}(B)})^{\textup{cong}}$ and that
$i^{\textup{cong}}(\mathfrak{c})=\mathfrak{d}$. Thus we have
$i^{\textup{cong}}(U_{\frac{a_{1}}{s^{n}},\frac{a_{2}}{s^{m}}})=U_{a_{1}s^{m},a_{2}s^{n}}$.

As the next step of the proof, we establish property (2). We note that it is
enough to prove property (2) for an affine open covering of $X$. Consider the
family $\mathcal{U}_{X}=\\{U_{i}\\}_{i\in I}$ of all affine open subschemes of
$X$. For all $i,j\in I$, consider principal affine opens $U_{ij}\subseteq
U_{i}$ and isomorphisms $\myvarphi_{ji}:U_{ij}\to U_{ji}$. As already shown,
this implies $U_{ij}^{\textup{cong}}$ are open subsets of
${U_{i}}^{\textup{cong}}$ for all $i,j\in I$. Moreover, since
$\myvarphi_{ki}=\myvarphi_{kj}\circ\myvarphi_{ji}$ on $U_{ij}\cap U_{jk}$ and
$\operatorname{Cong}:\operatorname{Mon}_{0}\to\operatorname{Top}$ is a functor
by Lemma 2.3, we have
$\myvarphi_{ki}^{\textup{cong}}=\myvarphi_{kj}^{\textup{cong}}\circ\myvarphi_{ji}^{\textup{cong}}$
on $U_{ij}^{\textup{cong}}\cap U_{jk}^{\textup{cong}}$. Thus the collection
$(I,{U_{i}^{\textup{cong}}}\\}_{i\in I},\\{{U_{ij}^{\textup{cong}}}\\}_{i,j\in
I},\\{\myvarphi_{ij}^{\textup{cong}}\\}_{i,j\in I})$ gives a gluing data of
topological spaces (in other words, ${{U}_{X}^{\textup{cong}}}$ is a
monodromy-free diagram in the category of topological spaces in the sense of
[21, §1]). The topological space obtained from this gluing data is
${X}^{\textup{cong}}$, by definition. Therefore for any
$U_{i}\in\mathcal{U}_{X}$, we have ${U_{i}}^{\textup{cong}}$ is an open subset
of ${X}^{\textup{cong}}$. Let $\\{X_{\myalpha}\\}_{\myalpha\in A}$ be an
affine open cover of $X$ and consider any $U\in\mathcal{U}_{X}$. Since every
affine open has a unique closed point and since $U$ is covered by $\\{U\cap
X_{\myalpha}\\}_{\myalpha\in A}$, we must have $U\subseteq X_{\myalpha}$ for
some $\myalpha\in A$. Therefore we have ${U}^{\textup{cong}}\subseteq
X_{\myalpha}^{\textup{cong}}$ for some $\myalpha\in A$. Since we have shown
that $\\{{U}^{\textup{cong}}\\}_{U\in\mathcal{U}_{X}}$ is an open cover for
${X}^{\textup{cong}}$, it follows that
$\\{X_{\myalpha}^{\textup{cong}}\\}_{\myalpha\in A}$ is an open cover of
${X}^{\textup{cong}}$.

We are all set to define $\myvarphi^{\textup{cong}}:Y^{\textup{cong}}\to
X^{\textup{cong}}$ for an arbitrary morphism $\myvarphi:Y\to X$. We choose an
affine open cover $\\{X_{\myalpha}\\}_{\myalpha\in A}$ of $X$ and an affine
open cover $\\{Y_{\myalpha}\\}_{\myalpha\in A}$ of $Y$ such that
$\myvarphi(Y_{\myalpha})\subseteq X_{\myalpha}$ for each $\myalpha\in A$. We
also choose an affine cover $\\{Y_{\myalpha\mybeta k}\\}_{k\in K}$ for each
$Y_{\myalpha\mybeta}=Y_{\myalpha}\cap Y_{\mybeta}$ and an affine cover
$\\{X_{\myalpha\mybeta k}\\}_{k\in K}$ for each
$X_{\myalpha\mybeta}=X_{\myalpha}\cap X_{\mybeta}$ such that
$\myvarphi(Y_{\myalpha\mybeta k})\subseteq X_{\myalpha\mybeta k}$ for each
$k\in K$. Let $\myvarphi_{\myalpha}:=\myvarphi|_{Y_{\myalpha}}:Y_{\myalpha}\to
X_{\myalpha}$ and $\myvarphi_{\myalpha\mybeta
k}:=\myvarphi|_{Y_{\myalpha\mybeta k}}:Y_{\myalpha\mybeta k}\to
X_{\myalpha\mybeta k}$. The corresponding morphisms of congruence spaces are
given by $\myvarphi_{\myalpha}^{\textup{cong}}:Y_{\myalpha}^{\textup{cong}}\to
X_{\myalpha}^{\textup{cong}}$ and $\myvarphi_{\myalpha\mybeta
k}^{\textup{cong}}:Y_{\myalpha\mybeta k}^{\textup{cong}}\to X_{\myalpha\mybeta
k}^{\textup{cong}}$. We define

$\displaystyle{\myvarphi}^{\textup{cong}}:{Y}^{\textup{cong}}=\bigcup_{\myalpha\in
A}Y_{\myalpha}^{\textup{cong}}$
$\displaystyle\longrightarrow{X}^{\textup{cong}}=\bigcup_{\myalpha\in
A}X_{\myalpha}^{\textup{cong}}\quad,\quad\mathfrak{q}$
$\displaystyle\mapsto\myvarphi_{\myalpha}^{\textup{cong}}(\mathfrak{q})\quad\text{
if\leavevmode\nobreak\ }\mathfrak{q}\in Y_{\myalpha}^{\textup{cong}}.$

Assume $\mathfrak{q}\in{Y_{\mybeta}}^{\textup{cong}}$ for a different choice
of affine open $Y_{\mybeta}$ of $Y$. Since $\\{Y_{\myalpha\mybeta k}\\}_{k\in
K}$ is an affine cover of $Y_{\myalpha\mybeta}$, we know that
$\\{Y_{\myalpha\mybeta k}^{\textup{cong}}\\}_{k\in K}$ is an open cover of
$Y_{\myalpha\mybeta}^{\textup{cong}}$. Let $\mathfrak{q}\in Y_{\myalpha\mybeta
k}^{\textup{cong}}$ for some $k\in K$. Since
$\operatorname{Cong}:\operatorname{Mon}_{0}\to\operatorname{Top}$ is a functor
by Lemma 2.3, we have
$\myvarphi_{\mybeta}^{\textup{cong}}(\mathfrak{q})=\myvarphi_{\myalpha\mybeta
k}^{\textup{cong}}(\mathfrak{q})=\myvarphi_{\myalpha}^{\textup{cong}}(\mathfrak{q})$.
This shows that there is no ambiguity in the choice of congruence that
represents the point $\mathfrak{q}$ in $Y^{\textup{cong}}$. We leave it to the
reader to verify that ${\myvarphi}^{\textup{cong}}$ is also independent of the
choice of affine covers considered in its definition by taking suitable common
refinements and using similar arguments as above. This shows that
$\myvarphi^{\textup{cong}}:Y^{\textup{cong}}\to X^{\textup{cong}}$ is well
defined as a map. It is clear that $\myvarphi^{\textup{cong}}$ is continuous
since each $\myvarphi_{\myalpha}^{\textup{cong}}$ is continuous by Lemma 2.3.

The only thing left to show is property (1) for general $X$ and $Y$. We note
that an open immersion $\myiota:Y\to X$ is a morphism of monoid schemes which
decomposes uniquely into an isomorphism of monoid schemes and the identity
inclusion of an open subscheme. We have already shown that for any open
subscheme $U$ of $X$, we have $U^{\textup{cong}}$ is an open subspace of
$X^{\textup{cong}}$ and we leave it to the reader to verify that any
isomorphism of monoid schemes induces a homeomorphism of congruence spaces. It
follows that $\myiota^{\textup{cong}}:Y^{\textup{cong}}\to X^{\textup{cong}}$
is an open topological embedding. ∎

######  Notation.

Given a morphism $\myvarphi:Y\to X$ of monoid schemes, we often write
$\widetilde{\myvarphi}:\widetilde{Y}\to\widetilde{X}$ for
$\myvarphi^{\textup{cong}}:Y^{\textup{cong}}\to X^{\textup{cong}}$.

###  2.5. Projection onto the monoid scheme

The congruence space $X^{\textup{cong}}$ of a monoid scheme $X$ comes equipped
with a continuous surjection $\mypi_{X}:X^{\textup{cong}}\to X$, which sends a
prime congruence ${\mathfrak{p}}$ to its nullideal $I_{\mathfrak{p}}$ in the
affine case. More precisely, $\mypi_{X}$ is characterized by the following
properties.

######  Proposition .

There are continuous maps $\mypi_{X}:X^{\textup{cong}}\to X$, one for every
monoid scheme $X$, which are uniquely determined by the following properties:

  1. (1)

If $X=\operatorname{MSpec}(A)$ is affine, then
$\mypi_{X}({\mathfrak{p}})=I_{\mathfrak{p}}$ for all prime congruences
${\mathfrak{p}}\in X^{\textup{cong}}$.

  2. (2)

The maps $\mypi_{X}$ are functorial in $X$, i.e.,

${Y^{\textup{cong}}}$${X^{\textup{cong}}}$${Y}$${X}$$\scriptstyle{\mypi_{Y}}$$\scriptstyle{\myvarphi^{\textup{cong}}}$$\scriptstyle{\mypi_{X}}$$\scriptstyle{\myvarphi}$

commutes for every morphism $\myvarphi:Y\to X$ of monoid schemes.

###### Proof.

Let $X=\operatorname{MSpec}(A)$ and let $\mathfrak{p}\in X^{\textup{cong}}$ be
any prime congruence on $A$. It is clear that $I_{\mathfrak{p}}$ is a prime
ideal of $A$ and that $\mypi_{X}$ is a well defined morphism of sets. Also for
any basic open $D(a)=\\{P\in X\mid a\notin P\\}$ of $\operatorname{MSpec}(A)$,
we have ${\mypi_{X}}^{-1}(D(a))=\\{\mathfrak{p}\in{X}^{\textup{cong}}\mid
a\notin I_{\mathfrak{p}}\\}=U_{a,0}$ and so $\mypi_{X}$ is a continuous map of
topological spaces. Moreover, for $X=\operatorname{MSpec}(A)$ and
$Y=\operatorname{MSpec}(B)$, the diagram in (2) commutes since for any
${\mathfrak{p}}$ in $(\operatorname{MSpec}(A))^{\textup{cong}}$, we have

$\mypi_{Y}(\myvarphi^{\textup{cong}}({\mathfrak{p}}))=\ \\{b\in
B\mid(\Gamma_{\myvarphi}(b),0)\in{\mathfrak{p}}\\}=\Gamma_{\myvarphi}^{-1}(I_{\mathfrak{p}})=\myvarphi(I_{\mathfrak{p}}).$

For a general monoid scheme $X$, every ${\mathfrak{p}}\in X^{\textup{cong}}$
is contained in $U^{\textup{cong}}$ for an affine open subscheme
$U=\operatorname{MSpec}(A)$ of $X$ and corresponds to a prime congruence
${\mathfrak{p}}$ of $A$. We define
$\mypi_{X}({\mathfrak{p}}):=I_{\mathfrak{p}}$. Let
$U^{\prime}=\operatorname{MSpec}(A)[s^{-1}]$ be an affine open of
$U=\operatorname{MSpec}(A)$ containing ${\mathfrak{p}}$. Let
${\mathfrak{p}}^{\prime}$ be the prime congruence of $A[s^{-1}]$ which
corresponds to the prime congruence ${\mathfrak{p}}$ of $A$. Since the
following diagram

${(\operatorname{MSpec}(A)[s^{-1}])^{\textup{cong}}}$${(\operatorname{MSpec}(A))^{\textup{cong}}}$${\operatorname{MSpec}(A)[s^{-1}]}$${\operatorname{MSpec}(A)}$$\scriptstyle{\mypi_{U^{\prime}}}$$\scriptstyle{\myvarphi^{\textup{cong}}}$$\scriptstyle{\mypi_{U}}$$\scriptstyle{\myvarphi}$

commutes, it shows that $I_{\mathfrak{p}}$ and $I_{{\mathfrak{p}}^{\prime}}$
represents the same point in $X$. Hence if ${\mathfrak{p}}$ is contained in
$V^{\textup{cong}}$ for some other affine open $V=\operatorname{MSpec}(B)$ of
$X$, by taking common affine refinement of $U$ and $V$ and using similar
argument as above, we can conclude that $\mypi_{X}:X^{\textup{cong}}\to X$ is
a well defined map. For general $X$ and $Y$, the morphism
$\myvarphi^{\textup{cong}}:Y^{\textup{cong}}\to X^{\textup{cong}}$ is defined
by taking restriction to suitable affine pieces and therefore (2) follows from
the affine case.

To check that $\mypi_{X}:X^{\textup{cong}}\to X$ is continuous consider an
affine open cover $\\{X_{\myalpha}\\}_{\myalpha\in A}$ of $X$. Then
$\\{X_{\myalpha}^{\textup{cong}}\\}_{\myalpha\in A}$ is an open cover of
$X^{\textup{cong}}$ and since each
$\mypi_{X_{\myalpha}}:X_{\myalpha}^{\textup{cong}}\to X_{\myalpha}$ is
continuous it follows that $\mypi_{X}:X^{\textup{cong}}\to X$ is continuous. ∎

######  Example .

In this example, we describe the congruence space of the affine $n$-space
${\mathbb{A}}^{n}_{{\mathbb{F}}_{1}}=\operatorname{MSpec}({{\mathbb{F}}_{1}}[t_{1},\dotsc,t_{n}])$.
By [7, Example 1.2], the prime ideals of
${{\mathbb{F}}_{1}}[t_{1},\dotsc,t_{n}]$ are of the form $P_{I}=\langle
t_{i}\rangle_{i\in I}$ where $I$ is a subset of $\\{1,\dotsc,n\\}$. Let
$J=\\{1,\dotsc,n\\}-I$. Then the quotient
${{\mathbb{F}}_{1}}[t_{1},\dotsc,t_{n}]/P_{I}$ is isomorphic to
${{\mathbb{F}}_{1}}[t_{j}\mid j\in J]$. The quotients of
${{\mathbb{F}}_{1}}[t_{j}\mid j\in J]$ by prime congruence ${\mathfrak{p}}$
with trivial nullideal $I_{\mathfrak{p}}=\langle 0\rangle$ are of the form

${\mathfrak{p}}_{H}\ =\ \Big{\langle}\Big{(}\ \prod_{j\in J}\ t_{j}^{a_{j}},\
\prod_{j\in J}\ t_{j}^{b_{j}}\ \Big{)}\ \Big{|}\ (a_{j}-b_{j})_{i\in J}\in
H\Big{\rangle}$

where $H$ is a subgroup of ${\mathbb{Z}}^{J}$. Piecing these two types of
quotients together, we see that the prime congruences of
${{\mathbb{F}}_{1}}[t_{1},\dotsc,t_{n}]$ are of the form

${\mathfrak{p}}_{I,H}\ =\ \big{\langle}\ (t_{i},0),(s,s^{\prime})\ \big{|}\
i\in I,\ (s,s^{\prime})\in{\mathfrak{p}}_{H}\ \big{\rangle}$

for subsets $I\subset\\{1,\dotsc,n\\}$ and subgroups $H$ of ${\mathbb{Z}}^{J}$
where $J=\\{1,\dotsc,n\\}-I$.

