submonoid of $\operatorname{Frac}A$ with respect to domination.

###### Proof.

Suppose $A$ is maximal in $\operatorname{Frac}A$ with respect to domination
and consider $a\in\operatorname{Frac}A-A$. We define
$B:=A[a]=\\{ca^{i}\in\operatorname{Frac}A\mid c\in A,i\geqslant 0\\}$. Since
$A$ is maximal with respect to domination and $A\neq B$, the inclusion
$A\hookrightarrow B$ is not a dominant morphism, i.e., $M_{A}\nsubseteq
M_{B}\cap A$. Therefore there exists some $d\in M_{A}-M_{B}=M_{A}\cap
B^{\times}$. Let $d=ca^{i}$ for some $c\in A$ and some integer $i\geqslant 0$.
If $i\geqslant 1$, we have $a\in B^{\times}$. If $i=0$, then $c=d\in M_{A}\cap
B^{\times}$ and so there exists $c^{-1}=c^{\prime}a^{j}\in B^{\times}$ for
some $c^{\prime}\in A$ and integer $j\geqslant 1$ (since $c\notin
A^{\times}$). This shows that $a\in B^{\times}$ also in the case $i=0$. Let
$a^{-1}=\tilde{c}a^{k}$ for some $\tilde{c}\in A$ and integer $k\geqslant 0$.
Thus $(a^{-1})^{k+1}=\tilde{c}\in A$ and
$A[a^{-1}]=\\{ca^{-i}\in\operatorname{Frac}A\mid c\in A,i\geqslant 0\\}$ is an
integral extension of $A$. Therefore by Lemma 5.1, the inclusion
$A\hookrightarrow A[a^{-1}]$ is a dominant morphism. Since $A$ is maximal with
respect to domination, we must have $A\operatorname{Cong}(A)[a^{-1}]$ which
implies $a\in A^{\times}$ as desired.

Conversely, suppose $A$ is a valuation monoid. Let $B$ be a pointed submonoid
of $\operatorname{Frac}A$ and $f:A\to B$ is a dominant morphism such that the
canonical injection $A\hookrightarrow\operatorname{Frac}A$ factorizes through
$f$. We want to show that $f$ is an isomorphism. We consider any $b\in
B-\\{0\\}\subseteq\operatorname{Frac}A$. Since $A$ is a valuation monoid,
either $b\in A$ or $b^{-1}\in A$ (we identify $A$ with $f(A)\subseteq B$). If
$b^{-1}\in A$, we have $b^{-1}\in B^{\times}\cap A=A^{\times}$ since $B$
dominates $A$. Thus $b=(b^{-1})^{-1}\in A$. ∎

######  Proposition .

Let $K$ be a pointed group and let $A\subseteq K$ be a pointed submonoid. Then
there exists a valuation monoid $A$ dominating $A$ such that $K$ is the
pointed group completion of $A$.

###### Proof.

We consider the collection $\mathscr{S}$ of all pointed submonoids of $K$
dominating $A$ as a partially ordered set using the relation of domination,
i.e., for any $B,C\in\mathscr{S}$, we say $B\leqslant C$ if and only if
$B\subseteq C$ and $M_{B}=M_{C}\cap B$. Suppose that $\\{A_{i}\\}_{i\in I}$ is
a totally ordered chain in $\mathscr{S}$. Then $D=\bigcup A_{i}$ is clearly a
pointed submonoid of $K$ which dominates all of the $A_{i}$ and hence $D$
dominates $A$ as well. Therefore by Zorn’s Lemma, $\mathscr{S}$ has a maximal
element, say $\overline{A}$. Since $\overline{A}$ is a maximal element in
$\mathscr{S}$, by Proposition 5.2, $A_{\operatorname{max}}$ is a valuation
monoid. We are left to check that $K$ is the pointed group completion of
$\overline{A}$. Suppose, if possible, $\operatorname{Frac}\overline{A}\neq K$.
We consider $t\in K$ such that $t\notin\operatorname{Frac}\overline{A}$. If
$t$ is algebraic over $\overline{A}$, there exists an integer $n\geqslant 1$
such that $t^{n}\in\overline{A}$. Thus $\overline{A}\to\overline{A}[t]$ is an
integral extension and therefore it is dominant by Lemma 5.1. This contradicts
that $\overline{A}$ is a maximal element in $\mathscr{S}$. On the other hand,
if $t$ is transcendental over $\overline{A}$, it follows that
$C:=\overline{A}[t]=\\{kt^{i}\in K\mid k\in\overline{A},i\geqslant
0\\}\subseteq K$ is a pointed submonoid of $K$ with $M_{C}=\langle
M_{\overline{A}},t\rangle$ and that $C$ dominates $\overline{A}$, i.e.,
$M_{C}\cap\overline{A}=M_{\overline{A}}$. It follows that $C$ dominates $A$.
Hence $C$ is a pointed submonoid of $K$ dominating $A$ and strictly containing
$\overline{A}$. This gives the required contradiction. ∎

##  6\. Universally closed morphisms

In this section, we introduce closed morphisms and universally closed
morphisms. Under the assumption of quasi-compactness, we characterize
universally closed morphisms in terms of a valuative criterion.

###  6.1. Closed images

######  Proposition .

Let $\myvarphi:Y\to X$ be a quasi-compact morphism of monoid schemes and
${\widetilde{\myvarphi}}:{\widetilde{Y}}\to{\widetilde{X}}$ be the associated
map between congruence spaces. Then ${\tilde{\myvarphi}}(\widetilde{Y})$ is
closed in $\widetilde{X}$ if and only if ${\tilde{\myvarphi}}(\widetilde{Y})$
is closed under specialization.

###### Proof.

Since every closed set is closed under specialization, we are left with the
converse implication. Assume that ${\tilde{\myvarphi}}(\widetilde{Y})$ is
closed under specialization. Without loss of generality, we can assume that
$X=\operatorname{MSpec}(B)$ is affine and that both $Y$ and $X$ are strongly
reduced. Consider
$\mathfrak{d}\in\overline{{\tilde{\myvarphi}}(\widetilde{Y})}$. We want to
show $\mathfrak{d}\in{\tilde{\myvarphi}}(\widetilde{Y})$. Since $\myvarphi$ is
quasi-compact, $Y=\bigcup_{i=1}^{n}Y_{i}$ for some affine opens $Y_{i}$. Thus
$\overline{{\tilde{\myvarphi}}(\widetilde{Y})}=\bigcup_{i=1}^{n}\overline{{\tilde{\myvarphi}}(\widetilde{Y_{i}})}$
and $\mathfrak{d}\in\overline{{\tilde{\myvarphi}}(\widetilde{Y_{i}})}$ for
some $i$. Hence we can assume that $Y=Y_{i}=\operatorname{MSpec}(A)$ is
affine. Let $f=\Gamma_{\myvarphi}:B\to A$ and let
$\mathfrak{c}_{0}=\operatorname{congker}(f)$. The surjective morphism $p:B\to
B/\mathfrak{c}_{0}$ induces a closed immersion
$p^{*}:\operatorname{MSpec}(B/\mathfrak{c}_{0})\hookrightarrow\operatorname{MSpec}(B)$.
By Proposition 4.1, we know that $\widetilde{p^{*}}$ is a closed topological
embedding. Since the diagram

$\textstyle{\operatorname{Cong}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\tilde{\myvarphi}}}$$\textstyle{\operatorname{Cong}(B)}$$\textstyle{\operatorname{Cong}(B/\mathfrak{c}_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{p^{*}}}$

commutes, we can assume that $p:B\xrightarrow{\cong}B/\mathfrak{c}_{0}$ is an
isomorphism and $f:=\Gamma_{\myvarphi}:B\to A$ is an injection. Let
$\mathfrak{d^{\prime}}\subseteq\mathfrak{d}$ be a minimal prime congruence of
$B$ (which exists by a standard application of Zorn’s lemma). Let
$S_{\mathfrak{d^{\prime}}}=B-\mathfrak{d}^{\prime}$. By Lemma 2.2,

