\section{Introduction}\label{section-intro}
The exactness and the integrality of morphisms of log schemes,
introduced by Kazuya Kato \cite{KK1}, have been widely accepted as
practically useful concepts in log geometry.
For instance, a log smooth morphism is underlain by a flat morphism if
it is integral, while the underlying morphism of a log smooth morphism
in general might be fairly pathological.

In this note we will prove that, under a few minor conditions, any
morphism of fs log schemes can be modified by so-called log
modifications to be an exact, and moreover an integral morphism.
Our main theorem is the following (Theorem \ref{thm-blowupbasechange2}
and Theorem \ref{thm-blowupbasechange3}):

\vspace{1ex}
\noindent
{\bf Theorem.} {\it Let $f\colon X\rightarrow Y$ be a morphism of
locally noetherian fs log schemes.
Suppose the underlying structure of $f$ and $Y$ are quasi-compact.}
\begin{enumerate}\vspace{-1ex}
\item {\it There exists a log-ideal $\mathcal{K}$ of $Y$ such that the base
      change of $f$ by the log blow-up of $Y$ along $\mathcal{K}$ is exact.} 
      \vspace{-1.7ex} 
\item {\it There exists a commutative diagram}
      $$
      \begin{array}{ccc}
      \til{X}&\longrightarrow&X\\
      \llap{$\scriptstyle{\til{f}}$}\vphantom{\bigg|}\Big\downarrow&&\vphantom{\bigg|}\Big\downarrow
      \rlap{$\scriptstyle{f}$}\\
      \til{Y}&\longrightarrow&Y
      \end{array}
      $$
      {\it of fs log schemes such that the horizontal arrows are log 
      modifications and $\til{f}$ is integral.}\vspace{-1ex}
\end{enumerate}

The log blow-up is also the concept which was already studied by Kazuya
Kato in his unpublished works (cf.\ \cite{KK2}\cite{KK3}). Roughly
speaking, a log blow-up is the composite of a usual blow-up centered
at an ideal which comes from an ideal of the log strtucture
(log-ideal) followed by normalization. The log modification is a
certain class of universally surjective log \'etale morphisms, which of
course includes log blow-up's. 

Notice that the second part of the theorem above (especially for a log
smooth morphism $f$) can be compared to a weak log-version of the
famous flattening theorem by Raynaud and Gruson \cite{RG1}.

\vspace{1ex}
{\it Acknowledgments.}
This work owes much to conversations with Professor Richard Pink. The
author expresses gratitude to him. The author thanks Max-Planck-Institut
f\"ur Mathematik Bonn for the nice hospitality.

\vspace{1ex}
{\it Notation and conventions.}
All sheaves are considered with respect to the \'etale topology. Log
schemes are denoted simply by capital letters. For a log scheme $X$, we
denote by $\under{X}$, $\O_X$, $\mathcal{M}_X$, and $\ovl{\mathcal{M}}_X$ the
underlying scheme, the structure sheaf of $\under{X}$, the log structure 
of $X$, and the characteristic $(=\mathcal{M}_X/\Oi_X)$ of $X$, respectively.
For a commutative and unitary monoid $P$ the associated group (resp.\
the subgroup of invertible elements) is denoted by $\gp{P}$ (resp.\
$\inv{P}$).  

\setcounter{equation}{0}
\section{Exact morphism and integral morphism}\label{section-imtegral}
In this section we recall the definition of exactness and integrality,
and basic properties of them, especially in relation to charts; we also
recall the notion of neat and good charts which is useful in the next
section. 
\begin{dfn}\label{dfn-exacthomo}
\begin{enumerate}\vspace{-1ex}
\item
A homomorphism $Q\rightarrow P$ of monoids 
is said to be {\it exact} if the diagram 
$$
\begin{array}{ccc}
Q&\longrightarrow&P\\
\vphantom{\bigg|}\Big\downarrow&&\vphantom{\bigg|}\Big\downarrow\\
\gp{Q}&\longrightarrow&\gp{P}
\end{array}
$$
is Cartesian in the category of sets.
\vspace{-1.7ex}
\item 
A morphism $f\colon X\rightarrow Y$ of log schemes is said to be {\it exact
at} $x\in X$ if the homomorphism $\ovl{\mathcal{M}}_{Y,\ovl{y}}\rightarrow
\ovl{\mathcal{M}}_{X,\ovl{x}}$ is exact, where $y=f(x)$.
({\it Here and hereafter $\ovl{\cdot}$ means the separable closure of points.})
The morphism $f$ is said to be {\it exact} if it is exact at any point of 
$X$.
\vspace{-1ex}
\end{enumerate}
\end{dfn}