The congruence spaces of the affine line ${\mathbb{A}}^{1}_{{\mathbb{F}}_{1}}$
and the affine plane ${\mathbb{A}}^{2}_{{\mathbb{F}}_{1}}$ over
${{\mathbb{F}}_{1}}$ are illustrated in Figure 2. The green areas indicate
closed subsets of the respective underlying topological spaces. While both
${\mathbb{A}}^{1}_{{\mathbb{F}}_{1}}$ and
${\mathbb{A}}^{2}_{{\mathbb{F}}_{1}}$ are finite, and all points and closed
subsets appear in the illustration, both
${\widetilde{{\mathbb{A}}}}^{1}_{{\mathbb{F}}_{1}}$ and
${\widetilde{{\mathbb{A}}}}^{2}_{{\mathbb{F}}_{1}}$ are infinite, i.e. the
illustrations only capture a few of their points and closed subsets.

The horizontal arrow is the diagonal embedding
$\Delta:{\mathbb{A}}^{1}_{{\mathbb{F}}_{1}}\to{\mathbb{A}}^{2}_{{\mathbb{F}}_{1}}$.
Remarkable is that the image of $\Delta$ (in orange) is not closed in
${\mathbb{A}}^{2}_{{\mathbb{F}}_{1}}$, but that the image of $\tilde{\Delta}$
(in light green) is closed in
${\widetilde{{\mathbb{A}}}}^{2}_{{\mathbb{F}}_{1}}$.

$P_{\emptyset}$$P_{1}$$P_{\emptyset}$$P_{2}$$P_{1}$$P_{12}$$\Delta$${\mathfrak{p}}_{\emptyset,{\mathbb{Z}}}$$\dotsb$${\mathfrak{p}}_{\emptyset,n{\mathbb{Z}}}$$\dotsb$${\mathfrak{p}}_{\emptyset,0}$${\mathfrak{p}}_{1,0}$${\mathfrak{p}}_{\emptyset,{\mathbb{Z}}^{2}}$${\mathfrak{p}}_{2,{\mathbb{Z}}}$${\mathfrak{p}}_{1,{\mathbb{Z}}}$${\mathfrak{p}}_{\emptyset,{\mathbb{Z}}\oplus
0}$${\mathfrak{p}}_{\emptyset,0\oplus{\mathbb{Z}}}$${\mathfrak{p}}_{12,0}$${\mathfrak{p}}_{2,0}$${\mathfrak{p}}_{1,0}$${\mathfrak{p}}_{\emptyset,\langle(1,-1)\rangle}$${\mathfrak{p}}_{\emptyset,0}$$\tilde{\Delta}$$\mypi_{{\mathbb{A}}^{1}_{{\mathbb{F}}_{1}}}$$\mypi_{{\mathbb{A}}^{2}_{{\mathbb{F}}_{1}}}$${\mathbb{A}}^{1}_{{\mathbb{F}}_{1}}$${\widetilde{{\mathbb{A}}}}^{1}_{{\mathbb{F}}_{1}}$${\mathbb{A}}^{2}_{{\mathbb{F}}_{1}}$${\widetilde{{\mathbb{A}}}}^{2}_{{\mathbb{F}}_{1}}$
Figure 2. The diagonal embeddings
$\Delta:{\mathbb{A}}^{1}_{{\mathbb{F}}_{1}}\to{\mathbb{A}}^{2}_{{\mathbb{F}}_{1}}$
and
$\tilde{\Delta}:{\widetilde{{\mathbb{A}}}}^{1}_{{\mathbb{F}}_{1}}\to{\widetilde{{\mathbb{A}}}}^{2}_{{\mathbb{F}}_{1}}$

###  2.6. Stalks and residue fields

The points of a monoid scheme $X$ come with the notion of a stalk and a
residue field, which are as follows: let ${\mathcal{U}}_{x}$ be the system of
all affine open neighbourhood of the point $x$ in $X$. The stalk at $x$ is the
pointed monoid

${\mathcal{O}}_{X,x}\ =\
\underset{U\in{\mathcal{U}}_{x}}{\operatorname{colim\,}}\ \Gamma U$

and the residue field at $x$ is the pointed group

$k(x)\ =\ {\mathcal{O}}_{X,x}\,/\,M_{x}$

where $M_{x}$ is the maximal ideal of ${\mathcal{O}}_{X,x}$. Stalks and
residue fields come with canonical morphisms

${\myomega_{x}:\operatorname{MSpec}({\mathcal{O}}_{X,x})\ \longrightarrow\
X\qquad\text{and}\qquad\mykappa_{x}:\operatorname{MSpec}(k(x))\
\longrightarrow\ X}$

that commute with the morphism
$\operatorname{MSpec}(k(x))\to\operatorname{MSpec}({\mathcal{O}}_{X,x})$ that
is induced by the quotient map ${\mathcal{O}}_{X,x}\to k(x)$.

Both stalks and residue fields generalize to invariants for points of the
congruence space of $X$ as follows.

######  Definition .

Let $X$ be a monoid scheme with congruence space ${\widetilde{X}}$ and
projection $\mypi_{X}:{\widetilde{X}}\to X$. Let $\tilde{x}\in{\widetilde{X}}$
and $x=\mypi_{X}(\tilde{x})$. The _stalk at $\tilde{x}$_ is the pointed monoid

${\mathcal{O}}_{{\widetilde{X}},\tilde{x}}\ =\
\underset{U\in{\mathcal{U}}_{x}}{\operatorname{colim\,}}\ \Gamma U\ =\
{\mathcal{O}}_{X,x}.$

where the colimit is a filtered colimit with respect to the restriction maps
$\operatorname{res}_{U,V}:\Gamma U\to\Gamma V$ if $V\subseteq U$.

For $U\in{\mathcal{U}}_{x}$, let $i_{U}:\Gamma
U\to{\mathcal{O}}_{X,x}={\mathcal{O}}_{{\widetilde{X}},\tilde{x}}$ the
canonical map into the colimit. Let ${\mathfrak{p}}_{\tilde{x},U}$ be the
prime congruence of $\Gamma U$ that corresponds to $\tilde{x}$ and define the
congruence

${\mathfrak{p}}_{\tilde{x}}\ =\
\big{\langle}\bigcup_{U\in{\mathcal{U}}_{x}}i_{U,\ast}({\mathfrak{p}}_{\tilde{x},U})\big{\rangle}$

of ${\mathcal{O}}_{{\widetilde{X}},\tilde{x}}$. The _residue field at
$\tilde{x}$_ is the pointed group

$k(\tilde{x})\ =\
{\mathcal{O}}_{{\widetilde{X}},\tilde{x}}\,/\,{\mathfrak{p}}_{\tilde{x}}.$

######  Lemma .

Let $X$ be a monoid scheme with congruence space ${\widetilde{X}}$. Let
$\tilde{x}\in{\widetilde{X}}$ and $x=\mypi_{X}(\tilde{x})$. Then
${\mathfrak{p}}_{\tilde{x}}$ is a prime congruence of
${\mathcal{O}}_{{\widetilde{X}},\tilde{x}}$ and $k(\tilde{x})$ is a pointed
group. More precisely,
${\mathfrak{p}}_{\tilde{x}}=i_{U,\ast}({\mathfrak{p}}_{\tilde{x},U})$ for any
$U\in{\mathcal{U}}_{x}$ and $k(\tilde{x})$ is isomorphic to
$\operatorname{Frac}(\Gamma U/{\mathfrak{p}}_{\tilde{x},U})$.

###### Proof.

For any $U,V\in\mathcal{U}_{x}$, we want to show that
$i_{U,\ast}({\mathfrak{p}}_{\tilde{x},U})=i_{V,\ast}({\mathfrak{p}}_{\tilde{x},V})$.
In fact, it is enough to check this for any basic affine open $V$ of $U$.
Since we have $i_{V}\circ\operatorname{res}_{U,V}=i_{U}$ it follows that
$i_{V,\ast}({\mathfrak{p}}_{\tilde{x},V})=i_{U,\ast}({\mathfrak{p}}_{\tilde{x},U})$.
Thus ${\mathfrak{p}}_{\tilde{x}}=i_{U,\ast}({\mathfrak{p}}_{\tilde{x},U})$.
Also, we have $k(\tilde{x})=\
{\mathcal{O}}_{{\widetilde{X}},\tilde{x}}\,/\,{\mathfrak{p}}_{\tilde{x}}=\
{\mathcal{O}}_{{\widetilde{U}},\tilde{x}}\,/\,i_{U,\ast}({\mathfrak{p}}_{\tilde{x},U})=S^{-1}A\,/\,S^{-1}({\mathfrak{p}}_{\tilde{x},U})$
for any $U=\operatorname{MSpec}(A)\in\mathcal{U}_{x}$ and
$S=A-\mypi_{U}({\mathfrak{p}}_{\tilde{x},U})=\\{a\in A\leavevmode\nobreak\
|\leavevmode\nobreak\ (a,0)\notin{\mathfrak{p}}_{\tilde{x},U}\\}$. Let
$\myvarphi:A\to A/{\mathfrak{p}}_{\tilde{x},U}$ denote the projection map.
Since localizations of monoids are exact (in the sense of Lemma 2.2), we have
$S^{-1}A\,/\,S^{-1}({\mathfrak{p}}_{\tilde{x},U})=\myvarphi(S)^{-1}(A\,/\,{\mathfrak{p}}_{\tilde{x},U})=\operatorname{Frac}(A\,/\,{\mathfrak{p}}_{\tilde{x},U})$.
∎

Note that a morphism $\myvarphi:X\to Y$ of monoid schemes induces morphisms

${{\tilde{\myvarphi}}}_{\mathfrak{p}}:\mathcal{O}_{\widetilde{Y},\mathfrak{q}}\longrightarrow\mathcal{O}_{\widetilde{X},\mathfrak{p}}\qquad\text{and}\qquad
k(\mathfrak{q})\longrightarrow k(\mathfrak{p})$

between stalks and residue fields for every $\mathfrak{p}\in\widetilde{X}$ and
${\tilde{\myvarphi}}(\mathfrak{p})=\mathfrak{q}\in\widetilde{Y}$. Note further
that the morphism $k(\mathfrak{q})\to k(\mathfrak{p})$ is injective in
contrast to the induced maps between residue fields associated with prime
ideals.

###  2.7. Fibres over the monoid scheme

Let $X$ be a monoid scheme with congruence space ${\widetilde{X}}$ and
projection $\mypi_{X}:{\widetilde{X}}\to X$. In this section, we study the
fibres $\mypi^{-1}_{X}(x)$ over the points $x$ of $X$.

######  Proposition .

Let $x\in X$ and $k(x)$ be the residue field at $x$. The canonical morphism
$\mykappa_{x}:\operatorname{MSpec}(k(x))\to X$ induces a homeomorphism
$\tilde{\mykappa}_{x}:\operatorname{Cong}(k(x))\to\mypi_{X}^{-1}(x)$.

###### Proof.

Since $\textup{im}(\mykappa_{x})=\\{x\\}$ and the diagram

${\operatorname{Cong}(k(x))}$${{\widetilde{X}}}$${\operatorname{MSpec}(k(x))}$${X}$$\scriptstyle{\mypi_{\operatorname{MSpec}(k(x))}}$$\scriptstyle{{\widetilde{\mykappa_{x}}}}$$\scriptstyle{\mypi_{X}}$$\scriptstyle{\mykappa_{x}}$

commutes, we have $im({\widetilde{\mykappa_{x}}})\subseteq\mypi_{X}^{-1}(x)$.
Moreover, we can assume that $X=\operatorname{MSpec}(A)$ is affine and $x$
corresponds to a prime ideal $P$ of $A$.

Since $A_{P}\rightarrow A_{P}/{PA_{P}}$ is a surjective morphism,
$\operatorname{MSpec}(A_{P}/{PA_{P}})\rightarrow\operatorname{MSpec}(A_{P})$
is a closed immersion and therefore
$\operatorname{Cong}(A_{P}/{PA_{P}})\rightarrow\operatorname{Cong}(A_{P})$ is
injective. By Proposition 2.2, we know that
$\operatorname{Cong}(A_{P})\rightarrow\operatorname{Cong}(A)$ is an injection.
Thus
${\widetilde{\mykappa_{x}}}:\operatorname{Cong}(k(x))\cong\operatorname{Cong}(A_{P}/{PA_{P}})\rightarrow\mypi_{X}^{-1}(P)\subseteq\mypi_{X}^{-1}(x)$
is an injection.

In order to show surjectivity, consider any
$\mathfrak{c}\in\mypi_{X}^{-1}(P)$. Since
$\mypi_{X}(\mathfrak{c})=I_{\mathfrak{c}}=P$, we have
$I_{\mathfrak{c}}\cap{(A-P)}=\emptyset$ and therefore
$\mathfrak{c}\in\operatorname{Cong}(A)_{P}$ by Proposition 2.2. Moreover,
since $(a,0)\in\mathfrak{c}$ for all $a\in P$, we have
$\mathfrak{c}\in\operatorname{Cong}(A_{P}/PA_{P})$ and this shows
${\widetilde{\mykappa}}_{x}$ is surjective.

Let $\myvarphi:A\to A_{P}\to A_{P}/PA_{P}$ be the natural map. Since
${\widetilde{\mykappa}}_{X}(U_{\myvarphi(a),\myvarphi(b)})=U_{a,b}\cap\mypi_{X}^{-1}(P)$
and $\myvarphi$ is surjective, it follows that ${\widetilde{\mykappa}}_{x}$ is
a homeomorphism. ∎

As a consequence of Proposition 2.7, we find two natural section
$\mysigma_{X}:X\to{\widetilde{X}}$ and $\mytau_{X}:X\to{\widetilde{X}}$ of
$\mypi_{X}$: the former section $\mysigma_{X}$ send a point $x$ of $X$ to the
image $\mykappa_{x}({\mathfrak{c}}_{\textup{triv}})$ of the trivial congruence
${\mathfrak{c}}_{\textup{triv}}$ on $k(x)$, which is prime since $k(x)$ is
integral; the latter section $\mytau_{X}$ sends $x$ to the image
${\widetilde{\mykappa}}_{x}({\mathfrak{m}})$ of congruence kernel
${\mathfrak{m}}$ of the unique morphism $k(x)\to{{\mathbb{F}}_{1}}$, which
sends all units of $k(x)$ to $1$.