$f_{\mathfrak{d}^{\prime}}:\ B_{\mathfrak{d}^{\prime}}\ =\
S_{\mathfrak{d}^{\prime}}^{-1}B\ \longrightarrow\
f(S_{\mathfrak{d}^{\prime}})^{-1}A\ =\ A_{\mathfrak{d}^{\prime}}$

is an injection. Since $\mathfrak{d}^{\prime}$ is minimal,
$B_{\mathfrak{d}^{\prime}}$ is a pointed group and
$\langle\mathfrak{d}^{\prime}\rangle_{B_{\mathfrak{d}^{\prime}}}$ is trivial.
Thus the natural composition

$\bar{f_{\mathfrak{d}^{\prime}}}:B_{\mathfrak{d}^{\prime}}\overset{f_{\mathfrak{d}^{\prime}}}{\longrightarrow}A_{\mathfrak{d}^{\prime}}\overset{q}{\longrightarrow}A_{\mathfrak{d}^{\prime}}/M_{A_{\mathfrak{d}^{\prime}}}=\\{0\\}\cup
A_{\mathfrak{d}^{\prime}}^{\times}$

is injective and

$\langle\mathfrak{d}^{\prime}\rangle_{B_{\mathfrak{d}^{\prime}}}=\widetilde{{\bar{f_{\mathfrak{d}^{\prime}}}}^{*}}(\mathfrak{c}_{\textup{triv}})=\widetilde{f_{\mathfrak{d}^{\prime}}^{*}}(\widetilde{q^{*}}(\mathfrak{c}_{\textup{triv}})).$

Let $g:A\to A_{\mathfrak{d}^{\prime}}$ and $h:B\to B_{\mathfrak{d}^{\prime}}$
denote the canonical localization maps. This and the commutative diagram

$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{A_{\mathfrak{d}^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\textstyle{A_{\mathfrak{d}^{\prime}}/M_{\mathfrak{d}^{\prime}}}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{h}$$\textstyle{B_{\mathfrak{d}^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{\mathfrak{d}^{\prime}}}$

imply that
$\mathfrak{d}^{\prime}=\widetilde{h^{*}}(\langle\mathfrak{d}^{\prime}\rangle_{B_{\mathfrak{d}^{\prime}}})=\widetilde{h^{*}}(\widetilde{f_{\mathfrak{d}^{\prime}}^{*}}(\widetilde{q^{*}}(\mathfrak{c}_{\textup{triv}})))=\widetilde{f^{*}}(\widetilde{g^{*}}(\widetilde{q^{*}}(\mathfrak{c}_{\textup{triv}}))$
and thus $\mathfrak{d}^{\prime}\in{\tilde{\myvarphi}}(\widetilde{Y})$. Since
${\tilde{\myvarphi}}(\widetilde{X})$ is closed under specialization, we have
$\mathfrak{d}\in{\tilde{\myvarphi}}(\widetilde{Y})$. ∎

###  6.2. The valuative criterion for universally closed morphisms

######  Definition .

A morphism $\myvarphi:Y\to X$ is _closed_ if the associated map
$\tilde{\myvarphi}:{\widetilde{Y}}\to{\widetilde{X}}$ between the respective
congruence spaces is a closed continuous map. A morphism $\myvarphi:Y\to X$ of
monoid schemes is _universally closed_ if for every morphism
$\mypsi:X^{\prime}\to X$, the base change
$\myvarphi^{\prime}:Y^{\prime}=X^{\prime}\times_{X}Y\to X^{\prime}$ of
$\myvarphi$ along $\mypsi$ is a closed morphism.

As a first example, we have the following.

######  Lemma .

Closed immersions are universally closed.

###### Proof.

Since surjective monoid morphisms are stable under cobase change, it follows
that closed immersions are stable under base change. Therefore the result
follows from Proposition 4.1. ∎

######  Definition .

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

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

where $A$ is a valuation monoid, $K=\operatorname{Frac}A$ and $\myiota=i^{*}$
is the inclusion $i:A\hookrightarrow K$. A _lift for the test diagram_ is a
morphism $\hat{\mynu}:T\to Y$ such that

${U}$${Y}$${T}$${X}$$\scriptstyle{\myeta}$$\scriptstyle{\myiota}$$\scriptstyle{\myvarphi}$$\scriptstyle{\mynu}$$\scriptstyle{\hat{\mynu}}$

commutes.

######  Remark .

The prime congruences $\mathfrak{c}$ of a pointed group $K$ correspond
bijectively to the subgroups $H_{\mathfrak{c}}:=\\{a\in
K\mid(a,1)\in\mathfrak{c}\\}$ of $K^{\times}$. Given a morphism of pointed
groups $f:K_{1}\to K_{2}$, we have an induced morphism
$\widetilde{f}_{*}:{\operatorname{Cong}(K_{1})}\to{\operatorname{Cong}(K_{2})}$
that sends a prime congruence $\mathfrak{c}$ on $K_{1}$ to its pushforward

$\widetilde{f}_{\ast}({\mathfrak{c}})\ =\
\big{\\{}\big{(}cf(a),cf(b)\big{)}\in K_{1}\times K_{1}\,\big{|}\,c\in
K_{2},(a,b)\in{\mathfrak{c}}\big{\\}},$

which is the prime congruence associated with
$f(H_{\mathfrak{c}})=H_{\widetilde{f}_{*}(\mathfrak{c})}$.

In the proof of the next result, we use the following shorthand notation.
Given a monoid scheme $X$ with congruence space ${\widetilde{X}}$ and
${\mathfrak{p}},{\mathfrak{q}}\in{\widetilde{X}}$ such that ${\mathfrak{q}}$
is a specialization of ${\mathfrak{p}}$, we denote by
${\mathcal{O}}_{{\widetilde{X}},{\mathfrak{q}}}/{\mathfrak{p}}$ the quotient
of the stalk ${\mathcal{O}}_{{\widetilde{X}},{\mathfrak{q}}}$ by the
congruence generated by ${\mathfrak{p}}$ where we interpret ${\mathfrak{p}}$
as a congruence on a monoid $A=\Gamma U$ for an affine open neighbourhood $U$
of ${\mathfrak{q}}$ in $X$. In other words,
${\mathcal{O}}_{{\widetilde{X}},{\mathfrak{q}}}/{\mathfrak{p}}=S^{-1}_{\mathfrak{q}}A/S^{-1}_{\mathfrak{q}}{\mathfrak{p}}$
for $S_{\mathfrak{q}}=A-{\mathfrak{q}}$.

######  Theorem 6.1.

Let $\myvarphi:X\to Y$ be a quasi-compact morphism of monoid schemes. Then
$\myvarphi:Y\to X$ is universally closed if and only if every test diagram

${U}$${Y}$${T}$${X}$$\scriptstyle{\myeta}$$\scriptstyle{\myiota}$$\scriptstyle{\myvarphi}$$\scriptstyle{\mynu}$

has a lift $\hat{\mynu}:T\to Y$.