If a homomorphism $h\colon Q\rightarrow P$ of monoids is exact and
$\inv{Q}=1$, then $h$ is injective.
In particular, if a morphism $f\colon X\rightarrow Y$ is exact at $x\in X$,
then $\ovl{\mathcal{M}}_{Y,\ovl{y}}\rightarrow\ovl{\mathcal{M}}_{X,\ovl{x}}$ is injective.

Let $P$ be a monoid and $A$ a ring.
We denote by $\mathop{\mathrm{S}}\nolimits(P)_A$ the log scheme underlain by the scheme $\Spec A[P]$
endowed with the log structure induced by $P\rightarrow A[P]$.
We write, in particular, $\mathop{\mathrm{S}}\nolimits(P)=\mathop{\mathrm{S}}\nolimits(P)_{\Z}$.
The log structure of $\mathop{\mathrm{S}}\nolimits(P)$ is denoted by $\mathcal{M}_P$.
(Caution: Even if $Q\rightarrow P$ is exact, $\mathop{\mathrm{S}}\nolimits(P)\rightarrow\mathop{\mathrm{S}}\nolimits(Q)$ is
not necessarily exact.)

Let $X$ be a log scheme.
A chart of $X$ modeled on a monoid $P$ is a strict morphism $\la\colon
X\rightarrow\mathop{\mathrm{S}}\nolimits(P)$ of log schemes.
The chart $\la$ is said to be good at a point $x\in X$ if $P\rightarrow
\ovl{\mathcal{M}}_{X,\ovl{x}}$ is an isomorphism.
Good charts always exist at any point, provided $X$ is fs.

Let $f\colon X\rightarrow Y$ be a morphism of log schemes.
A chart of $f$ modeled on a homomorphism $h\colon
Q\rightarrow P$ of monoids is a commutative diagram
$$
\begin{array}{ccc}
X&\longrightarrow&\mathop{\mathrm{S}}\nolimits(P)\\
\llap{$\scriptstyle{f}$}\vphantom{\bigg|}\Big\downarrow&&\vphantom{\bigg|}\Big\downarrow
\rlap{$\scriptstyle{\mathop{\mathrm{S}}\nolimits(h)}$}\\
Y&\longrightarrow&\mathop{\mathrm{S}}\nolimits(Q)
\end{array}
$$
with the horizontal arrows strict.
This chart of $f$ is said to be {\it neat} at $x\in X$ if $\gp{h}$ is
injective, $\gp{P}/\gp{Q}\rightarrow\ovl{\mathcal{M}}_{X/Y,\ovl{x}}$ is an
isomorphism, and $P\rightarrow\ovl{\mathcal{M}}_{X,\ovl{x}}$ is exact.
This chart of $f$ is said to be {\it good} at $x\in X$ if the charts 
$X\rightarrow \mathop{\mathrm{S}}\nolimits(P)$ and $Y\rightarrow\mathop{\mathrm{S}}\nolimits(Q)$ are good at $x$ and $f(x)$, 
respectively.
\begin{lem}\label{lem-neatchart}
(A.\ Ogus \cite[Theorem 2.13]{AG1})\ 
{\it Let $f\colon X\rightarrow Y$ be a morphism of fine log schemes and
$x\in X$ a point. 
Suppose there exists a chart} $Y\rightarrow\mathop{\mathrm{S}}\nolimits(Q)$ {\it of $Y$ modeled on
a fine monoid $Q$.}
{\it Then $f$ has fppf locally a neat chart at $x$ modeled on a homomorphism
of fine monoids extending the given chart} $Y\rightarrow\mathop{\mathrm{S}}\nolimits(Q)$.
{\it Furthermore, if the order of the torsion part of} 
$\gp{\ovl{\mathcal{M}}}_{X/Y,\ovl{x}}$ {\it is invertible in the residue field 
$k(\ovl{x})$, then the neat chart as above can be found \'etale locally.}
\end{lem}

\begin{cor}\label{cor-goodchart}
{\it If $f\colon X\rightarrow Y$ is a morphism of fs log schemes exact at 
$x\in $X, then $f$ allows fppf locally a good chart at $x$.}
\end{cor}