######  Remark .

The sections $\mysigma_{X}$ and $\mytau_{X}$ are less well-behaved than the
projection $\mypi_{X}$. Neither of them is continuous in general, as the
example $X=\operatorname{MSpec}(A)$ for $A=\\{0,e,1\\}$ with idempotent
$e=e^{2}$ shows, which is illustrated in Figure 3. The map
$\mypi_{X}:{\widetilde{X}}\to X$ is a bijection, which fails to be a
homeomorphism since ${\widetilde{X}}$ is discrete, but $X$ is not. Thus both
$\mysigma_{X}$ and $\mytau_{X}$ are equal to the inverse bijection to
$\mypi_{X}$ and fail to be continuous.

${\langle 0\rangle}$${\langle
e\rangle}$${\langle(e,1)\rangle}$${\langle(e,0)\rangle}$$\operatorname{MSpec}(A)$$\operatorname{Cong}(A)$$\mypi_{A}$
Figure 3. The projection
$\mypi_{A}:\operatorname{Cong}(A)\to\operatorname{MSpec}(A)$ for
$A=\\{0,e,1\\}$

It follows from Proposition 2.7 that $\mytau_{X}$ is functorial in $X$. The
section $\mysigma_{X}$ is functorial in open immersions, but fails to be
functorial in general, as the example of the diagonal embedding
$\Delta:{\mathbb{A}}^{1}_{{\mathbb{F}}_{1}}\to{\mathbb{A}}^{2}_{{\mathbb{F}}_{1}}$
shows:

${\widetilde{\Delta}}\circ\mysigma_{{\mathbb{A}}^{1}_{{\mathbb{F}}_{1}}}\big{(}\langle
0\rangle\big{)}={\widetilde{\Delta}}\big{(}\langle(0,0)\rangle\big{)}=\langle(t_{1},t_{2})\rangle\
\neq\ \langle(0,0)\rangle=\mysigma_{A^{2}_{{\mathbb{F}}_{1}}}\big{(}\langle
0\rangle\big{)}=\mysigma_{{\mathbb{A}}^{2}_{{\mathbb{F}}_{1}}}\circ\Delta\big{(}\langle
0\rangle\big{)}.$

Let $A$ be a pointed monoid. Taking congruence kernels defines an injection
$\mychi_{A}:\operatorname{Hom}(A,{{\mathbb{F}}_{1}})\to\operatorname{Cong}(A)$.
Let $X$ be a monoid scheme with congruence space ${\widetilde{X}}$ and
$X({{\mathbb{F}}_{1}})=\operatorname{Hom}(\operatorname{MSpec}({{\mathbb{F}}_{1}}),X)$.
Since the congruence kernel of a composition is the pullback of a congruence
kernel, the maps $\mychi_{A}$ are functorial in $A$. Therefore the injections

$\mychi_{\Gamma U}:\ U({{\mathbb{F}}_{1}})\ =\ \operatorname{Hom}(\Gamma
U,{{\mathbb{F}}_{1}})\ \longrightarrow\ \operatorname{Cong}(\Gamma U)\ =\
{\widetilde{U}}$

for affine open subschemes $U$ of $X$ glue to a functorial injection

$\mychi_{X}:\ X({{\mathbb{F}}_{1}})\ \longrightarrow\ {\widetilde{X}}.$

######  Proposition .

Let $X$ be a monoid scheme with congruence space ${\widetilde{X}}$.

  1. (1)

The image of $\mychi_{X}:X({{\mathbb{F}}_{1}})\to{\widetilde{X}}$ is the set
of closed points of ${\widetilde{X}}$.

  2. (2)

The composition $\mypi_{X}\circ\mychi_{X}:X({{\mathbb{F}}_{1}})\to X$ is a
bijection and $\mytau_{X}$ is the composition of its inverse bijection with
$\mychi_{X}$. In particular, $\mytau_{X}$ is functorial in $X$ and gives a
bijection between $X$ and the set of closed points of ${\widetilde{X}}$.

###### Proof.

All assertions are local in nature, so we can assume that
$X=\operatorname{MSpec}(A)$ is affine.

We begin with (1). Consider a congruence $\mathfrak{c}$ in the image of
$\mychi_{A}:\operatorname{Hom}(A,\mathbb{F}_{1})\to\operatorname{Cong}(A)$,
i.e. $\mathfrak{c}=\operatorname{congker}(f)$ for some
$f\in\operatorname{Hom}(A,\mathbb{F}_{1})$. Assume that $\mathfrak{c}$ is not
closed in ${\widetilde{X}}$. Then there exists a prime congruence
$\mathfrak{c^{\prime}}$ that properly contains $\mathfrak{c}$. Thus there is
an $(a_{1},a_{2})\in\mathfrak{c^{\prime}}$ with $f(a_{1})\neq f(a_{2})$.
Without loss of generality, let $f(a_{1})=0$ and $f(a_{2})=1$, i.e.
$(0,a_{1}),(a_{2},1)\in\mathfrak{c}\subseteq\mathfrak{c^{\prime}}$. By
transitivity, $(0,1)\in\mathfrak{c^{\prime}}$ which implies
$\mathfrak{c^{\prime}}=A\times A$ which yields the desired contradiction.
Hence $\mathfrak{c}$ is a closed point in $\operatorname{Cong}(A)$.

Conversely, let $\mathfrak{c}$ be any closed point in
$\operatorname{Cong}(A)$. It follows from the maximality of $\mathfrak{c}$
that $A/\mathfrak{c}\cong\mathbb{F}_{1}$ and thus
$\mathfrak{c}=\operatorname{congker}(A\to A/\mathfrak{c}\cong\mathbb{F}_{1})$.
This proves (1).

We continue with (2). Let $f,g\in X(\mathbb{F}_{1})$ satisfy that
$\mypi_{X}\circ\mychi_{X}(f)=\mypi_{X}\circ\mychi_{X}(g)$. This implies
$f^{-1}(0)=g^{-1}(0)$. Since $\textup{im}f=\textup{im}g=\\{0,1\\}$ has only
$2$ elements, this implies that $f=g$. Thus $\mypi_{X}\circ\mychi_{X}$ is
injective. For $P\in X$, we define the map $f_{P}:A\to\mathbb{F}_{1}$ by
$f_{P}(a)=0$ if $a\in P$ and $f_{P}(a)=1$ if $a\notin P$. Then
$\mypi_{X}\circ\mychi_{X}(f_{P})=P$, which shows that
$\mypi_{X}\circ\mychi_{X}$ is surjective. This shows that
$\mypi_{X}\circ\mychi_{X}:X({{\mathbb{F}}_{1}})\to X$ is a bijection.

Next we show that $\mytau_{X}=\mychi_{X}\circ(\mypi_{X}\circ\mychi_{X})^{-1}$.
Let $f\in X(\mathbb{F}_{1})$ and $P=\ker(f)$. Then
$\mytau_{X}\circ\mypi_{X}\circ\mychi_{X}(f)=\mytau_{X}(P)$. Let
$\myvarphi:A\to A_{P}\to A_{P}/PA_{P}\cong k(P)$ be the natural map and let
${\mathfrak{m}}$ denote the congruence kernel of the unique morphism
$k(P)\to{{\mathbb{F}}_{1}}$, which sends all units of $k(P)$ to $1$. Then

$\mytau_{X}(P)\ =\
{\widetilde{\mykappa}}_{P}({\mathfrak{m}})=\\{(a,a^{\prime})\in A\times
A\leavevmode\nobreak\ |\leavevmode\nobreak\ a,a^{\prime}\in
P\text{\leavevmode\nobreak\ or\leavevmode\nobreak\ }a,a^{\prime}\in A-P\\}\\\
=\\{(a,a^{\prime})\in A\times A\leavevmode\nobreak\ |\leavevmode\nobreak\
f(a)=f(a^{\prime})\\}\ =\ \operatorname{congker}(f)\ =\ \mychi_{X}(f).$

Thus $\mytau_{X}\circ\mypi_{X}\circ\mychi_{X}=\mychi_{X}$ as desired.

It follows that $\mytau_{X}$ gives a bijection between $X$ and the set of
closed points of ${\widetilde{X}}$. Since both $\mychi_{X}$ and $\mypi_{X}$
are functorial in $X$, so is $\mytau_{X}$. ∎

######  Remark .

The natural maps $\mychi_{X}:X({{\mathbb{F}}_{1}})\to{\widetilde{X}}$ and
$\mypi_{X}\circ\mychi_{X}:X({{\mathbb{F}}_{1}})\to X$ endow
$X({{\mathbb{F}}_{1}})$ with two distinct initial topologies, which differ in
general. Both topologies are, in fact, fine topologies that are induced by
topologies for ${{\mathbb{F}}_{1}}$.

The concept of fine topologies was introduced in [24] in the case of rational
point sets for usual schemes and later applied for semiring schemes and
ordered blue schemes in [23]. We recall the definition tailored to monoid
schemes.

Let $A$ be a monoid together with a topology and $X$ a monoid scheme. The
_fine topology for $X(A)$_ is the finest topology such that for every morphism
$U\to X$ from an affine monoid scheme $U$ into $X$, the induced map $U(A)\to
X(A)$ is continuous where $U(A)=\operatorname{Hom}(\Gamma U,A)$ carries the
compact open topology with respect to the discrete topology for $\Gamma U$.

The initial topology for $X({{\mathbb{F}}_{1}})$ with respect to
$\mychi_{X}:X({{\mathbb{F}}_{1}})\to{\widetilde{X}}$ coincides with the fine
topology for $X({{\mathbb{F}}_{1}})$ that is induced by the discrete topology
for ${{\mathbb{F}}_{1}}$.

The morphism $\mypi_{X}\circ\mychi_{X}:X({{\mathbb{F}}_{1}})\to X$ is a
bijection and thus the initial topology for $X({{\mathbb{F}}_{1}})$
corresponds to the Zariski topology of $X$. This equals the fine topology for
$X({{\mathbb{F}}_{1}})$ that is induced by the topology for
${{\mathbb{F}}_{1}}$ for which $\\{0\\}$ is closed, but $\\{1\\}$ is not.

##  3\. Strongly reduced monoid schemes

In this section, we develop the theory of (strong) reductions of monoids and
monoid schemes. Namely, as a consequence of the discrepancy between ideals and
congruences, we obtain to distinct concepts of reduction of a monoid: the
(weak) reduction quotients out the intersection of all prime ideals, the
strong reduction quotients out the intersection of all prime congruences.
These concepts carry over to the monoid schemes.

###  3.1. Radicals

######  Definition .

Let $A$ be a pointed monoid and $I$ an ideal of $A$. The _radical of $I$_ is
the ideal

$\sqrt{I}\ =\ \\{a\in A\mid a^{n}\in I\text{ for some }n\geqslant 0\\}.$

A _radical ideal of $A$_ is an ideal $I$ of $A$ with $\sqrt{I}=I$. The
_nilradical of $A$_ is the radical ideal
$\operatorname{Nil}(A)=\sqrt{\\{0\\}}$.

Let ${\mathfrak{c}}$ be a congruence on $A$. The _radical of ${\mathfrak{c}}$_
is the congruence

$\sqrt{{\mathfrak{c}}}\ =\ \big{\\{}(a,b)\in A\times
A\,\big{|}\,(aa^{n},ba^{n})\in{\mathfrak{c}}\text{ and
}(ab^{n},bb^{n})\in{\mathfrak{c}}\text{ for some }n\geqslant 0\big{\\}}.$

A _radical congruence_ is a congruence ${\mathfrak{c}}$ with
$\sqrt{{\mathfrak{c}}}={\mathfrak{c}}$. The _strong nilradical of $A$_ is the
radical
$\operatorname{\mathfrak{nil}}(A)=\sqrt{{\mathfrak{c}}_{\textup{triv}}}$ of
the trivial congruence ${\mathfrak{c}}_{\textup{triv}}$.

######  Proposition .

Let $A$ be a pointed monoid, $I$ an ideal of $A$ and ${\mathfrak{c}}$ a
congruence on $A$. Then

$\sqrt{I}\ =\ \bigcap_{\begin{subarray}{c}I\subset P,\\\ P\text{
prime}\end{subarray}}P,\qquad\sqrt{{\mathfrak{c}}}\ =\
\bigcap_{\begin{subarray}{c}{\mathfrak{c}}\subset{\mathfrak{p}},\\\
{\mathfrak{p}}\text{ prime}\end{subarray}}{\mathfrak{p}}\qquad\text{and}\qquad
I_{\sqrt{{\mathfrak{c}}}}\ =\ \sqrt{I_{\mathfrak{c}}}.$

###### Proof.

As a first step, we prove the claims for $I=\\{0\\}$ and
${\mathfrak{c}}={\mathfrak{c}}_{\textup{triv}}$, for which we have
$\sqrt{I}=\operatorname{Nil}(A)$ and
$\sqrt{{\mathfrak{c}}}=\operatorname{\mathfrak{nil}}(A)$. Let
$X=\operatorname{MSpec}(A)$ and ${\widetilde{X}}=\operatorname{Cong}(A)$. The
inclusions $\operatorname{Nil}(A)\subseteq\bigcap_{P\in X}P$ and
$\operatorname{\mathfrak{nil}}(A)\subseteq\bigcap_{{\mathfrak{p}}\in{\widetilde{X}}}{\mathfrak{p}}$
follow directly from the definitions.

Conversely consider $a\in A-\operatorname{Nil}(A)$. Then $0\notin
S_{a}=\\{a^{i}\\}_{i\geqslant 0}$ and $A[a^{-1}]=S_{a}^{-1}A\neq\\{0\\}$. Let
$q$ denote the unique maximal ideal
$M_{A[a^{-1}]}=A[a^{-1}]-A[a^{-1}]^{\times}$ of $A[a^{-1}]$ and let
$P=i_{a}^{-1}(Q)$ under the natural map $i_{a}:A\to A[a^{-1}]$. Then $P$ is a
prime ideal of $A$ and $a\notin P$. This proves that
$\operatorname{Nil}(A)=\bigcap_{P\in X}P$.

Next we consider $a,b\in A$ such that $(a,b)\notin\mathfrak{nil}(A)$. Without
loss of generality, we can assume $aa^{k}\neq ba^{k}$ for all integers
$k\geqslant 0$. Then $\frac{a}{1}\neq\frac{b}{1}$ in $A[a^{-1}]=S_{a}^{-1}(A)$
where $S_{a}=\\{a^{i}\\}_{i\geqslant 0}$. We consider the quotient map
$\mypi:A[a^{-1}]^{\times}\to A[a^{-1}]/M_{A[a^{-1}]}$ which is clearly
injective. Thus $\mypi(\frac{a}{1})\neq\mypi(\frac{b}{1})$ in
$A[a^{-1}]/M_{A[a^{-1}]}$. Since $A[a^{-1}]/M_{A[a^{-1}]}$ is a pointed group
(and therefore integral), the congruence
$\mathfrak{c}=\operatorname{congker}(A\to A[a^{-1}]/M_{A[a^{-1}]})$ is prime
and $(a,b)\notin\mathfrak{p}$. This proves that
$\mathfrak{nil}(A)=\bigcap_{\mathfrak{p}\in\widetilde{X}}\mathfrak{p}$.