###### Proof.

Assume $\myvarphi:Y\to X$ is universally closed and consider a test diagram as
in the theorem with $A=\Gamma T$ and $K=\operatorname{Frac}A$. This diagram
extends to

$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mylambda}$$\scriptstyle{\myiota}$$\scriptstyle{\myeta}$$\textstyle{T\times_{X}Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\mynu^{\prime}}$$\scriptstyle{\myvarphi^{\prime}}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\myvarphi}$$\textstyle{T\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mynu}$$\textstyle{X}$

Let $u_{0}=\langle 0\rangle\in U$ be the generic point of $U$,
$y_{0}=\mylambda(u_{0})\in Y_{T}=T\times_{X}Y$ and
$t_{0}=\myiota(u_{0})=\myvarphi^{\prime}(y_{0})\in T$ the generic point of
$T$. Then we obtain the following commutative diagram of residue fields:

$\textstyle{K=k(t_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\myvarphi^{\prime}_{T,t_{0}}}}$$\scriptstyle{id}$$\textstyle{L=k(y_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p=\overline{\mylambda_{Y_{T},y_{0}}}}$$\textstyle{K=k(u_{0})}$

Thus $p$ is surjective and $K\cong L/\mathfrak{c}_{0}$ for
$\mathfrak{c}_{0}=\operatorname{congker}(L\xrightarrow{p}K)$. Consider

${\mypsi:\quad{\operatorname{Cong}(L)}}$${{\operatorname{Cong}(K)}={\widetilde{U}}}$${\widetilde{Y_{T}}}$$\scriptstyle{\overline{\myvarphi^{\prime}_{T,t_{0}}}}$$\scriptstyle{\tilde{\mylambda}}$

and define $\mathfrak{q}_{0}=\mypsi({\mathfrak{c}}_{0})\in\widetilde{Y_{T}}$.
Since the diagram of maps

${{\operatorname{Cong}(L/\mathfrak{c}_{0})}}$${\widetilde{T}}$${{\operatorname{Cong}(L)}}$${{\widetilde{U}}}$${\widetilde{Y_{T}}}$${U}$${Y_{T}}$$\scriptstyle{\tilde{\myiota}}$$\scriptstyle{\overline{\myvarphi^{\prime}_{T,t_{0}}}}$$\scriptstyle{\tilde{\mylambda}}$$\scriptstyle{\mypi_{U}}$$\scriptstyle{\simeq}$$\scriptstyle{\mypi_{Y_{T}}}$$\scriptstyle{\tilde{\myvarphi}^{\prime}}$$\scriptstyle{\mylambda}$

commutes, we have $\mypi_{Y_{T}}(\mathfrak{q}_{0})=y_{0}$ in $Y_{T}$ and
$\mathfrak{z}_{0}:={\tilde{\myvarphi}^{\prime}}(\mathfrak{q}_{0})=\mathfrak{c}_{\textup{triv}}=\mysigma_{T}(t_{0})$
in $\widetilde{T}$. Let
$Z=\overline{\\{\mathfrak{q}_{0}\\}}\subseteq\widetilde{Y_{T}}$. Then
${\tilde{\myvarphi}^{\prime}}(Z)\subseteq\widetilde{T}$ is closed since
${\tilde{\myvarphi}}$ is universally closed. Since
$\mathfrak{z}_{0}\in{\tilde{\myvarphi}^{\prime}}(Z)$, we have
${\tilde{\myvarphi}^{\prime}}(Z)=\widetilde{T}$. Consider
$\mathfrak{z}_{1}:=\mysigma_{T}(M_{A})\in\widetilde{T}$. Then there exists
some $\mathfrak{q}_{1}\in Z$ such that
$\mathfrak{z}_{1}={\tilde{\myvarphi}^{\prime}}(\mathfrak{q}_{1})$. Thus we
have the induced map on stalks
${\tilde{\myvarphi}^{\prime}_{\mathfrak{q}_{1}}}:\mathcal{O}_{\widetilde{T},\mathfrak{z}_{1}}\to\mathcal{O}_{\widetilde{Y_{T}},\mathfrak{q}_{1}}$.
Let $B:=\mathcal{O}_{\widetilde{Y_{T}},\mathfrak{q}_{1}}/\mathfrak{q}_{0}$.
Then
$\mathfrak{q}:=S_{\mathfrak{q}_{0}}^{-1}\mathfrak{q}_{0}\in{\operatorname{Cong}(B)}$
and
$S_{\mathfrak{q}}^{-1}B=\mathcal{O}_{\widetilde{Y_{T}},\mathfrak{q}_{0}}/{\mathfrak{q}_{0}}$.
Let $s:B\to S_{\mathfrak{q}}^{-1}B$ denote the localization map. We obtain the
commutative diagram

$\textstyle{A=\mathcal{O}_{T,M_{A}}=\mathcal{O}_{\widetilde{T},\mathfrak{z}_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Gamma_{\myiota}}$$\scriptstyle{\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
{\tilde{\myvarphi}^{\prime}_{\mathfrak{q}_{1}}}}$$\textstyle{\mathcal{O}_{\widetilde{Y_{T}},\mathfrak{q}_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\textstyle{\mathcal{O}_{\widetilde{Y_{T}},\mathfrak{q}_{1}}/\mathfrak{q}_{0}=B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s}$
$\textstyle{K\cong
L/\mathfrak{c}_{0}=\mathcal{O}_{\widetilde{Y_{T}},\mathfrak{q}_{0}}/{\mathfrak{q}_{0}}}$

Since $s$ is a localization map of the integral monoid $B$, it must be
injective. Therefore the above diagram shows that $\operatorname{Frac}B\cong
K$. Let $f:=q\circ{\tilde{\myvarphi}^{\prime}_{\mathfrak{q}_{1}}}:A\to B$.
Then
$f^{-1}(M_{B})=f^{-1}(\mypi_{Y_{T}}(\mathfrak{q}_{1}))=\mypi_{T}(\mathfrak{z}_{1})=M_{A}$,
which shows $f:A\to B$ is dominant. Thus by Proposition 5.2, $f$ is an
isomorphism. Let $g:B\to A$ be its inverse. Then

$\hat{\mynu}:T=\operatorname{MSpec}(A)\overset{g^{*}}{\longrightarrow}\operatorname{MSpec}(B)\hookrightarrow
Y_{T}\overset{\mynu^{\prime}}{\longrightarrow}Y.$

is the required lift, which completes one direction of the proof.

Conversely suppose every test diagram for the morphism $\myvarphi:Y\to X$ has
a lift. We consider a base change

$\textstyle{Y^{\prime}=X^{\prime}\times_{X}Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\mypsi^{\prime}}$$\scriptstyle{\myvarphi^{\prime}}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\myvarphi}$$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mypsi}$$\textstyle{X}$

Let $Z\subseteq\widetilde{Y^{\prime}}$ be closed. We want to show that
${\tilde{\myvarphi}^{\prime}}(Z)\subseteq\widetilde{X^{\prime}}$ is closed.
Since $\myvarphi$ is quasi-compact, so is $\myvarphi^{\prime}$. By Proposition
6.1, it suffices to show that ${\tilde{\myvarphi}^{\prime}}(Z)$ is closed
under specialization. We consider any $\mathfrak{z}_{1}\in Z$ and let
$\mathfrak{p}_{1}=\tilde{\myvarphi}^{\prime}(\mathfrak{z}_{1})\in\widetilde{X^{\prime}}$.
For any $\mathfrak{p}_{0}\in\overline{\\{\mathfrak{p}_{1}\\}}$, we need to
show that $\mathfrak{p}_{0}\in{\tilde{\myvarphi}^{\prime}}(Z)$. As explained
in section 2.6, we obtain an injection

$A:=\mathcal{O}_{\widetilde{X^{\prime}},\mathfrak{p}_{0}}/\mathfrak{p}_{1}\overset{i}{\longrightarrow}\mathcal{O}_{\widetilde{X^{\prime}},\mathfrak{p}_{1}}/\mathfrak{p}_{1}=k(\mathfrak{p}_{1})\overset{{\tilde{\myvarphi}^{\prime}}_{\mathfrak{z}_{1}}}{\longrightarrow}k(\mathfrak{z}_{1})=:L.$

By Proposition 5.2 there exists a valuation monoid $\overline{A}$ dominating
$A$ such that $\operatorname{Frac}\overline{A}\cong L$. In other words, there
are morphisms $h:A\to\overline{A}$ and
$j:\overline{A}\hookrightarrow\operatorname{Frac}\overline{A}\cong L$ such
that $h$ is dominant, $\overline{A}$ is a valuation monoid and the diagram