\noindent{\bf Proof.}\hspace{2mm}
Take \'etale locally a good chart $Y\rightarrow\mathop{\mathrm{S}}\nolimits(Q)$ at $y=f(x)$ and 
extend it fppf locally to a neat chart $(X\rightarrow\mathop{\mathrm{S}}\nolimits(P),
Y\rightarrow\mathop{\mathrm{S}}\nolimits(Q),Q\rightarrow P)$ at $x$.
Since $\ovl{\mathcal{M}}_{Y,\ovl{y}}\rightarrow\ovl{\mathcal{M}}_{X,\ovl{x}}$ is injective,
the chart $X\rightarrow\mathop{\mathrm{S}}\nolimits(P)$ is automatically good at $x$.
\hfill$\square$
\begin{dfn}\label{dfn-integralhomo}
A homomorphism $h\colon Q\rightarrow P$ of integral monoids is said to be 
{\it integral} if for any integral monoid $Q'$ and a homomorphism 
$Q'\rightarrow Q$, the push-out $P\oplus_{Q}Q'$ in the category of monoids
is integral.
\end{dfn}

It is known that the integrality of $h\colon Q\rightarrow P$ is equivalent
to the following condition:
If $a_1,a_2\in Q$, $b_1,b_2\in P$ and $h(a_1)b_1=h(a_2)b_2$, 
there exist $a_3,a_4\in Q$ and $b\in P$ such that $b_1=h(a_3)b$, 
$b_2=h(a_4)b$, and $a_1a_3=a_2a_4$ (cf.\ \cite[(4.1)]{KK1}).
It is also known that $h\colon Q\rightarrow P$ is integral and injective
if and only if the map $\Z[h]\colon\Z[Q]\rightarrow\Z[P]$ is flat.
\begin{lem}\label{lem-integralexact}
\begin{enumerate}\vspace{-1ex}
\item {\it Let $h\colon Q\rightarrow P$ be an integral 
homomorphism of integral monoids such that} $h^{-1}(\inv{P})=\inv{Q}$.
{\it Then $h$ is exact.}
\vspace{-1.7ex}
\item {\it Consider the commutative diagram}
$$
\begin{array}{ccc}
P'&\stackrel{a}{\longrightarrow}&P\\
\llap{$\scriptstyle{h'}$}\vphantom{\bigg|}\Big\uparrow&&\vphantom{\bigg|}\Big\uparrow
\rlap{$\scriptstyle{h}$}\\
Q'&\longrightarrow&Q
\end{array}
$$
{\it of integral monoids such that the horizontal arrow are surjective.}
{\it Then}:
\begin{enumerate}\vspace{-1ex}
\item {\it If $h'$ is integral and there exists a submonoid $P''\subseteq P'$
such that} $P'/P''\cong P$, {\it then $h$ is integral.}
\vspace{-1ex}
\item {\it If $h$ is integral and the horizontal arrows are exact, then 
$h'$ is integral.}
\vspace{-1ex}
\end{enumerate}
({\it Here the quotient of a monoid $M$ by a submonoid $N$ is defined by the 
coset space $M/\sim$ with the induced monoid structure, where $a\sim b$ if 
and only if there exist $c,d\in N$ such that $ac=bd$.})
\vspace{-1ex}
\end{enumerate}
\end{lem}

\noindent{\bf Proof.}\hspace{2mm}
Straightforward.
\hfill$\square$
\begin{dfn}\label{dfn-integralmorphism}
A morphism $f\colon X\rightarrow Y$ of integral log schemes is said to be
{\it integral at} $x\in X$ if $\ovl{\mathcal{M}}_{Y,\ovl{y}}\rightarrow
\ovl{\mathcal{M}}_{X,\ovl{x}}$ is integral, where $y=f(x)$.
We say $f$ is {\it integral} if it is integral at any point of $X$.
\end{dfn}