Consider any $a\in I_{\mathfrak{nil}(A)}$. Then $(a,0)\in\mathfrak{nil}(A)$
which implies $a.a^{n}=0.a^{n}=0$ for some integer $n\geqslant 0$. Therefore
$a\in\operatorname{Nil}(A)$. Conversely, consider any
$a\in\operatorname{Nil}(A)$. Then $a^{n}=0$ for some $n\geqslant 0$. This
implies $a.a^{n-1}=0=0.a^{n-1}$ and $n.0^{n-1}=0=0.0^{n-1}$. Thus
$(a,0)\in\mathfrak{nil}(A)$, i.e. $a\in I_{\mathfrak{nil}(A)}$. This proves
that $I_{\mathfrak{nil}(A)}=\operatorname{Nil}(A)$.

We turn to the general case and consider an arbitrary ideal $I$ of $A$. The
previous arguments show that
$\operatorname{Nil}(A/I)=\bigcap_{P\in\operatorname{MSpec}(A/I)}\leavevmode\nobreak\
P$. Since the prime ideals of $A/I$ can be identified with the prime ideals of
$A$ containing $I$ and since $\operatorname{Nil}(A/I)$ can be identified with
$\sqrt{I}$ (see Proposition 2.1), it follows that

$\sqrt{I}\ =\ \bigcap_{\begin{subarray}{c}I\subset P,\\\ P\text{
prime}\end{subarray}}P.$

Let ${\mathfrak{c}}$ be an arbitrary congruence of $A$. The previous arguments
show that
$\mathfrak{nil}(A/I)=\bigcap_{{\mathfrak{p}}\in\operatorname{Cong}(A)/I}\leavevmode\nobreak\
{\mathfrak{p}}$. By Proposition 2.1, the prime congruences of
$A/{\mathfrak{c}}$ pull back to the prime congruences of $A$ that contain
${\mathfrak{c}}$. Thus $\mathfrak{nil}(A/{\mathfrak{c}})$ pulls back to
$\sqrt{{\mathfrak{c}}}$. Therefore

$\sqrt{{\mathfrak{c}}}\ =\
\bigcap_{\begin{subarray}{c}{\mathfrak{c}}\subset{\mathfrak{p}},\\\
{\mathfrak{p}}\text{ prime}\end{subarray}}{\mathfrak{p}}.$

To conclude the proof, note that $a\in I_{\sqrt{{\mathfrak{c}}}}$ if and only
if $(a,0)\in\sqrt{{\mathfrak{c}}}$, i.e. $(a^{n},0)\in{\mathfrak{c}}$ for some
$n>0$. This is equivalent to $a^{n}\in I_{\mathfrak{c}}$ or, in other words,
$a\in\sqrt{I_{\mathfrak{c}}}$. ∎

######  Definition .

Let $A$ be a pointed monoid. The _reduction of $A$_ is
$A^{\textup{red}}=A/\operatorname{Nil}(A)$. The _strong reduction of $A$_ is
$A^{\textup{sred}}=A/\mathfrak{nil}(A)$. We say that $A$ is _reduced_ if
$A=A^{\textup{red}}$ and that $A$ is _strongly reduced_ if
$A=A^{\textup{sred}}$.

######  Lemma .

Let $f:A\to B$ be a morphism of pointed monoids. If $B$ is reduced, then $f$
factors uniquely through $A\to A^{\textup{red}}$. If $B$ is strongly reduced,
then $f$ factors uniquely through $A\to A^{\textup{sred}}$. This defines
reflections

$(-)^{\textup{red}}:\ \operatorname{Mon}_{0}\ \longrightarrow\
\operatorname{Mon}_{0}\qquad\text{and}\qquad(-)^{\textup{sred}}:\
\operatorname{Mon}_{0}\ \longrightarrow\ \operatorname{Mon}_{0}$

onto the respective subcategories of reduced and strongly reduced pointed
monoids.

###### Proof.

Let $f:A\to B$ be a morphism of pointed monoids. If $a^{n}=0$ for some $a\in
A$ and $n\geqslant 0$, then $f(a)^{n}=0$ in $B$. Thus if $B$ is reduced, then
$f(a)=0$ and $f$ factors (necessarily uniquely) through $A^{\textup{red}}$. If
$aa^{n}=ba^{n}$ and $ab^{n}=bb^{n}$ for some $a,b\in A$ and $n\geqslant 0$,
then $f(a)f(a)^{n}=f(b)f(a)^{n}$ and $f(a)f(b)^{n}=f(b)f(b)^{n}$ in $B$. Thus
if $B$ is strictly reduced, then $f(a)=f(b)$ in $B$ and $f$ factors through
(necessarily uniquely) $A^{\textup{sred}}$.

These two universal properties show that the embeddings of the subcategories
of reduced and strongly reduced monoids into $\operatorname{Mon}$ have
respective left adjoints, which gives rise to the reflections
$(-)^{\textup{red}}:\operatorname{Mon}_{0}\to\operatorname{Mon}_{0}$ and
$(-)^{\textup{sred}}:\operatorname{Mon}_{0}\to\operatorname{Mon}_{0}$. ∎

######  Remark .

The difference between the nilradical and the strong nilradical extends the
difference of zero divisor free and integral monoids in the following sense.
If $A$ is a zero divisor free monoid, then $A$ is reduced and
$\operatorname{Nil}(A)=\\{0\\}$. On the other hand,
$\operatorname{\mathfrak{nil}}(A)=\operatorname{congker}(A\to\operatorname{Frac}A)$
might be non-trivial, and $A^{\textup{sred}}=A^{\overline{\textup{int}}}$.

###  3.2. Compatibility with localizations

######  Lemma .

Let $A$ be a pointed monoid and $\mypi:A\to A^{\textup{sred}}$ the quotient
map. Let $S$ be a multiplicative subset of $A$. Then
$(S^{-1}A)^{\textup{sred}}=\mypi(S)^{-1}A^{\textup{sred}}$.

###### Proof.

We denote the multiplicative set $\mypi(S)$ of $A^{\textup{sred}}$ by
$\bar{S}$. We claim that the canonical morphisms

$\begin{array}[]{cccc}f:&A&\longrightarrow&\bar{S}^{-1}A^{\textup{sred}}\\\
&a&\longmapsto&\frac{[a]}{[1]}\end{array}\qquad\qquad\begin{array}[]{cccc}g:&A&\longrightarrow&(S^{-1}A)^{\textup{sred}}\\\
&a&\longmapsto&[\frac{a}{1}]\end{array}$

descend to morphisms

$\bar{f_{s}}:(S^{-1}A)^{\textup{sred}}\longrightarrow\bar{S}^{-1}A^{\textup{sred}}\qquad\qquad\bar{g}_{\bar{s}}:\bar{S}^{-1}A^{\textup{sred}}\longrightarrow(S^{-1}A)^{\textup{sred}}$

which are mutually inverse to each other since both are induced from the
identity map $A\xrightarrow{id}A$. We proceed to show how to obtain the above
maps. Since $f(s)=\frac{[s]}{[1]}$ is invertible in
$\bar{S}^{-1}A^{\textup{sred}}$, there is a unique morphism
$f_{s}:S^{-1}A\to\bar{S}^{-1}A^{\textup{sred}}$ such that $f$ factors as
follows

$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\bar{S}^{-1}A^{\textup{sred}}}$$\textstyle{S^{-1}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{s}}$

The morphism $f_{s}:S^{-1}A\to\bar{S}^{-1}A^{\textup{sred}}$ is explicitly
given by $f_{s}(\frac{a}{s})=\frac{\mypi{(a)}}{\mypi{(s)}}$. Consider an
element $(\frac{a}{s^{i}},\frac{b}{s^{j}})\in\mathfrak{nil}(S^{-1}A)$. Since
$\frac{a}{s^{i}}=\frac{as^{j}}{s^{i+j}}$ and
$\frac{b}{s^{j}}=\frac{bs^{i}}{s^{i+j}}$, we can assume that any element of
$\mathfrak{nil}(S^{-1}A)$ is of the form $(\frac{a}{s^{i}},\frac{b}{s^{i}})$.
Thus $(\frac{a}{s^{i}},\frac{b}{s^{i}})\in\mathfrak{nil}(S^{-1}A)$ means that
$\frac{a}{s^{i}}(\frac{c}{s^{i}})^{N}=\frac{b}{s^{i}}(\frac{c}{s^{i}})^{N}$ in
$S^{-1}A$ for $c=a,b$ and for some integer $N\gg 0$. This implies
$s^{k}s^{i}a(s^{i}c)^{N}=s^{k}s^{i}b(s^{i}c)^{N}$ in $A$ for some integer
$k\geqslant 0$, $c=a,b$ and $N\gg 0$. In fact, we can assume $k=0$ if $N$ is
large enough, i.e., $s^{i}a(s^{i}c)^{N}=s^{i}b(s^{i}c)^{N}$ for $c=a,b$ and
$N\gg 0$. Then $(s^{i}a,s^{i}b)\in\mathfrak{nil}(A)$ and therefore
${\mypi(s)}^{i}\mypi(a)={\mypi(s)}^{i}\mypi(b)\text{\leavevmode\nobreak\
in\leavevmode\nobreak\ }A^{\textup{sred}}$ which implies
$\frac{\mypi(a)}{{\mypi(s)}^{i}}=\frac{\mypi(b)}{{\mypi(s)}^{i}}\text{\leavevmode\nobreak\
in\leavevmode\nobreak\ }\bar{S}^{-1}A^{\textup{sred}}$. Thus there is a unique
morphism
$\bar{f_{s}}:(S^{-1}A)^{\textup{sred}}\to\bar{S}^{-1}A^{\textup{sred}}$ such
that $f_{s}$ factors as follows

$\textstyle{S^{-1}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{s}}$$\textstyle{\bar{S}^{-1}A^{\textup{sred}}}$$\textstyle{(S^{-1}A)^{\textup{sred}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{f_{s}}}$

The morphism
$\bar{f_{s}}:(S^{-1}A)^{\textup{sred}}\to\bar{S}^{-1}A^{\textup{sred}}$ is
explicitly given by $\bar{f_{s}}([\frac{a}{s}])=\frac{\bar{a}}{\bar{s}}$.
Next, we consider $(a,b)\in\mathfrak{nil}(A)$, i.e., $ac^{N}=bc^{N}$ for
$c=a,b$ and an integer $N\gg 0$. This implies
$\frac{a}{1}(\frac{c}{1})^{N}=\frac{b}{1}(\frac{c}{1})^{N}\text{\leavevmode\nobreak\
in\leavevmode\nobreak\ }S^{-1}A$ and hence
$[\frac{a}{1}]=[\frac{b}{1}]\text{\leavevmode\nobreak\ in\leavevmode\nobreak\
}(S^{-1}A)^{\textup{sred}}$. Thus there is a unique morphism
$\bar{g}:A^{\textup{sred}}\to(S^{-1}A)^{\textup{sred}}$ such that $g$ factors
as follows

$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{(S^{-1}A)^{\textup{sred}}}$$\textstyle{A^{\textup{sred}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{g}}$

where the morphism $\bar{g}:A^{\textup{sred}}\to(S^{-1}A)^{\textup{sred}}$ is
explicitly given by $\bar{g}(\bar{a})=[\frac{a}{1}]$. Since
$\bar{g}(\bar{s})=[\frac{s}{1}]$ is invertible in $(S^{-1}A)^{\textup{sred}}$,
there is a unique morphism
$\bar{g}_{\bar{s}}:{\bar{S}}^{-1}A^{\textup{sred}}\to(S^{-1}A)^{\textup{sred}}$
such that $\bar{g}$ factors as follows

$\textstyle{A^{\textup{sred}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{g}}$$\textstyle{(S^{-1}A)^{\textup{sred}}}$$\textstyle{{\bar{S}}^{-1}A^{\textup{sred}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\bar{g}}_{\bar{s}}}$

The morphism
$\bar{g}_{\bar{s}}:{\bar{S}^{-1}}A^{\textup{sred}}\to(S^{-1}A)^{\textup{sred}}$
is explicitly given by
$\bar{g}_{\bar{s}}(\frac{\bar{a}}{\bar{s}^{i}})=[\frac{a}{s^{i}}]$. ∎

Proposition 2.2 extends to radical congruences in the following way.

######  Proposition .

Let $\myiota_{S}:A\to S^{-1}A$ be a localization of the pointed monoid $A$ at
$S$. Then the maps

${\Big{\\{}\begin{array}[]{c}\text{radical}\\\ \text{congruences on
}S^{-1}A\end{array}\Big{\\}}}$${\Big{\\{}\begin{array}[]{c}\text{radical
congruences}\\\ \text{${\mathfrak{c}}$ on $A$ with $I_{\mathfrak{c}}\cap
S=\emptyset$}\end{array}\Big{\\}}}$$\scriptstyle{\myiota_{S}^{\ast}}$$\scriptstyle{\myiota_{S,\ast}}$

are mutually inverse bijections.

###### Proof.

Let ${\mathfrak{d}}$ be a radical congruence on $S^{-1}A$. Then by Proposition
3.1,
${\mathfrak{d}}=\bigcap_{\begin{subarray}{c}{\mathfrak{d}}\subset{\mathfrak{p}},\\\
{\mathfrak{p}}\text{ prime}\end{subarray}}{\mathfrak{p}}$. Thus we have

$\myiota_{S}^{\ast}({\mathfrak{d}})=(\myiota_{S},\myiota_{S})^{-1}(\underset{\begin{subarray}{c}{\mathfrak{d}}\subset{\mathfrak{p}},\\\
{\mathfrak{p}}\text{
prime}\end{subarray}}{\bigcap}{\mathfrak{p}})=\underset{\begin{subarray}{c}{\mathfrak{d}}\subset{\mathfrak{p}},\\\
{\mathfrak{p}}\text{
prime}\end{subarray}}{\bigcap}(\myiota_{S},\myiota_{S})^{-1}({\mathfrak{p}})=\underset{\begin{subarray}{c}{\mathfrak{d}}\subset{\mathfrak{p}},\\\
{\mathfrak{p}}\text{
prime}\end{subarray}}{\bigcap}\myiota_{S}^{\ast}({\mathfrak{p}})=\underset{\begin{subarray}{c}\myiota_{S}^{\ast}({\mathfrak{d}})\subset{\mathfrak{q}},\\\
{\mathfrak{q}}\text{ prime}\end{subarray}}{\bigcap}{\mathfrak{q}}$

where the last equality follows from Proposition 2.2. This shows that
$\myiota_{S}^{\ast}({\mathfrak{d}})$ is a radical congruence on $A$.