${A}$${K}$${\overline{A}}$${L}$$\scriptstyle{i}$$\scriptstyle{h}$$\scriptstyle{{\tilde{\myvarphi}^{\prime}}_{\mathfrak{z}_{1}}}$$\scriptstyle{j}$

commutes. We obtain the commutative diagram

${U:=\operatorname{MSpec}(L)}$${Y^{\prime}}$${Y}$${T:=\operatorname{MSpec}(\overline{A})}$${\operatorname{MSpec}(A)}$${X^{\prime}}$${X}$$\scriptstyle{\myeta}$$\scriptstyle{\myiota=j^{*}}$$\scriptstyle{\myeta^{\prime}}$$\scriptstyle{\mypsi^{\prime}}$$\scriptstyle{\myvarphi^{\prime}}$$\scriptstyle{\myvarphi}$$\scriptstyle{\mynu}$$\scriptstyle{\mynu^{\prime}}$$\scriptstyle{\leavevmode\nobreak\
\leavevmode\nobreak\ h^{*}}$$\scriptstyle{\myalpha}$$\scriptstyle{\mypsi}$

Its outer square is a test diagram, which has a lift $\hat{\mynu}:T\to Y$ by
assumption. This lift defines a unique morphism $\hat{\mynu}^{\prime}:T\to
Y^{\prime}$ such that the diagram

$\textstyle{T\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{\mynu}^{\prime}}$$\scriptstyle{\mynu^{\prime}}$$\scriptstyle{\hat{\mynu}}$$\textstyle{Y^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mypsi^{\prime}}$$\scriptstyle{\myvarphi^{\prime}}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\myvarphi}$$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mypsi}$$\textstyle{X}$

commutes. By construction, we have

$\mathfrak{z}_{1}=\widetilde{\hat{\mynu}^{\prime}}(\mathfrak{c}_{\textup{triv}}),\qquad\widetilde{\mynu^{\prime}}(\mathfrak{c}_{\textup{triv}})=\mathfrak{p}_{1}={\tilde{\myvarphi}^{\prime}}(\mathfrak{z}_{1}),\qquad\mynu^{\prime}(M_{\overline{A}})=h^{*}(M_{\overline{A}})=M_{A}=\mypi_{X^{\prime}}(\mathfrak{p}_{0}).$

Since $Z$ is closed and $\mathfrak{c}_{\textup{triv}}\in\widetilde{T}$ is
generic, $\widetilde{\hat{\mynu}^{\prime}}(\widetilde{T})\subseteq Z$. Since
$A\overset{h}{\to}{\overline{A}}$ is an injection and since
$h^{-1}(M_{\overline{A}})=M_{A}$, the induced map
$\overline{h}:A/M_{A}\to{\overline{A}}/M_{\overline{A}}$ is an injection.
Consider the map
$\widetilde{\overline{h}}_{*}:{\operatorname{Cong}(A/M_{A})}\to{\operatorname{Cong}({\overline{A}}/M_{\overline{A}})}$,
as defined in section 2.6. Then
$\widetilde{\overline{h}^{*}}(\widetilde{\overline{h}_{*}}(\mathfrak{p}_{0}))=\mathfrak{p}_{0}$.
Then
${\tilde{\myvarphi}^{\prime}}\big{(}\widetilde{\hat{\mynu}^{\prime}}(\widetilde{\overline{h}_{*}}(\mathfrak{p}_{0})\big{)}={\tilde{\mynu}^{\prime}}\big{(}\widetilde{\overline{h}_{*}}(\mathfrak{p}_{0})\big{)}=\mathfrak{p}_{0}$,
which shows that $\mathfrak{p}_{0}\in{\tilde{\myvarphi}^{\prime}}(Z)$. ∎

##  7\. Separated morphisms

In this section, we develop a topological characterization of separated
morphisms and extend the characterization in terms of the valuative criterion
from finite type morphisms, as treated in [7], to arbitrary morphisms.

######  Definition .

A morphism $\myvarphi:Y\to X$ of monoid schemes is _separated_ if the diagonal
$\Delta_{\myvarphi}\colon Y\to Y\times_{X}Y$ is a closed immersion. A morphism
$\myvarphi:Y\to X$ of monoid schemes is _quasi-separated_ if the diagonal
$\Delta_{\myvarphi}:Y\to Y\times_{X}Y$ is quasi-compact.

###  7.1. Locally closed immersions

Our topological characterization of separated morphisms relies on some general
facts for locally closed immersions.

######  Definition .

A morphism $\myvarphi:Y\to X$ is a _locally closed immersion_ if it factors
into a closed immersion $Y\to U$ followed by an open immersion $U\to X$.

######  Lemma .

Let $\myvarphi:Y\to X$ be a quasi-separated morphism of monoid schemes with
diagonal $\Delta_{\myvarphi}:Y\to Y\times_{X}Y$ and associated map
$\tilde{\Delta}_{\myvarphi}:{\widetilde{Y}}\to(Y\times_{X}Y)^{\textup{cong}}$
of congruence spaces. Then $\Delta_{\myvarphi}$ is a locally closed immersion
and the topological closure of $\textup{im}(\tilde{\Delta}_{\myvarphi})$ is a
vanishing set of $(Y\times_{X}Y)^{\textup{cong}}$.

###### Proof.

If $U=\operatorname{MSpec}(A)$ is affine open of $Y$, then $\myvarphi(U)$ is
contained in an affine open $V=\operatorname{MSpec}(B)$ of $X$ and the
restriction of the diagonal morphism to $\Delta_{\myvarphi}:U\to U\times_{V}U$
corresponds to the surjective monoid morphism $A\otimes_{B}A\to A$. This shows
that $\Delta_{\myvarphi}:U\to U\times_{V}U=U\times_{X}U$ is a closed
immersion.

Note that $U\times_{V}U=U\times_{X}U$ is an open subset of $Y\times_{X}Y$. If
${\mathcal{U}}$ is an affine open covering of $Y$, then
$W=\bigcup_{U\in{\mathcal{U}}}U\times_{X}U$ is open in $Y\times_{X}Y$ and
$\Delta_{\myvarphi}:Y\to W$ is a closed immersion. Thus
$\Delta_{\myvarphi}:Y\to Y\times_{X}Y$ is a locally closed immersion, which
establishes the first claim of the lemma.

We turn to the second claim. Since $\Delta_{\myvarphi}$ is quasi-compact,
Lemma 4.6 shows that the vanishing closure of
$Z=\textup{im}(\tilde{\Delta}_{\myvarphi})$ is the vanishing set of the
congruence kernel ${\mathfrak{D}}=\operatorname{Congker}(\Delta_{\myvarphi})$
of $\Delta_{\myvarphi}$. We need to show that every point ${\mathfrak{q}}\in
Z$ is in the image of $\tilde{\Delta}_{\myvarphi}$.

The fibre product $Y\times_{X}Y$ is covered by affine opens of the form
$U_{1}\times_{V}U_{2}$ where $U_{1}$ and $U_{2}$ are affine opens of $Y$ and
$V$ is an affine open of $X$ that contains both $\myvarphi(U_{1})$ and
$\myvarphi(U_{2})$. If ${\mathfrak{q}}\in U_{1}\times_{V}U_{2}$, then
$U_{1}\times_{V}U_{2}$ is non-trivial and thus
$W=\myvarphi^{-1}(U_{1}\times_{V}U_{2})=U_{1}\cap U_{2}$ is a nonempty open
subset of $Y$.

The congruence kernel of $\Gamma(U_{1}\times_{V}U_{2})\to\Gamma W$ is

${\mathfrak{d}}\ =\ {\mathfrak{D}}(U_{1}\times_{V}U_{2})\ =\ \\{(a_{1}\otimes
a_{2},\ b_{1}\otimes b_{2})\mid a_{1}a_{2}=b_{1}b_{2}\text{ in }\Gamma W\\}$

where $a_{1}\otimes a_{2}$ and $b_{1}\otimes b_{2}$ are elements of
$\Gamma(U_{1}\times_{V}U_{2})=\Gamma U_{1}\otimes_{\Gamma V}\Gamma U_{2}$ and
where $a_{1}a_{2}$ is a shorthand notation for
$\operatorname{res}_{U_{1},W}(a_{1})\cdot\operatorname{res}_{U_{2},W}(a_{2})$,
and similar for $b_{1}b_{2}$. By our assumptions, ${\mathfrak{q}}\in
V_{\mathfrak{d}}$, i.e., ${\mathfrak{d}}\subseteq{\mathfrak{q}}$ as congruence
on $\Gamma U_{1}\otimes_{\Gamma V}\Gamma U_{2}$.