Note that the integrality of $\ovl{\mathcal{M}}_{Y,\ovl{y}}\rightarrow
\ovl{\mathcal{M}}_{X,\ovl{x}}$ is equivalent to that of
$\mathcal{M}_{Y,\ovl{y}}\rightarrow\mathcal{M}_{X,\ovl{x}}$.
By Lemma \ref{lem-integralexact} (1), integral morphisms are exact.
If $h\colon Q\rightarrow P$ is integral, then by Lemma \ref{lem-integralexact}
(2) (a), the morphism $\mathop{\mathrm{S}}\nolimits(h)\colon\mathop{\mathrm{S}}\nolimits(P)\rightarrow\mathop{\mathrm{S}}\nolimits(Q)$ is integral.

In what follows, all push-outs are taken in the category of fs monoids,
unless otherwise explicitely specified.
Accordingly, all fiber products of fs log schemes are taken in the category 
fs log schemes, unless otherwise specified.

Let $P\leftarrow Q\rightarrow Q'$ be homomorphisms of fs monoids and
$\gp{P}\leftarrow\gp{Q}\rightarrow\gp{Q'}$ the induced morphisms, which
gives rise to the push-out $\gp{P}\oplus_{\gp{Q}}\gp{Q'}$ in the
category of finitely generated abelian groups.
\begin{lem}\label{lem-pushoutfs}
{\it The push-out of $P\leftarrow Q\rightarrow Q'$ in the category of fs 
monoids is given by the smallest
fs submonoid in} $\gp{P}\oplus_{\gp{Q}}\gp{Q'}$ {\it containing 
those images of $P$ and $Q'$, or what amounts to the same, the saturation of 
the submonoid in} $\gp{P}\oplus_{\gp{Q}}\gp{Q'}$ {\it generated by those 
images of $P$ and $Q'$.}
\end{lem}

\noindent{\bf Proof.}\hspace{2mm}
It is enough to prove that the push-out in the category of fine monoids is
the submonoid in $\gp{P}\oplus_{\gp{Q}}\gp{Q'}$ generated by the 
images of $P$ and $Q'$, which is easy to verify.
\hfill$\square$
\begin{cor}\label{cor-pushoutfs}
{\it For a homomorphism $Q\rightarrow P$ of fs monoids such that $\gp{Q}\cong
\gp{P}$, the push-out of
$P\leftarrow Q\rightarrow P$ is given by $P$.}
\end{cor}
	
We denote the push-out of $P\leftarrow Q\rightarrow Q'$ by $P_{Q'}$ if
the monoid $Q$ is evident in the context. 
\begin{lem}\label{lem-exactstable}
{\it For an exact homomorphism $Q\rightarrow P$ of fs monoids 
and a homomorphism $Q\rightarrow Q'$ of fs monoids,
the induced homomorphism $Q'\rightarrow P_{Q'}$ is exact.}
\end{lem}

\noindent{\bf Proof.}\hspace{2mm}
It is straightfoward to see that the induced homomorphism $Q'\rightarrow
P'$ is exact, where $P'$ is the submonoid in $\gp{P}\oplus_{\gp{Q}}\gp{{Q'}}$
generated by the images of $P$ and $Q'$.
The monoid $P_{Q'}$ is the saturation of $P'$.
Since $Q'$ is saturated, we see that $Q'\rightarrow P_{Q'}$ is also exact.
\hfill$\square$
\begin{cor}\label{cor-exactstable}
{\it Exactness is stable under base change.}
\end{cor}
\setcounter{equation}{0}
\section{Log blow-up and log modification}\label{section-logblowup}
\begin{dfn}\label{dfn-idealmon}
Let $Q$ be a monoid. A subset $K\subseteq Q$ (possibly empty) is said to 
be an ideal of $Q$ if $ax\in K$ for any $a\in Q$ and $x\in K$.
\end{dfn}