Conversely, let ${\mathfrak{c}}$ be a radical congruence on $A$ such that
$I_{\mathfrak{c}}\cap S=\emptyset$. Consider any
$(\frac{a}{s},\frac{a^{\prime}}{s^{\prime}})\in\sqrt{\myiota_{S,*}({\mathfrak{c}})}$.
Since $\frac{a}{s}$ is equivalent to $\frac{as^{\prime}}{ss^{\prime}}$ and
$\frac{a^{\prime}}{s^{\prime}}$ is equivalent to
$\frac{a^{\prime}s}{ss^{\prime}}$ in $S^{-1}A$, we have
$(\frac{a}{s},\frac{a^{\prime}}{s^{\prime}})\in\sqrt{\myiota_{S,*}({\mathfrak{c}})}$
if and only if
$(\frac{as^{\prime}}{1},\frac{a^{\prime}s}{1})\in\sqrt{\myiota_{S,*}({\mathfrak{c}})}$.
Therefore we have
$(\frac{as^{\prime}}{1}(\frac{as^{\prime}}{1})^{n},\frac{a^{\prime}s}{1}(\frac{as^{\prime}}{1})^{n})\in{\myiota_{S,*}({\mathfrak{c}})}$
and
$(\frac{as^{\prime}}{1}(\frac{a^{\prime}s}{1})^{n},\frac{a^{\prime}s}{1}(\frac{a^{\prime}s}{1})^{n})\in{\myiota_{S,*}({\mathfrak{c}})}$
for some $n\gg 0$. This implies there exists some $r\in S$ such that
$(as^{\prime}a^{n}s^{\prime n}r,a^{\prime}sa^{n}s^{\prime
n}r)\in{\mathfrak{c}}$ and $(as^{\prime}a^{\prime
n}s^{n}r,a^{\prime}sa^{\prime n}s^{n}r)\in{\mathfrak{c}}$. Hence we have
$(as^{\prime}r(as^{\prime}r)^{n},a^{\prime}sr(as^{\prime}r)^{n})\in{\mathfrak{c}}$
and
$(as^{\prime}r(a^{\prime}sr)^{n},a^{\prime}sr(a^{\prime}sr)^{n})\in{\mathfrak{c}}$.
Since ${\mathfrak{c}}$ is radical it follows that
$(as^{\prime}r,\leavevmode\nobreak\ a^{\prime}sr)\in{\mathfrak{c}}$ which
implies
$(\frac{as^{\prime}}{1},\frac{a^{\prime}s}{1})\in\myiota_{S,*}({\mathfrak{c}})$
or equivalently,
$(\frac{a}{s},\frac{a^{\prime}}{s^{\prime}})\in{\myiota_{S,*}({\mathfrak{c}})}$.
This shows that ${\myiota_{S,*}({\mathfrak{c}})}$ is radical.

We have therefore shown that both maps are well defined. The proof that the
maps are mutually bijective is similar to Proposition 2.2. ∎

###  3.3. Strong reduction of monoid schemes

Let $X=\operatorname{MSpec}(A)$ be an affine monoid scheme. We define
$X^{\textup{sred}}=\operatorname{MSpec}(A^{\textup{sred}})$. By Lemma 3.1, a
morphism $\myvarphi:\operatorname{MSpec}(A)\to\operatorname{MSpec}(B)$ of
affine monoid schemes induces a morphism
$\myvarphi^{\textup{sred}}:\operatorname{MSpec}(B^{\textup{sred}})\hookrightarrow\operatorname{MSpec}(A^{\textup{sred}})$
between the respective strong reductions.

######  Lemma .

Let $i:\operatorname{MSpec}(B)\hookrightarrow\operatorname{MSpec}(A)$ be an
open immersion of affine monoid schemes. Then
$i^{\textup{sred}}:\operatorname{MSpec}(B^{\textup{sred}})\hookrightarrow\operatorname{MSpec}(A^{\textup{sred}})$
is an open immersion of strongly reduced affine monoid schemes.

###### Proof.

Since $i:\operatorname{MSpec}(B)\hookrightarrow\operatorname{MSpec}(A)$ is an
open immersion, it follows from [7, Lemma 1.3] and [7, Lemma 2.4] that we must
have $\operatorname{MSpec}(B)\cong\operatorname{MSpec}(A[s^{-1}])$ for some
$s\in A$. Therefore by Lemma 3.2, it follows that
$i^{\textup{sred}}:\operatorname{MSpec}(B^{\textup{sred}})\hookrightarrow\operatorname{MSpec}(A^{\textup{sred}})$
is an open immersion. ∎

######  Proposition .

The functor $(-)^{\textup{sred}}$ on affine monoid schemes extends uniquely to
a functor $(-)^{\textup{sred}}:\operatorname{MSch}\to\operatorname{MSch}$,
which preserves affine open covers.

###### Proof.

Since morphisms are determined by their restrictions to open affines, any
endofunctor on $\operatorname{MSch}$ is determined by its restriction to
affine monoid schemes, which shows the uniqueness of $(-)^{\textup{sred}}$.

Next we construct $(-)^{\textup{sred}}$ on objects. Let $X$ be a monoid
scheme. Consider the family $\mathcal{U}_{X}=\\{U_{i}\\}_{i\in I}$ of all
affine open subschemes of $X$. For all $i,j\in I$, consider principal affine
opens $U_{ij}\subseteq U_{i}$ and isomorphisms $\myvarphi_{ji}:U_{ij}\to
U_{ji}$. Then by Lemma 3.3, we have $U_{ij}^{\textup{sred}}$ are open subsets
of $U_{i}^{\textup{sred}}$ for all $i,j\in I$. By Lemma 3.1, we know that
$(-)^{\textup{sred}}:\operatorname{Mon}_{0}\to\operatorname{Mon}_{0},\leavevmode\nobreak\
A\mapsto A^{\textup{sred}}$ is a functor and therefore it follows that the
collection $(I,\\{{U_{i}}^{\textup{sred}}\\}_{i\in
I},\\{U_{ij}^{\textup{sred}}\\}_{i,j\in
I},\\{\myvarphi_{ij}^{\textup{sred}}\\}_{i,j\in I})$ gives a gluing data of
monoid schemes (in other words,
$\mathcal{U}_{X}^{\textup{sred}}=\\{U_{i}^{\textup{sred}}\\}_{i\in I}$ is a
monodromy-free diagram in the category of monoid schemes in the sense of [21,
§1]). We define $X^{\textup{sred}}$ as the monoid scheme obtained from this
gluing data.

In particular, every open subscheme $U_{i}\in\mathcal{U}_{X}$ defines an open
subscheme ${U_{i}}^{\textup{sred}}$ of ${X}^{\textup{sred}}$ and
$X^{\textup{sred}}=\bigcup_{U_{i}\in\mathcal{U}_{X}}U_{i}^{\textup{sred}}$.
This shows that $(-)^{\textup{sred}}$ preserves affine open covers.

We continue with the construction of $(-)^{\textup{sred}}$ on morphisms. Let
$f:Y\to X$ be a morphism of monoid schemes. We choose an affine open cover
$\\{X_{\myalpha}\\}_{\myalpha\in A}$ of $X$ and an affine open cover
$\\{Y_{\myalpha}\\}_{\myalpha\in A}$ of $Y$ such that
$f(Y_{\myalpha})\subseteq X_{\myalpha}$ for each $\myalpha\in A$. We also
choose an affine cover $\\{Y_{\myalpha\mybeta k}\\}_{k\in K}$ for each
$Y_{\myalpha\mybeta}=Y_{\myalpha}\cap Y_{\mybeta}$ and an affine cover
$\\{X_{\myalpha\mybeta k}\\}_{k\in K}$ for each
$X_{\myalpha\mybeta}=X_{\myalpha}\cap X_{\mybeta}$ such that
$f(Y_{\myalpha\mybeta k})\subseteq X_{\myalpha\mybeta k}$ for each $k\in K$.
Let $f_{\myalpha}:=f|_{Y_{\myalpha}}:Y_{\myalpha}\to X_{\myalpha}$ and
$f_{\myalpha\mybeta k}:=f|_{Y_{\myalpha\mybeta k}}:Y_{\myalpha\mybeta k}\to
X_{\myalpha\mybeta k}$. The corresponding morphisms of strongly reduced affine
monoid schemes are respectively given by
$f_{\myalpha}^{\textup{sred}}:Y_{\myalpha}^{\textup{sred}}\to
X_{\myalpha}^{\textup{sred}}$ and $f_{\myalpha\mybeta
k}^{\textup{sred}}:Y_{\myalpha\mybeta k}^{\textup{sred}}\to X_{\myalpha\mybeta
k}^{\textup{sred}}$. We define

$\displaystyle f^{\textup{sred}}:{Y}^{\textup{sred}}=\bigcup_{\myalpha\in
A}Y_{\myalpha}^{\textup{sred}}$
$\displaystyle\longrightarrow\bigcup_{\myalpha\in
A}X_{\myalpha}^{\textup{sred}}={X}^{\textup{sred}}$
$\displaystyle\mathfrak{q}$ $\displaystyle\mapsto
f_{\myalpha}^{\textup{sred}}(\mathfrak{q})\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{ if\leavevmode\nobreak\
}\mathfrak{q}\in Y_{\myalpha}^{\textup{sred}}.$

To show that $f^{\textup{sred}}$ is well defined, it is enough to check that
$\mathfrak{q}\in Y_{\myalpha\mybeta k}^{\textup{sred}}\subseteq
Y_{\myalpha}^{\textup{sred}}$ implies $f_{\myalpha\mybeta
k}^{\textup{sred}}(\mathfrak{q})=f_{\myalpha}^{\textup{sred}}(\mathfrak{q})$
which is clear from the functoriality described in Lemma 3.1. It is also clear
that $f^{\textup{sred}}$ is continuous. ∎

For any affine monoid scheme $U=\operatorname{MSpec}(A)$, the morphism
$U^{\textup{sred}}\to U$ is a closed immersion since $A\to
A^{\textup{sred}}=A/\mathfrak{nil}(A)$ is surjective. Since being a closed
immersion is a local property, it follows from the definition of
$X^{\textup{sred}}$ that the natural morphism
$i^{\textup{sred}}:X^{\textup{sred}}\to X$ is a closed immersion for any
general monoid scheme $X$.

We call a monoid scheme $X$ _strongly reduced_ if
$i^{\textup{sred}}:X^{\textup{sred}}\to X$ is an isomorphism of monoid
schemes.

######  Proposition .

Let $X$ be a monoid scheme. Then the following hold.

  1. (1)

The maps $i^{\textup{sred}}:X^{\textup{sred}}\to X$ and
$\widetilde{i^{\textup{sred}}}:\widetilde{X^{\textup{sred}}}\to\widetilde{X}$
are homeomorphisms.

  2. (2)

Given a closed immersion $\myvarphi:Z\to X$ such that
${\tilde{\myvarphi}}:\widetilde{Z}\to\widetilde{X}$ is a homeomorphism, there
exists $\mypsi:X^{\textup{sred}}\to Z$ such that the following diagram
commutes

$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\myvarphi}$$\textstyle{X}$$\textstyle{X^{\textup{sred}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mypsi}$$\scriptstyle{i^{\textup{sred}}}$

  3. (3)

If $X=X^{\textup{sred}}=\operatorname{MSpec}(A)$ and
$\widetilde{V_{a,b}}=\widetilde{X}$, then $a=b$.

###### Proof.

We begin with (1). Since $i^{\textup{sred}}:X^{\textup{sred}}\to X$ is a
closed immersion, we know that by definition $i^{\textup{sred}}$ is a
homeomorphism onto its image. Also
$\widetilde{i^{\textup{sred}}}:\widetilde{X^{\textup{sred}}}\to\widetilde{X}$
is a homeomorphism onto its image by Proposition 4.1. By Proposition 2.5, the
diagram

$\textstyle{\widetilde{X^{\textup{sred}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{i^{\textup{sred}}}}$$\scriptstyle{\mypi_{X^{\textup{sred}}}}$$\textstyle{\widetilde{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mypi_{X}}$$\textstyle{X^{\textup{sred}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{\textup{sred}}}$$\textstyle{X}$

commutes. Since $\mypi_{X^{\textup{sred}}}$ and $\mypi_{X}$ are both
surjective, it is enough to show that $\widetilde{i^{\textup{sred}}}$ is
surjective to prove $i^{\textup{sred}}$ is surjective. Since the question is
local in $X$, we can assume that $X=\operatorname{MSpec}(A)$. Let $p:A\to
A^{\textup{sred}}$ be the quotient map. Since every
$\mathfrak{c}\in\widetilde{X}$ contains $\mathfrak{nil}(A)$, Proposition 2.1
implies that the map
$\tilde{p}^{\ast}:{\widetilde{X}}^{\textup{sred}}\to{\widetilde{X}}$ is
surjective. This shows (1).

We continue with (2). Let $\myvarphi:Z\to X$ be a closed immersion such that
${\tilde{\myvarphi}}:\widetilde{Z}\to\widetilde{X}$ is a homeomorphism. Since
the question is local in $X$ and $\myvarphi$ is a closed immersion, we can
assume that $X=\operatorname{MSpec}(A)$ and
$Z=\operatorname{MSpec}(A/\mathfrak{d})$ for some
$\mathfrak{d}\in\operatorname{Cong}(A)$. By assumption,
${\tilde{\myvarphi}}:\widetilde{Z}\to\widetilde{X}$ is surjective and
therefore every $\mathfrak{c}\in\widetilde{X}$ contains $\mathfrak{d}$. Thus
$\mathfrak{d}\subseteq\bigcap_{\mathfrak{c}\in\widetilde{X}}\mathfrak{c}=\mathfrak{nil}(A)$
and so there exists $p:A/\mathfrak{d}\to A/\mathfrak{nil}(A)$ such that the
following diagram commutes

$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A/\mathfrak{d}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{A/\mathfrak{nil}(A)}$

which yields (2).

We continue with (3). If $\widetilde{V_{a,b}}=\widetilde{X}$ for
$X=X^{\textup{sred}}=\operatorname{MSpec}(A)$ , then
$\mathfrak{nil}(A)=\mathfrak{c}_{\textup{triv}}$ and $(a,b)\in\mathfrak{c}$
for all $\mathfrak{c}\in\widetilde{X}$. Thus
$(a,b)\in\bigcap_{\mathfrak{c}\in\widetilde{X}}\mathfrak{c}=\mathfrak{nil}(A)=\mathfrak{c}_{\textup{triv}}$
which implies $a=b$. This shows (3) and concludes the proof. ∎

##  4\. Closed immersions and vanishing sets

In this section, we investigate the supports of closed subschemes of a monoid
scheme $X$ as subsets of its congruence space ${\widetilde{X}}$. A first
insight is that these are closed subsets of ${\widetilde{X}}$. However, not
every closed subset of ${\widetilde{X}}$ is the support of a closed subscheme.
The aim of this section is to characterize supports of closed subschemes in
terms of the vanishing sets of quasi-coherent congruence sheaves. This study
leads eventually to a topological characterization of closed immersions.

###  4.1. Closed subschemes

Closed immersions of monoid schemes were introduced in [7]. We recall the
definition.