In the next step, we find a prime congruence
${\mathfrak{q}}^{\prime}\in\textup{im}({\widetilde{\Delta}}_{\myvarphi})$ with
${\mathfrak{q}}^{\prime}\subseteq{\mathfrak{q}}$. Let $Q=I_{\mathfrak{q}}$ be
the vanishing ideal of ${\mathfrak{q}}$, which is a point of
$U_{1}\times_{V}U_{2}$. Let $\textup{pr}_{1}:Y\times_{X}Y\to Y$ be the
canonical projection onto the first factor. Then $Q_{1}=\textup{pr}_{1}(Q)\in
U_{1}$. Let $Q_{1}^{\prime}\subseteq Q_{1}$ be a prime ideal that is in $W$
and ${\mathfrak{q}}_{1}^{\prime}=\mysigma_{W}(Q_{1}^{\prime})$ the minimal
prime congruence in ${\widetilde{W}}$ with vanishing ideal $Q_{1}^{\prime}$.
Then
${\mathfrak{q}}^{\prime}={\widetilde{\Delta}}_{\myvarphi}({\mathfrak{q}}^{\prime}_{1})\in
W\times_{V}W$ is the minimal prime congruence in
$\textup{im}({\widetilde{\Delta}}_{\myvarphi})$ with vanishing ideal

$I_{{\mathfrak{q}}^{\prime}}\ =\ \\{a_{1}\otimes a_{2}\mid a_{1}a_{2}\in
Q_{1}^{\prime}\text{ in }\Gamma W\\}.$

We claim that ${\mathfrak{q}}^{\prime}\subseteq{\mathfrak{q}}$ as congruences
on $\Gamma U_{1}\otimes_{\Gamma V}\Gamma U_{2}$. By the minimality of
${\mathfrak{q}}^{\prime}$, it suffices to show that
$Q^{\prime}=I_{{\mathfrak{q}}^{\prime}}$ is contained in $Q=I_{\mathfrak{q}}$.
Consider $a\otimes b\in Q^{\prime}$, i.e.,

$\operatorname{res}_{U_{1},W}(a)\cdot\operatorname{res}_{U_{2},W}(b)\ \in\
Q_{1}^{\prime}\ \subseteq\ Q_{1}.$

The restriction $\operatorname{res}_{U_{2},W}(b)$ is the multiple of an
element of the form $\operatorname{res}_{U_{1},W}(b^{\prime})$ by an
invertible element in $\Gamma W$. Since $Q_{1}$ is invariant under the
multiplication by units, we conclude that
$\operatorname{res}_{U_{1},W}(ab^{\prime})\in Q_{1}$ and thus $ab\otimes 1\in
Q$, as desired. This shows that
${\mathfrak{q}}^{\prime}\subseteq{\mathfrak{q}}$.

By our assumptions $\textup{im}(\tilde{\Delta}_{\myvarphi})$ is closed and
thus contains every specialization of ${\mathfrak{q}}^{\prime}$. This shows
that ${\mathfrak{q}}$ is in the image of $\tilde{\Delta}_{\myvarphi}$, which
concludes the proof. ∎

######  Lemma .

Let $\myvarphi:Y\to X$ be a locally closed immersion and
$\tilde{\myvarphi}:{\widetilde{Y}}\to{\widetilde{X}}$ the associated map of
congruence spaces. If $\textup{im}(\tilde{\myvarphi})$ is a vanishing set of
$X$, then $\myvarphi$ is a closed immersion.

###### Proof.

Since $\myvarphi:Y\to X$ is a locally closed immersion, there exists an open
subset $U$ of $X$ such that $\myvarphi$ factors into

$\myvarphi:Y\xrightarrow{\myvarphi_{U}}U\xrightarrow{\myiota_{U}}X.$

where $\myvarphi_{U}$ is a closed immersion and $\myiota_{U}$ is an open
immersion. Let $Z:=\textup{im}(\tilde{\myvarphi})$. Since $Z$ is a vanishing
set of $X$, Theorem 4.1 implies that there exists a strongly reduced closed
subscheme $Y^{\prime}\overset{j}{\hookrightarrow}X$ such that
$Z=\textup{im}(\widetilde{j}:\widetilde{Y^{\prime}}\to\widetilde{X})$. Since
$\textup{im}(\tilde{\myvarphi})\subset{\widetilde{U}}$, $\widetilde{j}$
factors into

$\widetilde{j}:\widetilde{Y^{\prime}}\overset{\widetilde{j_{U}}}{\longrightarrow}\widetilde{U}\overset{\widetilde{\myiota_{U}}}{\longrightarrow}\widetilde{X}.$

Thus

$j\ =\ \mypi_{X}\circ\widetilde{j}\circ\mysigma_{Y^{\prime}}\ =\
\mypi_{X}\circ\widetilde{\myiota_{U}}\circ\widetilde{j_{U}}\circ\mysigma_{Y^{\prime}}\
=\ \myiota_{U}\circ\mypi_{U}\circ\widetilde{j_{U}}\circ\mysigma_{Y^{\prime}},$

which shows that $j$ factors through $U$ via the map
$j_{U}:=\mypi_{U}\circ\widetilde{j_{U}}\circ\mysigma_{Y^{\prime}}:Y^{\prime}\to
U$. Since $Y^{\prime}$ is strongly reduced (and therefore final among all
closed subschemes with support $Z$), $\myvarphi_{U}$ factors through
$Y^{\prime}$, i.e., there exists some morphism $\myiota^{\prime}:Y\to
Y^{\prime}$ such that the diagram

${Y}$${U}$${Y^{\prime}}$${X}$$\scriptstyle{\myiota^{\prime}}$$\scriptstyle{\myvarphi_{U}}$$\scriptstyle{\myiota_{U}}$$\scriptstyle{\leavevmode\nobreak\
\leavevmode\nobreak\ j}$$\scriptstyle{j_{U}}$

commutes. Since $j=\myiota_{U}\circ j_{U}$ is a closed immersion and
$\myiota_{U}$ is separated as an open immersion, $j_{U}$ is a closed
immersion. Therefore $j_{U}$ is separated. Since $\myvarphi_{U}=j_{U}\circ
i^{\prime}$ is a closed immersion, it follows that $\myiota^{\prime}$ is a
closed immersion. Since composition of closed immersions is a closed
immersion, we conclude that $\myvarphi=j\circ\myiota^{\prime}$ is a closed
immersion. ∎

######  Remark .

It is not enough to ask for $\textup{im}(\tilde{\myvarphi})$ to be closed for
Lemma 7.1 to hold. For example, let
$X=\operatorname{MSpec}({{\mathbb{F}}_{1}}[x,y]/{\langle xy\rangle})$ and
consider the locally closed immersion

$\myvarphi:\
Y=\operatorname{MSpec}({{\mathbb{F}}_{1}})\myamalg\operatorname{MSpec}({{\mathbb{F}}_{1}})\stackrel{{\scriptstyle\myiota_{Y}}}{{\longrightarrow}}\mathbb{G}_{m,{{\mathbb{F}}_{1}}}\myamalg\mathbb{G}_{m,{{\mathbb{F}}_{1}}}\stackrel{{\scriptstyle\myiota_{U}}}{{\longrightarrow}}\\{\langle
x\rangle,\langle y\rangle\\}=U\subseteq X$

where the map $\myiota_{Y}$ is induced by maps of the form
${{\mathbb{F}}_{1}}[t^{\pm 1}]\to{{\mathbb{F}}_{1}}$ given by $t\mapsto 1$ and
the map $\myiota_{U}$ is induced by isomorphisms of the form
${{\mathbb{F}}_{1}}[s,t^{\pm 1}]/\langle s\rangle\to{{\mathbb{F}}_{1}}[t^{\pm
1}]$. It follows that $\myiota_{Y}$ is a closed immersion and $\myiota_{U}$ is
an open immersion and hence $\myvarphi$ is a locally closed immersion. The
image $\widetilde{\myvarphi}(\widetilde{Y})=V_{x,1}\cup V_{y,1}$ is a closed
subset of $\widetilde{X}$, but $\myvarphi$ is not a closed immersion since it
is not affine.