Any ideal of a finitely generated monoid is finitely generated as an ideal.
Products and intersections of ideals are obviously ideals.
The correspondence $K\mapsto\ovl{K}\colon=\pi(K)\cdot\ovl{Q}$ gives rise to 
the natural bijection between the set of all ideals in $Q$ and the set of all
ideals in $\ovl{Q}$, where $\pi\colon Q\rightarrow\ovl{Q}=Q/\inv{Q}$ is the
natural projection. 
\begin{ntn}
Let $Q$ be an fs monoid.
For a subset $E\subseteq\gp{Q}$ we denote by $Q\langle E\rangle$ the 
smallest fs submonoid in $\gp{Q}$ which contains $Q$ and $E$.
\end{ntn}
\begin{dfn}
Let $X$ be a log scheme.
A {\it log-ideal} of $X$ is a sheaf of ideals $\mathcal{K}$ of $\mathcal{M}_X$.
\end{dfn}
\begin{ntn}
\begin{enumerate}\vspace{-1ex}
\item For an ideal $K\subseteq Q$ we denote by $\til{K}$ the log-ideal of
      $\mathop{\mathrm{S}}\nolimits(Q)$ generated by the image of $K$.
\vspace{-1.7ex}
\item Let $f\colon X\rightarrow Y$ be a morphism of log schemes and
      $\mathcal{K}$ a log-ideal of $Y$. Then we denote by $f^{\bullet}\mathcal{K}$
      the log-ideal of $X$ generated by the image of $f^{-1}\mathcal{K}$.
\vspace{-1ex}
\end{enumerate}
\end{ntn}

Clearly, for $f\colon X\rightarrow Y$ and $g\colon Y\rightarrow Z$ and a 
log-ideal $\mathcal{K}$ of $Z$, we have $(g\circ f)^{\bullet}\mathcal{K}=f^{\bullet}
g^{\bullet}\mathcal{K}$.

A log scheme $X$ is called {\it locally noetherian} if the underlying scheme
$\under{X}$ is locally noetherian.
\begin{lem}\label{lem-coherentideal}
(A.\ Ogus \cite[Proposition 2.6]{AG1})\ 
{\it Let $X$ be a locally noetherian fine log scheme and $\mathcal{K}$ a log-ideal.
Then the following conditions are equivalent}:
\begin{enumerate}\vspace{-1ex}
\item {\it For any $x,y\in X$ such that $x$ belongs to the topological closure 
of $\{y\}$, the natural map $\mathcal{K}_{\ovl{x}}\rightarrow\mathcal{K}_{\ovl{y}}$ is
surjective.}
\vspace{-1.7ex}
\item {\it For any point $x\in X$ and any \'etale neighborhood $\iota\colon
U\rightarrow X$ of $x$ with chart} $\la\colon U\rightarrow\mathop{\mathrm{S}}\nolimits(P)$ {\it set}
$K=\beta_{\ovl{x}}^{-1}(\mathcal{K}_{\ovl{x}})$, {\it where} $\beta\colon
P\rightarrow\mathcal{M}_X|_U$.
{\it Then replacing $U$ by a sufficiently small \'etale neighborhood of $x$,
we have}
$\iota^{\bullet}\mathcal{K}=\la^{\bullet}\til{K}$.
\vspace{-1ex}
\end{enumerate}
\end{lem}

\begin{dfn}\label{dfn-coherentideal}
Let $X$ be a locally noetherian fine log scheme.
Then a log-ideal $\mathcal{K}$ of $X$ is said to be {\it coherent} if it 
satisfies the conditions in Lemma \ref{lem-coherentideal}.
\end{dfn}
\begin{dfn}\label{dfn-locallyprincipal}
A coherent log-ideal $\mathcal{K}$ of $X$ is said to be {\it locally principal}
if it is \'etale locally generated by a single global section.
\end{dfn}
\begin{dfn}\label{dfn-logblowupuniversal}
Let $X$ be a locally noetherian fs log scheme and $\mathcal{K}$ a coherent log-ideal
of $X$.
Then the {\it log blow-up} $\rho_{\mathcal{K}}\colon\mathop{\mathrm{B}}\nolimits_{\mathcal{K}}(X)\rightarrow X$
of $X$ along $\mathcal{K}$ is a morphism of locally noetherian fs log schemes
satisfying the following conditions:
\begin{enumerate}\vspace{-1ex}
\item The log ideal $\rho_{\mathcal{K}}^{\bullet}\mathcal{K}$ of $\mathop{\mathrm{B}}\nolimits_{\mathcal{K}}(X)$ is 
locally principal.\vspace{-1.7ex}
\item For any morphism $f\colon X'\rightarrow X$ of locally 
noetherian fs log schemes such that $f^{\bullet}\mathcal{K}$ is locally principal, 
there exists a unique morphism $\til{f}\colon X'\rightarrow\mathop{\mathrm{B}}\nolimits_{\mathcal{K}}(X)$ 
such that $f=\rho_{\mathcal{K}}\circ\til{f}$.
\vspace{-1ex}
\end{enumerate}
\end{dfn}