######  Definition .

A morphism $\myvarphi:Y\to X$ of monoid schemes is _affine_ if the inverse
image $\myvarphi^{-1}(U)$ of every affine open subset $U$ of $X$ is affine. A
_closed immersion_ is an affine morphism $\myvarphi:Y\to X$ for which the
pullback $\Gamma U\to\Gamma V$ of sections is surjective for every affine open
$U$ of $X$ and $V=\myvarphi^{-1}(U)$. A _closed subscheme of $X$_ is an
isomorphism class of closed immersions into $X$. The _support of a closed
subscheme_ represented by a closed immersion $\myvarphi:Y\to X$ is the image
of ${\widetilde{\myvarphi}}$ in ${\widetilde{X}}$.

Note that isomorphic closed immersions have the same image in
${\widetilde{X}}$. Therefore the support of a closed subscheme does not depend
on the choice of representative. In fact, every closed subscheme of $X$ has a
canonical representative $\myvarphi:Z\to X$, which consists of the inclusion
of a topological subspace $Z$ of $X$ together with a quotient sheaf of
$({\mathcal{O}}_{X})|_{Z}$. In this case, ${\widetilde{Z}}$ appears as a
subset of ${\widetilde{X}}$ and is thus equal to the support of the closed
subscheme $Z$.

The relevance of congruence spaces is that they have enough closed subsets to
include the supports of closed subschemes. This is made precise in the
following result.

######  Proposition .

Let $\myvarphi:Y\to X$ be a morphism of monoid schemes and
$\widetilde{\myvarphi}:\widetilde{Y}\to\widetilde{X}$ the associated map of
congruence spaces. If $\myvarphi$ is a closed immersion, then
$\widetilde{\myvarphi}$ is a closed topological embedding. In particular, the
support of a closed subscheme of $X$ is closed in ${\widetilde{X}}$.

###### Proof.

The claim of the proposition is local in $X$, so we can assume that
$X=\operatorname{MSpec}(A)$ and $Y=\operatorname{MSpec}(A/\mathfrak{c})$ for
some congruence $\mathfrak{c}$ on $A$. We consider the morphism
$\Gamma_{\myiota}:A\to A/\mathfrak{c}$ of pointed monoids corresponding to the
morphism
$\myiota:\operatorname{MSpec}(A/\mathfrak{c})\to\operatorname{MSpec}(A)$. We
have ${\widetilde{\myiota}}(\mathfrak{d})=\\{(a,b)\in A\times
A\mid(\Gamma_{\myiota}(a),\Gamma_{\myiota}(b))\in\mathfrak{d}\\}$ for
$\mathfrak{d}\in{\widetilde{Y}}$. If
${\widetilde{\myiota}}(\mathfrak{d})={\widetilde{\myiota}}(\mathfrak{d^{\prime}})$
for $\mathfrak{d},\mathfrak{d^{\prime}}\in{\widetilde{X}}$, then
$(\Gamma_{\myiota}(a),\Gamma_{\myiota}(b))\in\mathfrak{d}$ if and only if
$(\Gamma_{\myiota}(a),\Gamma_{\myiota}(b))\in\mathfrak{d^{\prime}}$ for all
$(a,b)\in A\times A$. This implies $\mathfrak{d}=\mathfrak{d^{\prime}}$ since
$\Gamma_{\myiota}$ is surjective. Hence ${\widetilde{\myiota}}$ is injective.
To prove our claim it is enough to check that ${\widetilde{\myiota}}$ is
closed. Consider $\bar{a},\bar{b}\in A/\mathfrak{c}$ where
$\bar{a}=\Gamma_{\myiota}(a),\bar{b}=\Gamma_{\myiota}(b)$. Then

$\displaystyle{\widetilde{\myiota}}(Z_{\bar{a},\bar{b}})\ $ $\displaystyle=\
\\{\mathfrak{e}\in{\widetilde{Y}}\mid\mathfrak{e}={\widetilde{\myiota}}(\mathfrak{d})\text{\leavevmode\nobreak\
with\leavevmode\nobreak\
}(\bar{a},\bar{b})\in\mathfrak{d}\text{\leavevmode\nobreak\ for
some\leavevmode\nobreak\ }\mathfrak{d}\in{\widetilde{X}}\\}$ $\displaystyle=\
\\{\mathfrak{e}\in{\widetilde{Y}}\mid\mathfrak{c}\subseteq\mathfrak{e}\text{\leavevmode\nobreak\
and\leavevmode\nobreak\ }(a,b)\in\mathfrak{e}\\}=\
Z_{a,b}\cap(\bigcap_{(c,d)\in\mathfrak{c}}Z_{c,d}).\qed$

######  Remark .

Note that the definition of closed immersion $\myvarphi:Y\to X$ in [7] asks in
addition that $\myvarphi$ is a homeomorphism onto its image. We can omit this
condition since it follows from the other properties of a closed immersion.

Indeed, since the question is local in $X$, we can assume that
$X=\operatorname{MSpec}(A)$ and its inverse image $Y=\operatorname{MSpec}(B)$
are affine. By the definition of a closed immersion, $B$ is a quotient of $A$.
Thus pulling back prime ideals from $B$ to $A$ is injective. It is a
homeomorphism onto its image since
$\myvarphi(U_{[h]})=U_{h}\cap\textup{im}\myvarphi$ for every $h\in A$ with
class $[h]$ in $B$.

###  4.2. Congruence sheaves

We recall some basic notions around modules over monoids from [3]. A pointed
monoid $A$ acts naturally on $A\times A$ via the diagonal multiplication. A
_submodule of $A\times A$_ is a nonempty subset $M$ of $A\times A$ such that
$(ca,cb)\in M$ for all $c\in A$ and $(a,b)\in M$. Given a morphism $f:A\to B$,
the _base extension of $M$ along $f$_ is the submodule

$M\otimes_{A}B\ =\ \big{\\{}(a,b)\in B\times B\leavevmode\nobreak\
\big{|}\leavevmode\nobreak\ a=cf(a^{\prime})\text{ and }b=cf(b^{\prime})\text{
for some }a^{\prime},b^{\prime}\in A,\,c\in B\big{\\}}$

of $B\times B$. The _quasi-coherent ${\mathcal{O}}_{X}$-module on
$X=\operatorname{MSpec}(A)$ associated with $M$_ is the sheaf of monoids
$M^{\sim}$ with

$M^{\sim}(U)\ =\ M\otimes_{A}\Gamma U$

for all affine opens $U$ of $X$ with respect to the restriction map
$\operatorname{res}_{X,U}\colon A=\Gamma X\to\Gamma U$.

######  Definition .

Let $X$ be a monoid scheme. A _congruence sheaf on $X$_ is a subsheaf
${\mathfrak{C}}$ of $\mathcal{O}_{X}\times\mathcal{O}_{X}$ such that
$\mathfrak{C}(U)$ is a submodule of
$\mathcal{O}_{X}(U)\times\mathcal{O}_{X}(U)$ and an equivalence relation for
all open subsets $U$ of $X$. A congruence sheaf ${\mathfrak{C}}$ is _quasi-
coherent_ if ${\mathfrak{C}}|_{U}\simeq{{\mathfrak{C}}(U)}^{\sim}$ for every
affine open subset $U$ of $X$.

Let $\myvarphi:Y\to X$ be a morphism of monoid schemes. The _congruence kernel
of $\myvarphi$_ is the congruence sheaf $\operatorname{Congker}(\myvarphi)$ on
$X$ given by

$\operatorname{Congker}(\myvarphi)(U)\ =\
\operatorname{congker}\big{(}\myvarphi^{\\#}(U):\Gamma
U\to\Gamma\myvarphi^{-1}(U)\big{)}$

for every open subset $U$ of $X$.

Note that $\operatorname{Congker}(\myvarphi)$ inherits the sheaf property from
${\mathcal{O}}_{X}\times{\mathcal{O}}_{X}$ and is thus indeed a sheaf. It is
in general not quasi-coherent. It is, however, for closed immersions, as the
next result shows, and more generally for quasi-compact morphisms, as proven
in Lemma 4.6.

######  Lemma .

Let $\myvarphi:Y\to X$ be a closed immersion of monoid schemes. Then
$\operatorname{Congker}(\myvarphi)$ is a quasi-coherent congruence sheaf.
Conversely, any quasi-coherent congruence sheaf determines a closed immersion
of monoid schemes.

###### Proof.

Let $\myvarphi:Y\to X$ be a closed immersion of monoid schemes. The claim of
the proposition is local in $X$, so we can assume that
$X=\operatorname{MSpec}(A)$ and $Y=\operatorname{MSpec}(A/\mathfrak{c})$ for
some congruence $\mathfrak{c}$ on $A$. We will show that
$\operatorname{Congker}(\myvarphi)={\mathfrak{c}}^{\sim}$. Since
$\operatorname{Congker}(\myvarphi)$ is a sheaf, it is enough to check that
$\operatorname{Congker}(\myvarphi)(U)={\mathfrak{c}}^{\sim}(U)$ for every
affine open $U$ of $X$. Since every affine open of $X$ is a principal open, we
can assume that $U=\operatorname{MSpec}((A)[s^{-1}])$ for some $s\in A$. We
note that by universal property of pullback it follows
$\Gamma\myvarphi^{-1}(\operatorname{MSpec}((A)[s^{-1}]))\cong\Gamma\operatorname{MSpec}((A/{\mathfrak{c}})[\myvarphi(s)^{-1}])\operatorname{Cong}(A)/{\mathfrak{c}}[\myvarphi(s)^{-1}]$.
Since localization of monoids is exact by Lemma 2.2, we have

$\displaystyle\operatorname{Congker}(\myvarphi)(U)\ $ $\displaystyle=\
\operatorname{congker}\big{(}\myvarphi^{\\#}(U):A[s^{-1}]\to
A/{\mathfrak{c}}[\myvarphi(s)^{-1}]\big{)}$
$\displaystyle=S^{-1}\big{(}\operatorname{congker}(A\to
A/{\mathfrak{c}})\big{)}\qquad\text{where\leavevmode\nobreak\
}S=\\{1,s,s^{2},s^{3},\ldots\\}$
$\displaystyle=S^{-1}{\mathfrak{c}}={\mathfrak{c}}\otimes_{A}A[s^{-1}]={\mathfrak{c}}^{\sim}(U).$

Conversely, let ${\mathfrak{C}}$ be a quasi-coherent congruence sheaf on $X$.
For any affine open $U=\operatorname{MSpec}(A)$ of $X$, we know that the
surjection $A\to A/{\mathfrak{C}}(U)$ determines a closed immersion. Since
${\mathfrak{C}}$ is quasi-coherent, we obtain a closed immersion
$\myvarphi:Y\to X$ with
$\myvarphi|_{\myvarphi^{-1}(U)}:\operatorname{MSpec}(A/{\mathfrak{C}}(U))\to
U=\operatorname{MSpec}(A)$. ∎

###  4.3. Vanishing sets

Let $A$ be a pointed monoid with spectrum $X=\operatorname{MSpec}(A)$ and
congruence space ${\widetilde{X}}$. Recall that the topology of closed subsets
of ${\widetilde{X}}$ is generated by the closed subsets
$V_{a,b}=\\{{\mathfrak{p}}\in{\widetilde{X}}\mid(a,b)\in{\mathfrak{p}}\\}$
with $a,b\in A$. For a subset $S$ of $A\times A$, we define

$V_{S}\ =\ \bigcap_{(a,b)\in S}\ V_{a,b}.$

Note that not every closed subset of ${\widetilde{X}}$ is of the form $V_{S}$;
cf. Example 2.3.

######  Lemma .

Let $A$ be a pointed monoid, $S$ a subset of $A\times A$ and
${\mathfrak{c}}=\langle S\rangle$ the congruence generated by $S$. Then

$V_{S}\ =\ V_{\mathfrak{c}}\ =\ V_{\sqrt{{\mathfrak{c}}}}.$

###### Proof.

The inclusions $V_{\sqrt{{\mathfrak{c}}}}\subseteq V_{\mathfrak{c}}\subseteq
V_{S}$ follow from elementary set theoretic considerations. If
${\mathfrak{p}}$ is a prime congruence containing $S$, then we also have
${\mathfrak{c}}=\langle S\rangle\subseteq{\mathfrak{p}}$ and therefore $V_{S}\
\subseteq\ V_{\mathfrak{c}}$. Since
$\sqrt{{\mathfrak{c}}}=\bigcap_{{\mathfrak{c}}\subseteq{\mathfrak{p}}}{\mathfrak{p}}$
it follows that $V_{\mathfrak{c}}\subseteq V_{\sqrt{{\mathfrak{c}}}}$. ∎

######  Lemma .

Let $\myiota_{S}:A\to S^{-1}A$ be a localization of a pointed monoid $A$ at
$S$ and let
$\myiota_{S}^{*}:\operatorname{Cong}(S^{-1}A)\to\operatorname{Cong}(A)$
denotes the corresponding morphism of congruence spaces. Then
${\myiota_{S}^{*}}^{-1}(V_{\mathfrak{c}})=V_{\myiota_{S,*}({\mathfrak{c}})}$.

###### Proof.

We have
${\myiota_{S}^{*}}^{-1}(V_{\mathfrak{c}})=\\{{\mathfrak{d}}\in\operatorname{Cong}(B)\leavevmode\nobreak\
|\leavevmode\nobreak\
{\mathfrak{c}}\subseteq\myiota_{S}^{*}({\mathfrak{d}})\\}$. By Proposition
2.2, we know that $\myiota_{S,*}$ and $\myiota_{S}^{*}$ are mutually inverse
bijections. It follows that
$\myiota_{S,*}(\myiota_{S}^{*}({\mathfrak{d}}))={\mathfrak{d}}$ and
$\leavevmode\nobreak\ {\mathfrak{c}}\subseteq\myiota_{S}^{*}({\mathfrak{d}})$
if and only if $\myiota_{S,*}({\mathfrak{c}})\subseteq{\mathfrak{d}}$. Thus we
have $\myvarphi^{-1}(V_{\mathfrak{c}})=V_{\myiota_{S,*}({\mathfrak{c}})}$. ∎

######  Definition .

Let $X$ be a monoid scheme with congruence space ${\widetilde{X}}$. A
_vanishing set of $X$_ is a subset $Z$ of ${\widetilde{X}}$ such that for
every affine open $U$ of $X$ with congruence space ${\widetilde{U}}$, the
intersection $Z\cap{\widetilde{U}}$ is of the form $V_{\mathfrak{c}}$ for some
congruence ${\mathfrak{c}}$ on $\Gamma U$.

######  Lemma .

Let $X$ be a monoid scheme with congruence space ${\widetilde{X}}$ and
${\mathcal{U}}$ an affine open covering of $X$. Then a subset $Z$ of
${\widetilde{X}}$ is a vanishing set if for every $U\in{\mathcal{U}}$ with
congruence space ${\widetilde{U}}$, the intersection $Z\cap{\widetilde{U}}$ is
of the form $V_{\mathfrak{c}}$ for some congruence ${\mathfrak{c}}$ on $\Gamma
U$.