###  7.2. Topological characterization of separated morphisms

######  Theorem 7.1.

Let $\myvarphi:Y\to X$ be a morphism of monoid schemes with diagonal
$\Delta_{\myvarphi}:Y\to Y\times_{X}Y$ and associated map
$\tilde{\Delta}_{\myvarphi}:{\widetilde{Y}}\to(Y\times_{X}Y)^{\textup{cong}}$
of congruence spaces. 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 $(Y\times_{X}Y)^{\textup{cong}}$.

###### Proof.

Suppose $\myvarphi$ is separated. Then $\tilde{\Delta}_{\myvarphi}$ is a
closed immersion and therefore quasi-compact and by Proposition 4.1 the image
of $\tilde{\Delta}_{\myvarphi}$ is a closed subset of
$(Y\times_{X}Y)^{\textup{cong}}$. This shows that (1) implies (2).

Assume (2). Then by Lemma 7.1, we know that $\Delta_{\myvarphi}:Y\to
Y\times_{X}Y$ is a locally closed immersion and
$\textup{im}(\tilde{\Delta}_{\myvarphi})$ is a vanishing set of
$(Y\times_{X}Y)^{\textup{cong}}$. Therefore $\Delta_{\myvarphi}$ is a closed
immersion by Lemma 7.1. This shows that (2) implies (1) and completes the
proof. ∎

###  7.3. The valuative criterion for separated morphisms

######  Theorem 7.2.

A morphism $\myvarphi:Y\to X$ of monoid schemes is separated if and only if

  1. (1)

the morphism $\myvarphi$ is quasi-separated, and

  2. (2)

every test diagram

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

has at most one lift.

###### Proof.

Suppose $\myvarphi:Y\to X$ satisfies (1) and (2). Then
$\Delta_{\myvarphi}:Y\to Y\times_{X}Y$ is quasi-compact. By Theorems 6.1 and
7.1, it suffices to show that the morphism $\Delta_{\myvarphi}:Y\to
Y\times_{X}Y$ satisfies the existence part of the valuative criterion.
Consider a test diagram

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

for $\Delta_{\myvarphi}$. Let $p_{1},p_{2}:Y\times_{X}Y\longrightarrow Y$ be
the canonical projections. Then both maps $a=p_{1}\circ\mynu$ and
$b=p_{2}\circ\mynu$ are lifts for the test diagram

${U}$${Y}$${T}$${X}$$\scriptstyle{\myeta}$$\scriptstyle{\myiota}$$\scriptstyle{\myvarphi}$$\scriptstyle{\myvarphi\circ
a=\myvarphi\circ b}$

and consequently equal by (2). Hence
$a=b:\operatorname{MSpec}(A)\longrightarrow Y$ is a lift for the the test for
$\Delta_{\myvarphi}$. This shows that $\myvarphi$ is separated.

Conversely, let $\myvarphi:Y\to X$ be a separated morphism. By Theorem 7.1,
$\myvarphi$ is quasi-separated, which establishes (1). Consider two lifts
$a,b:\operatorname{MSpec}(A)\longrightarrow Y$ for the test diagram in (2).
Let $Z\subseteq\operatorname{MSpec}(A)$ be the equalizer of $a$ and $b$. In
other words, $Z$ is the fibre product of $Y$ and $T$ over $Y\times_{X}Y$. Thus
the projection $p_{2}:Z=Y\times_{Y\times_{X}Y}T\to T$ is the base change of
the closed immersion $\Delta_{\myvarphi}$ along $\mynu:T\to Y\times_{X}Y$ and
therefore itself a closed immersion. By Proposition 4.1, $\widetilde{Z}$ is
homeomorphic to a closed subset of $\operatorname{Cong}(A)$.

Since $a\circ\myiota=\myeta=b\circ\myiota$, the image of
$\tilde{\myiota}:{\widetilde{U}}\to{\widetilde{T}}$ is contained in
${\widetilde{Z}}$. Thus the generic point of ${\widetilde{T}}$, which is in
${\widetilde{U}}$, is in ${\widetilde{Z}}$. We conclude that
$\widetilde{Z}={\widetilde{T}}$ and
$Z=\mypi_{Z}(\widetilde{Z})=\mypi_{T}({\widetilde{T}})=T$. Hence $a=b$, which
establishes (2). ∎

##  8\. Proper morphisms

In this section, we introduce proper morphisms of monoid schemes and
characterize them in terms of the valuative criterion for properness in the
finite type case. This shows, in particular, that our notion of proper
morphisms extends that of Cortiñaz, Haesemeyer, Walker and Weibel; cf. [7,
Definition 8.4].

######  Definition .

A morphism $\myvarphi:Y\to X$ of monoid schemes is _proper_ if it is
separated, of finite type and universally closed.

###  8.1. The valuative criterion of properness

######  Theorem 8.1.

Let $\myvarphi:Y\to X$ be a finite type morphism of monoid schemes. Then
$\myvarphi$ is proper if and only if every test diagram

${U}$${Y}$${T}$${X}$$\scriptstyle{\myeta}$$\scriptstyle{\myiota}$$\scriptstyle{\myvarphi}$$\scriptstyle{\mynu}$

has a unique lift.

###### Proof.

By Theorem 7.2, $\myvarphi$ is separated if and only if every test diagram has
at most one lift. Since $\myvarphi$ is of finite type, it is quasi-compact.
Thus we can apply Theorem 6.1,
${\tilde{\myvarphi}}:\widetilde{Y}\to\widetilde{X}$, which shows that
$\myvarphi$ is universally closed if and only if every test diagram has a
lift. ∎

######  Corollary .

Let $X$ be a monoid scheme and let $k$ be an algebraically closed field. Then
$X\to\operatorname{MSpec}(\mathbb{F}_{1})$ is proper if and only if $X_{k}$ is
a complete $k$-variety.

###### Proof.

By Theorem 8.1, the morphism $X\to\operatorname{MSpec}(\mathbb{F}_{1})$ is
proper if and only if every test diagram for
$X\to\operatorname{MSpec}(\mathbb{F}_{1})$ has a unique lift and if $X$ is of
finite type. By [7, Theorem 8.9], every test diagram for
$X\to\operatorname{MSpec}(\mathbb{F}_{1})$ has a unique lift if and only if
every test diagram for $X_{k}\to\operatorname{Spec}(k)$ has a unique lift. By
[3, Lemma 3.5], $X$ is of finite type if and only if $X_{k}$ is of finite
type. By the valuative criterion for properness for usual schemes, this is, in
turn, equivalent to $X_{k}\to\operatorname{Spec}(k)$ being proper, i.e.
$X_{k}$ being a complete $k$-variety. ∎

######  Lemma .

  1. (1)

Closed immersions are proper.

  2. (2)

Proper morphisms are closed under composition and base change.

  3. (3)

Properness is local on the base.

###### Proof.

We begin with (1). By definition, closed immersions are of finite type. By
Lemma 6.2, closed immersions are universally closed. Closed immersions are
affine and thus separated. Thus closed immersions are proper.

We continue with (2). Finite type morphisms and universally closed morphisms
are clearly stable under composition. If $Y\to X$ and $X\to Z$ are separated,
then the composition

$Y\xrightarrow{\Delta_{Y/X}}Y\times_{X}Y\longrightarrow Y\times_{Z}Y$

is a closed immersion because the composition of affine morphisms is affine
and the composition of surjections of sheaves is surjective. This shows that
proper morphisms are closed under composition. Proper morphisms are also
closed under base change because the property of being finite type, separated
and universally closed are all stable under base change.

Finally we turn to (3). All defining properties of proper morphisms, finite
type, separated and universally closed, are local on the base and therefore so
is properness. ∎

We recall from [7, §7] that a morphism $Y\to X$ of monoid schemes is
_projective_ , if locally on $X$, it factors as a closed immersion
$Y\to{\mathbb{P}^{n}_{X}}$ for some $n\geqslant 0$ followed by the projection
${\mathbb{P}^{n}_{X}}\to X$.

######  Corollary .

Projective morphisms are proper.

###### Proof.

Since closed immersions are proper by Lemma 8.1, it is enough to show that
morphisms ${\mathbb{P}^{n}_{X}}\to X$ are proper. Since
$\mathbb{P}^{n}_{X}\cong\mathbb{P}^{n}_{\mathbb{F}_{1}}\times_{\mathbb{F}_{1}}X$,
it also suffices to show that morphisms
${\mathbb{P}^{n}_{\mathbb{F}_{1}}}\to{\mathbb{F}_{1}}$ are proper. We know
that ${\mathbb{P}^{n}_{\mathbb{C}}}\to{\mathbb{C}}$ are proper morphisms.
Therefore the result follows from Corollary 8.1. ∎

##  Appendix A Alternatives, ramifications, and why not

Certain concepts from ring theory and algebraic geometry allow for different
generalizations to monoids and monoid schemes. From the outset, it is not
clear which choices to make—and this is one of the struggles in the
development of ${{\mathbb{F}}_{1}}$-geometry.

We comment in this appendix on alternative perspectives on the congruence
space and why we choose to dismiss them.

###  A.1. Prime congruences

There are alternative concepts for prime congruences, which alters the notion
of points of the congruence space.

A more restrictive choice of prime congruences is considered by Manoel Jarra
in [18] and [19], which leads to subspaces of the congruence space
${\widetilde{X}}$ of a monoid scheme $X$. This theory has different goals in
mind than the present text: Jarra shows that his subspaces reflect suitably
the dimension of monoid schemes and link to Smirnov’s approach to the ABC-
conjecture.

There is also a more lax notion of prime congruence. To facilitate the
language in our discussion, we say that a congruence ${\mathfrak{c}}$ on a
monoid $A$ is _weak prime_ if $A/{\mathfrak{c}}$ is without zero divisors. As
a first observation, note that every prime congruence is weakly prime, but not
vice versa. For example, the trivial congruence on $A=\\{0,e,1\\}$ with
idempotent $e=e^{2}$ is weak prime, but not prime.

We define the _weak congruence space of $A$_ as the topological space
$\operatorname{Cong}^{w}(A)$ of all weak prime congruence of $A$ together with
the topology generated by the open subsets of the form

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

with $a,b\in A$. It comes with a topological embedding
$\myiota_{A}:\operatorname{Cong}(A)\hookrightarrow\operatorname{Cong}^{w}(A)$.

A large part of the theory of this paper transfer mutatis mutandis to weak
congruence spaces: the weak congruence space generalizes to monoid schemes $X$
in terms of open coverings, which yields a topological space
${\widetilde{X}}^{w}$ and a commutative diagram of continuous maps

${{{\widetilde{X}}}}$${{{\widetilde{X}}^{w}}}$${X}$$\scriptstyle{\myiota_{X}}$$\scriptstyle{\mypi_{X}}$$\scriptstyle{\mypi_{X}^{w}}$

Open (closed) immersions induce open (closed) topological embeddings of weak
congruence spaces. The topological characterization of closed immersions and
separated morphisms carries over to weak congruence spaces.

######  Remark  (Uncertainty about closed morphisms and valuative criteria).

At the point of writing, we do not know, however, if the weak congruence space
characterizes the same class of closed morphisms as the congruence space. It
seems very possible that they do, and in this case, the weak congruence space
would be a natural extension of the congruence space.

If not, then this infers a discrepancy for (universally) closed and proper
morphisms. The valuative criteria for these classes of morphisms serve as a
strong argument that the definitions of this paper are satisfactory. This is
our primary reason to work with the congruence space ${\widetilde{X}}$ in this
paper opposed to the weak congruence space ${\widetilde{X}}^{w}$.

Zooming in into the proof of the valuative criteria of universally closed and
proper morphisms, we see that a direct transfer of the arguments to the
setting of weak congruence spaces faces the difficulty that the congruence
kernel of morphisms into valuation monoids are prime congruences. This means
that weak prime congruence that are not prime are not seen by test diagrams in
the sense of this paper.

A natural variant that might come to mind is a larger class of test objects,
such as pointed monoids without zero divisors whose integral closure is a
valuation monoid. A simple example of this nature is the projection
$f:A=\\{0,e,1\\}\to{{\mathbb{F}}_{1}}$, which sends the idempotent element
$e=e^{2}$ of $A$ to $f(e)=1$. However, this particular choice of test object
fails to satisfy the lifting property for the induced closed immersion
$\myvarphi=f^{\ast}:U=\operatorname{MSpec}({{\mathbb{F}}_{1}})\to\operatorname{MSpec}(A)=T$
and the tautological test diagram

${U}$${U}$${T}$${T}$$\scriptstyle{\textup{id}_{U}}$$\scriptstyle{\myvarphi}$$\scriptstyle{\myvarphi}$$\scriptstyle{\textup{id}_{T}}$

since $\myvarphi:U\to T$ is not an isomorphism.

To conclude, it might be possible that the weak congruence space leans itself
for a characterization of closed morphisms (in the sense of this paper) and
therefore is _a posteriori_ a suitable extension of the notion of congruence
space. _A priori_ , we do not know whether this holds and therefore refrain
from a consideration of weak congruence spaces in the main body of this paper.

###  A.2. Structure sheaf for the congruence space

There are approaches to endow the congruence space
${\widetilde{X}}=\operatorname{Cong}A$ of a monoid $A$ with a structure sheaf
of local coordinate monoids. In contrast to the structure sheaf of the prime
spectrum $X=\operatorname{MSpec}A$, this is more subtle since it is not clear
what the coordinate monoids for the open subsets of the form
$U_{a,b}=\\{{\mathfrak{p}}\mid(a,b)\notin{\mathfrak{p}}\\}$ should be.

There are different solutions to this, which we discuss in the following.
However, in every case the conclusion is that a refinement of the structure
sheaf of ${\mathcal{O}}_{X}$ to a sheaf ${\mathcal{O}}_{{\widetilde{X}}}$ on
${\widetilde{X}}$ results in the fact that for some pointed monoids $A$, the
canonical map $A\to{\mathcal{O}}_{{\widetilde{X}}}({\widetilde{X}})$ is not an
isomorphism.

Topos theory sheds light on this phenomenon: if we endow the category
${\mathcal{A}}$ of affine monoid schemes with the Grothendieck topology of all
covering families $\\{U_{i}\to X\\}_{i\in I}$ by principal opens that appear
in the canonical topology for ${\mathcal{A}}$, then we find that each such
covering has an $i\in I$ for which $U_{i}\to X$ is an isomorphism. As a
consequence, the locale of open subschemes of $X$ is the locale of the prime
spectrum of $A=\Gamma X$; in essence this follows from Marty’s work in [30].
This means that any refinement of the topology of $\operatorname{MSpec}(A)$
towards $\operatorname{Cong}(A)$ destroys the property of
$A\to{\mathcal{O}}_{X}(X)$ being an isomorphism.

This insight is the reason for us to consider the congruence space merely as a
topological space and to dismiss the notion of a structure sheaf.
Nevertheless, we describe some natural sheaves on the congruence space in the
following. The main reason for the existence of different natural choices is
that there are several reasonable alternatives for coordinate monoids of open
subsets of the form $U_{a,b}$ of the congruence space $\operatorname{Cong}(A)$
of a pointed monoid.