###### Proof.

Consider an affine open $U$ of $X$. Then the unique maximal ideal $M_{\Gamma
V}$ of $\Gamma V$ is its unique closed point, which is contained in some
$W\in{\mathcal{U}}$. Since $W$ is open, it contains the whole subset $U$. Thus
$\Gamma U=S^{-1}\Gamma W$ for $S=\\{a\in\Gamma
W\mid\operatorname{res}_{W,U}(a)\notin M_{\Gamma U}\\}$. If
$Z\cap{\widetilde{W}}=V_{\mathfrak{c}}$, as required by the hypothesis of the
lemma, then $Z\cap{\widetilde{U}}=V_{S^{-1}{\mathfrak{c}}}$ by Proposition
3.2. This shows that $Z$ is a vanishing set. ∎

###  4.4. Supports of closed subschemes

######  Definition .

Let $X$ be a monoid scheme with congruence space ${\widetilde{X}}$ and
${\mathfrak{C}}$ a quasi-coherent congruence sheaf on $X$. A _vanishing set of
${\mathfrak{C}}$_ is a vanishing set $Z$ of ${\widetilde{X}}$ such that
$Z\cap{\widetilde{U}}=V_{{\mathfrak{C}}(U)}$ for every affine open subset $U$
of $X$ with congruence space ${\widetilde{U}}$.

######  Theorem 4.1.

Let $X$ be a monoid scheme with congruence space ${\widetilde{X}}$ and $Z$ a
subset of ${\widetilde{X}}$. Then the following are equivalent:

  1. (1)

$Z$ is the support of a closed subscheme of $X$.

  2. (2)

$Z$ is the vanishing set of a quasi-coherent congruence sheaf on $X$.

  3. (3)

$Z$ is a vanishing set.

###### Proof.

Let $Z$ be the support of a closed subscheme of $X$ represented by a closed
immersion $\myvarphi:Y\to X$, i.e.,
$Z={\widetilde{\myvarphi}}({\widetilde{Y}})$. By Lemma 4.2, we know that the
$\operatorname{Congker}(\myvarphi)$ is a quasi-coherent congruence sheaf of
$X$. We claim that $Z$ is the vanishing set of the quasi-coherent congruence
sheaf ${\mathfrak{C}}:=\operatorname{Congker}(\myvarphi)$. Let
$U=\operatorname{MSpec}(A)$ be an affine open of $X$. Then
$\myvarphi^{-1}(U)=\operatorname{MSpec}(A/{\mathfrak{C}}(U))$ and
$Z\cap{\widetilde{U}}={\widetilde{\myvarphi}}(\operatorname{Cong}(A/{\mathfrak{C}}(U)))=\\{{\mathfrak{p}}\in\operatorname{Cong}(A)\leavevmode\nobreak\
|\leavevmode\nobreak\
{\mathfrak{C}}(U)\subseteq{\mathfrak{p}}\\}=V_{{\mathfrak{C}}(U)}$. This
proves our claim and shows that
$\eqref{char1}\implies\eqref{char2}\implies\eqref{char3}$.

Let $Z$ be a vanishing set. For any affine open subset $U$ of $X$, we consider
the strongly reduced congruence ${\mathfrak{c}}_{U}$ of $\Gamma U$ with
$V_{{\mathfrak{c}}_{U}}=Z\cap{\widetilde{U}}$. It follows from Lemma 4.3 that
we have a quasi-coherent congruence sheaf ${\mathfrak{C}}$ given by
${\mathfrak{C}}(U)={\mathfrak{c}}_{U}$. Therefore by Lemma 4.2, we have a
closed immersion $\myvarphi:Y\to X$ with
$\myvarphi|_{\myvarphi^{-1}(U)}:\operatorname{MSpec}(A/{\mathfrak{c}}_{U})\to
U=\operatorname{MSpec}(A)$ for every affine $U$ of $X$. We have
$(\myvarphi|_{\myvarphi^{-1}(U)})^{\textup{cong}}(\operatorname{Cong}(A/{\mathfrak{c}}_{U}))=V_{{\mathfrak{c}}_{U}}=Z\cap{\widetilde{U}}$.
Thus it follows ${\widetilde{\myvarphi}}({\widetilde{Y}})=Z$. This shows
$\eqref{char3}\implies\eqref{char1}$. ∎

######  Corollary .

Every quasi-coherent congruence sheaf has a vanishing set.

###### Proof.

Any quasi-coherent congruence sheaf on $X$ determines a closed immersion
$\myvarphi:Y\to X$ by Lemma 4.2 and the support of the closed subscheme
$[\myvarphi:Y\to X]$ is a vanishing set by the implication (1)$\Rightarrow$(2)
of Theorem 4.1. ∎

We denote the vanishing set of a quasi-coherent congruence sheaf
${\mathfrak{C}}$ on $X$ by $V_{\mathfrak{C}}$ and we note that for every
affine $U$ of $X$, we have
$V_{\mathfrak{C}}\cap{\widetilde{U}}=V_{{\mathfrak{C}}(U)}$.

######  Theorem 4.2.

Let $X$ be a monoid scheme. Then there are mutually inverse bijections

${\big{\\{}\text{closed subschemes of
$X$}\big{\\}}}$${\Big{\\{}\begin{array}[]{c}\text{quasi-coherent}\\\
\text{congruence sheaves on
$X$}\end{array}\Big{\\}}}$$\scriptstyle{\Phi}$$\scriptstyle{\Psi}$

that are given by $\Phi([\myvarphi:Y\to X])=\operatorname{Congker}(\myvarphi)$
and $\Psi({\mathfrak{C}})=[\myvarphi:Y\to X]$ where
$Y=\mypi_{X}(V_{\mathfrak{C}})$, embedded via $\myvarphi$ as a subspace of
$X$, and ${\mathcal{O}}_{Y}=({\mathcal{O}}_{X}/{\mathfrak{C}})|_{Y}$.

###### Proof.

As the first part of the proof, we very that $\Phi$ and $\Psi$ are well-
defined. If $[\myvarphi:Y\to X]$ is a closed subscheme, then we know by Lemma
4.2 that $\operatorname{Congker}(\myvarphi)$ is a quasi-coherent congruence
sheaf. Thus $\Phi$ is well defined.

Let $\mathfrak{C}$ be any quasi-coherent congruence sheaf on $X$. Then by
Lemma 4.2, we have a closed immersion $\myvarphi:Y\longrightarrow X$ with
$\myvarphi|_{\myvarphi^{-1}(U)}:\operatorname{MSpec}(A/{\mathfrak{C}}(U))\to
U=\operatorname{MSpec}(A)$. As a subspace of $X$, we indeed have
$Y=\mypi_{X}(V_{\mathfrak{C}})$ since for any affine open
$U=\operatorname{MSpec}(A)$ of $X$ we have

$\mypi_{X}(V_{\mathfrak{C}})\cap U\ =\
\mypi_{X}(V_{\mathfrak{C}}\cap\widetilde{U})\ =\
\mypi_{X}(V_{\mathfrak{C}(U)})\\\ =\
\mypi_{X}(\operatorname{Cong}(A/\mathfrak{C}(U)))\ =\
\operatorname{MSpec}(A/\mathfrak{C}(U)).$

In addition, $({\mathcal{O}}_{X}/{\mathfrak{C}})|_{U}$ is the quasi-coherent
sheaf of monoids $({A/\mathfrak{C}(U)})^{\sim}$ associated with the $A$-module
$A/\mathfrak{C}(U)$. Therefore

$({\mathcal{O}}_{X}/{\mathfrak{C}})|_{\operatorname{MSpec}(A/\mathfrak{C}(U))}=({A/\mathfrak{C}(U)\otimes_{A}A/\mathfrak{C}(U)})^{\sim}=({A/\mathfrak{C}(U)})^{\sim}={\mathcal{O}}_{\operatorname{MSpec}(A/\mathfrak{C}(U))}$

where $A/\mathfrak{C}(U)$ is viewed as an $A/\mathfrak{C}(U)$-module. Since an
affine cover of $Y$ is given by
$(\operatorname{MSpec}(A/\mathfrak{C}(U)),{\mathcal{O}}_{\operatorname{MSpec}(A/\mathfrak{C}(U))})$,
it follows that ${\mathcal{O}}_{Y}=({\mathcal{O}}_{X}/{\mathfrak{C}})|_{Y}$.
Therefore $\Psi$ is also well defined.

We turn to the proof that $\Phi$ and $\Psi$ are mutually inverse bijections.
Let $\mathfrak{C}$ be a quasi-coherent congruence sheaf on $X$. For any affine
open $U=\operatorname{MSpec}(A)$ of $X$, we have
$\Phi(\Psi(\mathfrak{C}))|_{U}=\operatorname{Congker}(\myvarphi)|_{U}$ where
$[\myvarphi:\mypi_{X}(V_{\mathfrak{C}})\longrightarrow X]$ denotes the closed
subscheme associated with $\mathfrak{C}$. Since

$\operatorname{Congker}(\myvarphi)|_{U}\ =\
{\operatorname{congker}(\myvarphi^{\\#}(U))}^{\sim}\ =\
{\mathfrak{C}(U)}^{\sim}\ =\ \mathfrak{C}|_{U},$

it follows that $\Phi(\Psi(\mathfrak{C}))=\mathfrak{C}$.

Conversely, let $[\myvarphi:Y\to X]$ be a closed subscheme of $X$. By Theorem
4.1$(1)$, we know that
$\widetilde{\myvarphi}(\widetilde{Y})=V_{\operatorname{Congker}(\myvarphi)}$.
Therefore
$\mypi_{X}(V_{\operatorname{Congker}(\myvarphi)})=\mypi_{X}(\widetilde{\myvarphi}(\widetilde{Y}))=\myvarphi(\mypi_{Y}(\widetilde{Y}))=\myvarphi(Y)$.
Thus $\Psi(\Phi([\myvarphi:Y\longrightarrow X]))=[\myvarphi:Y\longrightarrow
X]$, which completes the proof. ∎

######  Remark .

The results of this section can be summarized in the following diagram:

${\big{\\{}\text{vanishing sets of
$X$}\big{\\}}}$${\Big{\\{}\begin{array}[]{c}\text{strongly reduced}\\\
\text{closed subschemes of
$X$}\end{array}\Big{\\}}}$${\Big{\\{}\begin{array}[]{c}\text{quasi-
coherent}\\\ \text{congruence sheaves on
$X$}\end{array}\Big{\\}}}$${\big{\\{}\text{closed subschemes of
$X$}\big{\\}}}$bijectivebijective

The upper horizontal arrow identifies a vanishing set with the unique strongly
reduced closed subscheme supported on it and the lower horizontal arrow stands
for the bijection from Theorem 4.2. The downwards arrow on the left hand side
sends a vanishing set to the corresponding reduced quasi-coherent congruence
sheaf, the upwards arrow sends a quasi-coherent congruence sheaf to its
vanishing set. The downwards arrow on the right hand side is the natural
inclusion, the upwards arrow sends a closed subscheme to its strong reduction.

All of these arrows are functorial in, respectively, inclusions of subspaces,
surjections of sheaves and closed immersions of closed subschemes. The
horizontal arrows become equivalences of categories and the vertical arrows
are mutual adjoints (one is an embedding, the other a retract of categories).
The horizontal arrows commute with the downwards arrows and with the upwards
arrows.

###  4.5. Quasi-compact morphisms

######  Definition .

A morphism $\myvarphi:Y\to X$ of monoid schemes is _quasi-compact_ if
$\myvarphi^{-1}(U)$ is quasi-compact for all affine open subsets $U$ of $X$.

Note that an affine monoid scheme $Y=\operatorname{MSpec}(A)$ is always quasi-
compact since it has a unique closed point. Therefore affine morphisms are
quasi-compact. Since finite unions of affine opens are also quasi-compact,
morphisms of finite type are quasi-compact. In general, quasi-compactness of
morphisms can be verified on a chosen open affine covering ${\mathcal{U}}$ of
$X$. That is, $\myvarphi:Y\to X$ is quasi-compact if $\myvarphi_{i}^{-1}(U)$
is quasi-compact for all $U\in{\mathcal{U}}$.

###  4.6. Vanishing closure

######  Definition .

Let $X$ be a monoid scheme with congruence space ${\widetilde{X}}$. Let $Z$ be
a subset of ${\widetilde{X}}$. The _vanishing closure of $Z$_ is the vanishing
set

$Z^{\textup{vcl}}\ =\ \bigcap_{\begin{subarray}{c}Z\subset V\\\ \text{$V$
vanishing set}\end{subarray}}V.$

Note that $Z^{\textup{vcl}}$ is indeed a vanishing set due to its affine
nature and the relation $V_{\mathfrak{c}}\cap
V_{{\mathfrak{c}}^{\prime}}=V_{\langle{\mathfrak{c}}\cup{\mathfrak{c}}^{\prime}\rangle}$
given by Lemma 4.3.

######  Lemma .

The congruence kernel of a quasi-compact morphism of monoid schemes
$\myvarphi:Y\to X$ is quasi-coherent.

###### Proof.

Since the question is local in $X$, we can assume that
$X=\operatorname{MSpec}(B)$ is affine. Consider a principal affine open
$U=\operatorname{MSpec}(B[h^{-1}])\subseteq X$ and let $V$ denote the open
subset $\myvarphi^{-1}(U)=U\times_{X}Y$ of $Y$. Let

$\mathfrak{c}=\operatorname{congker}(\myvarphi^{\\#}(X):B\longrightarrow\Gamma
Y)\quad\text{and}\quad\mathfrak{c}_{h}=\operatorname{congker}(\myvarphi^{\\#}(U):B[h^{-1}]\longrightarrow\Gamma
V).$

We need to show $\mathfrak{c}_{h}=S^{-1}\mathfrak{c}$ for
$S=\\{h^{i}\\}_{i\geqslant 0}$. Since $\myvarphi$ is quasi-compact, we have
$Y=\bigcup_{i\in I}W_{i}$ for affine open $W_{i}=\operatorname{MSpec}(C_{i})$
where $I$ is a finite set. Therefore the corresponding morphism of global
sections $\Gamma Y\to\prod_{i}C_{i}$ is injective. If
$\bar{h}=\myvarphi^{\\#}(X)(h)\in\Gamma Y$ and
$\bar{h_{i}}=\operatorname{res}_{Y,W_{i}}(\bar{h})\in C_{i}$ then
$C_{i}\otimes_{B}B[h^{-1}]=C[\bar{h_{i}}^{-1}]$. Thus
$W_{i}\times_{X}U=\operatorname{MSpec}(C_{i}[\bar{h_{i}}^{-1}])$ and we have
the following commutative diagram

$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mylambda_{h}}$$\scriptstyle{\myvarphi^{\\#}(X)}$$\textstyle{\Gamma
Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\myzeta}$$\scriptstyle{\operatorname{res}_{Y,V}}$$\textstyle{\prod_{i\in
I}C_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\prod\mylambda_{\bar{h_{i}}}}$$\scriptstyle{\mylambda_{\bar{h}}}$$\textstyle{B[h^{-1}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\myvarphi^{\\#}(U)}$$\textstyle{\Gamma
V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\myeta}$$\textstyle{\prod_{i\in
I}C_{i}[\bar{h_{i}}^{-1}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\myiota}$$\textstyle{(\prod_{i\in
I}C_{i})[\bar{h}^{-1}]}$

where $\myiota$ is an isomorphism since $I$ is finite. Both $\myeta$ and
$\myzeta$ are injections. Thus if $\mygamma=\myzeta\circ\myvarphi^{\\#}(X)$
and $\mydelta=\myiota\circ\myeta\circ\myvarphi^{\\#}(U)$, then we have
$\mathfrak{c}=\operatorname{congker}(\mygamma)$ and
$\mathfrak{c}_{h}=\operatorname{congker}(\mydelta)$.