####  A.2.1. Pullback sheaf

The first candidate for a structure sheaf for the congruence space
${\widetilde{X}}$ of a monoid scheme $X$ that might come to mind is the
pullback ${\mathcal{O}}_{{\widetilde{X}}}=\mypi_{X}^{\ast}{\mathcal{O}}_{X}$
of the structure sheaf of $X$. Explicitly ${\mathcal{O}}_{{\widetilde{X}}}$ is
the sheafification of the presheaf with values

${\mathcal{O}}_{{\widetilde{X}}}(V)\ =\
\operatorname{colim\,}{\mathcal{O}}_{X}(U)$

where $U$ ranges over all open subsets of $X$ that contain $\mypi_{X}(V)$,
modulo the identifications coming from the restriction maps.

For general $X$, a sheafification of the presheaf pullback is necessary. This
means, in particular, that for affine $X=\operatorname{MSpec}(A)$, the
canonical morphism
$A={\mathcal{O}}_{X}(X)\to{\mathcal{O}}_{{\widetilde{X}}}({\widetilde{X}})$ is
in general not an isomorphism, as the following example shows.

######  Example .

Consider $A=\\{0,e,1\\}$ with idempotent $e^{2}=e$. Then
$X=\operatorname{MSpec}(A)$ has two prime ideals $\langle 0\rangle=\\{0\\}$
and $\langle e\rangle=\\{0,e\\}$ and ${\widetilde{X}}$ has two prime
congruences ${\mathfrak{p}}_{0}=\langle(e,1)\rangle$ and
${\mathfrak{p}}_{e}=\langle(e,0)\rangle$. The map
$\mypi_{X}:{\widetilde{X}}\to X$ is a bijection with
$\mypi_{X}({\mathfrak{p}}_{0})=\langle 0\rangle$ and
$\mypi_{X}({\mathfrak{p}}_{e})=\langle e\rangle$, but not a homeomorphism
since $U_{e,1}=\\{{\mathfrak{p}}_{e}\\}$ is open in ${\widetilde{X}}$, but
$\mypi_{X}(U_{e,1})=\\{\langle e\rangle\\}$ is not open in $X$.

In particular, ${\widetilde{X}}=U_{e,0}\myamalg
U_{e,1}=\\{{\mathfrak{p}}_{e}\\}\myamalg\\{{\mathfrak{p}}_{0}\\}$ is discrete
and thus
${\mathcal{O}}_{{\widetilde{X}}}({\widetilde{X}})={\mathcal{O}}_{{\widetilde{X}}}(U_{e,0})\times{\mathcal{O}}_{{\widetilde{X}}}(Y_{e,1})$.
We find that

${\mathcal{O}}_{{\widetilde{X}}}(U_{e,0})\ =\ {\mathcal{O}}_{X}(\\{\langle
0\rangle\\})\ =\
{{\mathbb{F}}_{1}}\qquad\text{and}\qquad{\mathcal{O}}_{{\widetilde{X}}}(U_{e,1})\
=\ {\mathcal{O}}_{X}(X)\ =\ A.$

We see that in this case the canonical map
$A={\mathcal{O}}_{X}(X)\to{\mathcal{O}}_{{\widetilde{X}}}({\widetilde{X}})={{\mathbb{F}}_{1}}\times
A$ is injective, but not surjective.

####  A.2.2. Restriction to a coarser topology

The failure of the presheaf pullback of ${\mathcal{O}}_{X}$ to
${\widetilde{X}}$ from section A.2.1 is caused by open subsets of
${\widetilde{X}}$ that are not inverse images of open subsets of $X$.

There is a trivial solution to fix this failure: we consider ${\widetilde{X}}$
together with the topology generated by the inverse image of open subsets of
$X$. In the affine case $X=\operatorname{MSpec}(A)$, these are unions of open
subsets of ${\widetilde{X}}$ of the form $U_{a,0}$ with $a\in A$.

For this coarser topology on ${\widetilde{X}}$, the presheaf pullback
${\mathcal{O}}_{{\widetilde{X}}}^{\prime}$ of ${\mathcal{O}}_{X}$ along
$\mypi_{X}$ is a sheaf and satisfies

${\mathcal{O}}_{{\widetilde{X}}}^{\prime}(U_{a,0})\ =\
{\mathcal{O}}_{X}(U_{a})\ =\ A[a^{-1}].$

In particular, we have
${\mathcal{O}}_{{\widetilde{X}}}^{\prime}({\widetilde{X}})=A$ in the affine
case ${\widetilde{X}}=\operatorname{Cong}(A)$. This point of view does,
however, not provide any additional insight into the nature of monoid schemes
and is therefore not of interest.

####  A.2.3. Berkovich’s approach: open subsets with coordinate monoid

In contrast to the sheaf ${\mathcal{O}}_{{\widetilde{X}}}$ from section A.2.1,
which endows open subsets of ${\widetilde{X}}$ with (colimits of) local
coordinate monoids for $X$, Berkovich takes another route in [1].

Namely, there are more open subsets of ${\widetilde{X}}$ than the inverse
images of opens in $X$ (as considered in section A.2.2) that come with a
canonical coordinate monoid. Let us call an open subset $U$ of
${\widetilde{X}}$ _affine-like_ if there is a monoid morphism $f:A\to A_{U}$
such that the induced morphism $\myvarphi=f^{\ast}:Y\to X$ with
$Y=\operatorname{MSpec}A_{U}$ satisfies the following properties:

  1. (1)

The image of the associated map
$\tilde{\myvarphi}:{\widetilde{Y}}\to{\widetilde{X}}$ between congruence
spaces has image $U$.

  2. (2)

Every morphism $\mypsi:Z\to X$ with $\textup{im}(\tilde{\mypsi})\subset U$
factors uniquely through $\myvarphi:Y\to X$.

  3. (3)

For every $y\in{\widetilde{Y}}$ and $x=\myvarphi(y)$, the induced morphism
$k(y)\to k(x)$ be the respective residue fields is an isomorphism.

For such open subsets $U$ of ${\widetilde{U}}$, the monoid $A_{U}$ is a
canonical choice of coordinate monoid. In fact, affine-like open subsets of
${\widetilde{X}}$ are closed under finite intersections and form a basis for a
topology on ${\widetilde{X}}$ that is in general strictly coarser than the
topology of all open subsets and strictly finer than the topology induced from
$X$ (as considered in section A.2.2).

If we attempt to extend the sheaf ${\mathcal{O}}_{{\widetilde{X}}}$ to this
finer topology, then we gain only a presheaf that fails in general to satisfy
the sheaf axiom. Sheafification defines a sheaf of monoids on this finer
topology, whose global sections do not necessarily agree with $A$, as the
following example shows.

######  Example .

We reconsider the monoid $A=\\{0,e,1\\}$ with idempotent $e=e^{2}$ from
Example A.2.1. The only opens of ${\widetilde{X}}=\operatorname{Cong}A$ that
stem from $X$ are $\emptyset$, ${\widetilde{X}}$ and $U_{e,0}=\\{\langle e\sim
1\rangle\\}$. The open subset $U_{e,1}=\\{\langle e\sim 0\rangle\\}$ is
affine-like as witnessed by the morphism $A\to A/\langle e\sim
0\rangle\simeq{{\mathbb{F}}_{1}}$. Thus the coordinate monoid of $U_{e,1}$ is
${{\mathbb{F}}_{1}}$, equally to
${\mathcal{O}}_{{\widetilde{X}}}(U_{e,0})\simeq{{\mathbb{F}}_{1}}$. The monoid
of global sections of the sheaf on affine-like opens is
${{\mathbb{F}}_{1}}\times{{\mathbb{F}}_{1}}$, which contains $A$ as a proper
submonoid.

####  A.2.4. Deitmar: sheaf for all opens via enriched structures

Deitmar uses in [12] a richer structure, called sesquiads, that incorporate
additive relations, which allows him to define localizations
$A[\frac{1}{a-b}]$ for difference of elements $a$ and $b$ in $A$. This
generalized localization $A[\frac{1}{a-b}]$ serves as the coordinate object
for $U_{a,b}$. Also this theory fails to be subcanonical; in particular, the
global sections of the resulting structure sheaf on $\operatorname{Cong}A$ do
in general not equal $A$.

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