For to show that $S^{-1}\mathfrak{c}=\mathfrak{c}_{h}$, it is enough to check
that any element of the form $(\frac{b}{1},\frac{b^{\prime}}{1})$ in
$S^{-1}\mathfrak{c}$ also belongs to $\mathfrak{c}_{h}$ and vice versa. If
$(\frac{b}{1},\frac{b^{\prime}}{1})\in S^{-1}\mathfrak{c}$, then there exists
$t\in S$ such that $(bt,b^{\prime}t)\in\mathfrak{c}$ and therefore we have
$(\mylambda_{\bar{h}}\circ\mygamma)(bt)=(\mylambda_{\bar{h}}\circ\mygamma)(b^{\prime}t)$.
It follows from the commutativity of the above diagram that
$\mydelta(\frac{bt}{1})=\mydelta(\frac{b^{\prime}t}{1})$. This implies that
$(\frac{bt}{1},\frac{b^{\prime}t}{1})\in\mathfrak{c}_{h}$ or, equivalently,
$(\frac{b}{1},\frac{b^{\prime}}{1})\in\mathfrak{c}_{h}$.

Conversely, consider any $(\frac{b}{1},\frac{b^{\prime}}{1})$ in
$\mathfrak{c}_{h}$. Then we have
$(\mydelta\circ\mylambda_{h})(b)=(\mydelta\circ\mylambda_{h})(b^{\prime})$. It
follows from the commutativity of the above diagram that
$\frac{\mygamma(b)}{1}=\frac{\mygamma(b^{\prime})}{1}$, i.e., there exists
some $i\in\mathbb{Z}$ such that
$\mygamma(b)\bar{h}^{i}=\mygamma(b^{\prime})\bar{h}^{i}$. In other words, we
have $\mygamma(bh^{i})=\mygamma(b^{\prime}h^{i})$, i.e.,
$(bh^{i},b^{\prime}h^{i})\in\mathfrak{c}$. Thus
$(\frac{b}{1},\frac{b^{\prime}}{1})$ in $S^{-1}\mathfrak{c}$. ∎

######  Lemma .

Let $\myvarphi:Y\to X$ be a quasi-compact morphism of monoid schemes and
$\tilde{\myvarphi}:{\widetilde{Y}}\to{\widetilde{X}}$ the associated map of
congruence spaces. Then

$V_{\operatorname{Congker}(\myvarphi)}\ =\
\textup{im}(\tilde{\myvarphi})^{\textup{vcl}}.$

###### Proof.

Since $\myvarphi$ is quasi-compact, the congruence kernel
$\operatorname{Congker}(\myvarphi)$ defines a quasi-coherent congruence sheaf
by Section 4.6, which means that it defines a closed subscheme $Z$ of $X$ with
vanishing set $V_{\operatorname{Congker}(\myvarphi)}$.

The closed subscheme $Z$ is, by definition of the congruence kernel and its
quasi-coherence, the smallest closed subscheme of $X$ such that
${\widetilde{Z}}$ contains the image of $\tilde{\myvarphi}$. Since every
vanishing set is the support of the congruence space of some closed subscheme
of $X$, this establishes the claim of the lemma. ∎

###  4.7. A topological characterization of closed immersions

######  Theorem 4.3.

Let $\myvarphi:Y\to X$ be a morphism of monoid scheme and
$\tilde{\myvarphi}:{\widetilde{Y}}\to{\widetilde{X}}$ the associated map of
congruence spaces. Then the following are equivalent.

  1. (1)

$\myvarphi:Y\to X$ is a closed immersion.

  2. (2)

$\myvarphi:Y\to X$ is a quasi-compact topological embedding,
$\myvarphi^{\\#}:{\mathcal{O}}_{X}\to\myvarphi_{\ast}{\mathcal{O}}_{Y}$ is
surjective and the image of $\tilde{\myvarphi}$ is a vanishing set.

###### Proof.

By Proposition 4.1, (1) implies (2). To prove the converse implication, assume
(2) and let ${\mathfrak{C}}$ be the congruence kernel of $\myvarphi$. By Lemma
4.6, the image of $\tilde{\myvarphi}$ is equal to the vanishing set
$V_{\mathfrak{C}}$ of ${\mathfrak{C}}$. Thus $\myvarphi$ factors through the
closed immersion $\mypsi:Z\to X$ corresponding to ${\mathfrak{C}}$. After
substituting $X$ by $Z$, we can thus assume that
$\myvarphi^{\\#}:{\mathcal{O}}_{X}\to\myvarphi_{\ast}{\mathcal{O}}_{Y}$ is an
isomorphism of sheaves and that $\tilde{\myvarphi}$ is a homeomorphism.

Since the problem is local in $X$, we can assume that $X$ is affine, i.e.
$X=\operatorname{MSpec}(\Gamma X)\simeq\operatorname{MSpec}(\Gamma Y)$. This
reduces us to the situation that $\myvarphi$ is the canonical morphism
$\myvarphi:Y\to\operatorname{MSpec}(\Gamma Y)=X$. Under these assumptions, it
suffices to show that $Y$ is affine, since this implies that
$Y=\operatorname{MSpec}(\Gamma Y)=X$ and that $\myvarphi:Y\to X$ is the
identity.

Let $M$ be the maximal prime ideal of $B=\Gamma X$. Then its fibre
$\mypi_{X}^{-1}(M)$ in ${\widetilde{X}}$ has a unique closed point
${\mathfrak{m}}$, which is the congruence kernel of the composition $B\to
M/B\to{{\mathbb{F}}_{1}}$ of the quotient map $B\to B/M$ with the unique map
$B/M\to{{\mathbb{F}}_{1}}$ that maps all units of the pointed group $B/M$ to
$1$. Let $Z=\myvarphi^{-1}({\mathfrak{m}})$ and
${\mathfrak{n}}\in{\widetilde{Y}}$ be the unique point that maps to
${\mathfrak{m}}$. Since
$\mypi_{X}\circ\tilde{\myvarphi}=\myvarphi\circ\mypi_{Y}$ and since
$\tilde{\myvarphi}$ is a homeomorphism, ${\mathfrak{n}}$ is the unique closed
point of $\mypi_{Y}^{-1}(Z)$. Let $N=\mypi_{Y}({\mathfrak{n}})$.

Let $y$ be any closed point of $Y$. Since $\myvarphi$ is a topological
embedding, $\myvarphi(y)=M$ and thus $y\in Z$. Since $y$ is closed,
$\mypi_{Y}^{-1}(y)$ is closed in ${\widetilde{Y}}$ and thus contains
${\mathfrak{n}}$. This shows that $y=N$. We conclude that $Y$ has a unique
closed point and is therefore affine, which concludes the proof. ∎

###  4.8. Dominant morphisms

######  Definition .

Let $X$ be a monoid scheme and ${\widetilde{X}}$ its congruence space. A
subset $Z$ of ${\widetilde{X}}$ is _strictly dense_ if its vanishing closure
is ${\widetilde{X}}$. A morphism $\myvarphi:Y\to X$ of monoid schemes is
_dominant_ if the associated map
$\tilde{\myvarphi}:{\widetilde{Y}}\to{\widetilde{X}}$ of congruence spaces has
a strictly dense image.

######  Lemma .

If $\myvarphi:Y\to X$ is a dominant morphism of monoid schemes, then
$\myvarphi(Y)$ is dense in $X$.

###### Proof.

Since the morphisms $\mypi_{Y}:\widetilde{Y}\to Y$ are functorial and
$\mypi_{Y}\circ\mysigma_{Y}=\textup{id}_{Y}$, the diagram

${\widetilde{Y}}$${\widetilde{X}}$${Y}$${X}$$\scriptstyle{{\tilde{\myvarphi}}}$$\scriptstyle{\mypi_{X}}$$\scriptstyle{\mysigma_{Y}}$$\scriptstyle{\myvarphi}$

commutes. By definition, $\mysigma_{Y}(p)$ is minimal in $\mypi_{Y}^{-1}(p)$
for all $p\in Y$, i.e.,
$\mypi_{Y}^{-1}(p)\subseteq\overline{\\{\mysigma_{Y}(p)\\}}$. We consider any
$\mathfrak{c}\in{\tilde{\myvarphi}}(\widetilde{Y})$ and let
$\mathfrak{c}={\tilde{\myvarphi}}(\mathfrak{d})$. Let $V$ be any open in
$\widetilde{X}$ such that $\mathfrak{c}\in V$. If $q=\mypi_{Y}(\mathfrak{d})$,
then we have
$\mathfrak{d}\in\mypi_{Y}^{-1}(q)\subseteq\overline{\\{\mysigma_{Y}(q)\\}}$
which implies $\mysigma_{Y}(q)\in{\tilde{\myvarphi}}^{-1}(V)$. Thus
${\tilde{\myvarphi}}(\mysigma_{Y}(q))\in V$ and so
$V\cap{\tilde{\myvarphi}}(\mysigma_{Y}(Y))\neq\emptyset$. Therefore it follows
that
$\overline{{\tilde{\myvarphi}}(\widetilde{Y})}=\overline{{\tilde{\myvarphi}}\circ\mysigma_{Y}(Y)}$.
Hence
$\widetilde{X}=\overline{{\tilde{\myvarphi}}(\widetilde{Y})}=\overline{{\tilde{\myvarphi}}\circ\mysigma_{Y}(Y)}\subseteq\mypi_{X}^{-1}(\overline{\myvarphi(Y)})$.
This shows that $\overline{\myvarphi(Y)}=X$. ∎

######  Remark .

Note that $\myvarphi:Y\to X$ does not need to be dominant if $\myvarphi(Y)$ is
dense in $X$. For example, the diagonal
$\Delta:{\mathbb{A}}^{1}_{{\mathbb{F}}_{1}}\to{\mathbb{A}}^{2}_{{\mathbb{F}}_{1}}$
has a dense image, but is not dominant.

On the other hand, a strictly dense image of the associated map
$\tilde{\myvarphi}:{\widetilde{Y}}\to{\widetilde{X}}$ of congruence spaces
does not imply that the image of $\tilde{\myvarphi}$ is dense, as attested by
the morphism
$\myvarphi:\operatorname{MSpec}({{\mathbb{F}}_{1}})\myamalg\operatorname{MSpec}({{\mathbb{F}}_{1}})\to{\mathbb{A}}^{1}_{{\mathbb{F}}_{1}}$
that stems from the two different maps
${{\mathbb{F}}_{1}}[t]\to{{\mathbb{F}}_{1}}$, one which sends $t$ to $0$, the
other which sends $t$ to $1$. The image of $\tilde{\myvarphi}$ consists of the
two prime congruences $\langle(t,0)\rangle$ and $\langle(t,1)\rangle$ on
${\mathbb{A}}^{1}_{{\mathbb{F}}_{1}}$, which form a proper closed subset of
$\operatorname{Cong}({\mathbb{A}}^{1}_{{\mathbb{F}}_{1}})$. Thus the image is
not dense, but its vanishing closure is all of
$\operatorname{Cong}({\mathbb{A}}^{1}_{{\mathbb{F}}_{1}})$.

######  Lemma .

Let $\myvarphi:Y\to X$ be a dominant morphism of monoid schemes and $X$ be
strongly reduced. Then $\myvarphi^{\\#}:\Gamma X\to\Gamma Y$ is injective.

###### Proof.

Since the image of the associated map
$\tilde{\myvarphi}:{\widetilde{Y}}\to{\widetilde{X}}$ has a strictly dense
image and since $X$ is strictly reduced, it follows from Lemma 4.6 and
Proposition 3.3 that the congruence kernel of $\myvarphi$ is trivial. Thus
$\myvarphi^{\\#}:\Gamma X\to\Gamma Y$ is injective. ∎

##  5\. Valuation monoids

The notion of a valuation monoid was first introduced in [7].

######  Definition .

A _valuation monoid_ is an integral monoid $A$ such that for every non-zero
element $\myalpha\in\operatorname{Frac}A$, either $\myalpha\in A$ or
$\myalpha^{-1}\in A$.

For our study, we require an existence result of valuation monoids that
dominate a given integral monoid. All of this notions are introduced in the
following.

###  5.1. Dominant maps of monoids

######  Definition .

A morphism $f:A\to B$ of pointed monoids is _dominant_ , and $B$ _dominates
$A$_, if $f^{-1}(M_{B})=M_{A}$ where $M_{A}$ is the maximal ideal of $A$ and
$M_{B}$ is the maximal ideal of $B$.

Note that for any morphism $f:A\to B$ of pointed monoids, it is always true
that $f(A^{\times})\subseteq B^{\times}$ and so $f^{-1}(M_{B})\subseteq
M_{A}$. Thus the morphism $f$ is dominant if and only if
$f^{-1}(B^{\times})\subseteq A^{\times}$. Also note that composition of
dominant morphisms is dominant.

######  Definition .

Let $B$ be a pointed monoid and let $A$ be an integral submonoid of $B$. An
element $b$ of $B$ is _algebraic over $A$_ if there is an $a\in A$ and an
integer $n\geqslant 1$ such that $ab^{n}\in A$. Elements of $B$ that are not
algebraic over $A$ are called _transcendental over $A$_.

An _integral extension of monoids_ is an injective morphism $f:A\to B$ of
pointed monoids such that for any $b\in B$ there is an integer $n\geqslant 1$
such that $b^{n}\in A$.

######  Lemma .

An integral extension $f:A\to B$ is dominant.

###### Proof.

We want to show $B^{\times}\cap A\subseteq A^{\times}$ (we identify $A$ with
$f(A)\subseteq B$). We consider $a\in B^{\times}\cap A$, i.e., $ab=1$ for some
$b\in B$. Since $A\hookrightarrow B$ is an integral extension, there exists
some integer $n\geqslant 1$ such that $c:=b^{n}\in A$. Thus
$a(a^{n-1}c)=a(a^{n-1}b^{n})=(ab)^{n}=1$ which shows $a\in A^{\times}$. ∎

###  5.2. Characterization and existence of valuation monoids

######  Proposition .

An integral monoid $A$ is a valuation monoid if and only if $A$ is a maximal