The log blow-up along $\mathcal{K}$ is unique up to isomorphisms if it exists.
We are going to construct $\rho_{\mathcal{K}}\colon\mathop{\mathrm{B}}\nolimits_{\mathcal{K}}(X)\rightarrow X$.
We should first start with the case $X=\mathop{\mathrm{S}}\nolimits(P)$ for an fs monoid $P$.

The following construction is due to Kazuya Kato \cite{KK3}:
Let $K\subseteq P$ be an ideal. Then the log blow-up 
$\mathop{\mathrm{B}}\nolimits_K(P)\rightarrow\mathop{\mathrm{S}}\nolimits(P)$ of $\mathop{\mathrm{S}}\nolimits(P)$ along $K$ is constructed as follows:
Let $I(K)$ be the ideal of $\Z[P]$ generated by $K$.
Consider the natural morphism $\mathop{\mathrm{Proj}}\nolimits\oplus_nI(K)^n\rightarrow
\Spec\Z[P]$.
$\mathop{\mathrm{Proj}}\nolimits\oplus_nI(K)^n$ has the affine open covering
$$
\mathop{\mathrm{Proj}}\nolimits\oplus_nI(K)^n=\bigcup_{p\in K}\Spec\Z[P\langle p^{-1}K
\rangle].
$$
(It is enough for $p$ to run over the generators of $K$.)
The canonical log structures given by $P\langle p^{-1}K\rangle\rightarrow
\Z[P\langle p^{-1}K\rangle]$ obviously glue to a fine log structure on 
$\mathop{\mathrm{Proj}}\nolimits\oplus_nI(K)^n$.
The fs log scheme $\mathop{\mathrm{B}}\nolimits_K(P)$ is defined by the saturation of this resulting 
log scheme.
The morphisms $\mathop{\mathrm{S}}\nolimits(P\langle p^{-1}K\rangle)\rightarrow\mathop{\mathrm{S}}\nolimits(P)$ give rise to
the desired morphism $\mathop{\mathrm{B}}\nolimits_K(P)\rightarrow\mathop{\mathrm{S}}\nolimits(P)$, which turns out to be the
log blow-up in this case.

Let $X$ be an fs log scheme underlain by a locally noetherian scheme and
$\mathcal{K}$ a coherent log-ideal.
Suppose $X$ has a chart $\la\colon X\rightarrow\mathop{\mathrm{S}}\nolimits(P)$ modeled on an fs monoid 
$P$, and that the log-ideal $\mathcal{K}$ is generated by global sections.
Take a point $x\in X$ and set $K=\beta_{\ovl{x}}^{-1}(\mathcal{K}_{\ovl{x}})$, where
$\beta\colon P\rightarrow\mathcal{M}_X$.
Since $\mathcal{K}$ is generated by global sections, the ideal $K$ does not depend
on the choice of $x$.
In this situation we define the log blow-up $\mathop{\mathrm{B}}\nolimits_{\mathcal{K}}(X)\rightarrow X$ by
$$
\mathop{\mathrm{B}}\nolimits_{\mathcal{K}}(X)\colon=X\times_{\mathop{\mathrm{S}}\nolimits(P)}\mathop{\mathrm{B}}\nolimits_K(P)\longrightarrow X.
$$
It can be checked without so much pain that this morphism satisfies the 
conditions in Definition \ref{dfn-logblowupuniversal}.
In particular, it does not depend on the choice of the chart $\la$.

In general we take an \'etale covering $\bigcup_{i\in I}U_i\rightarrow X$
such that each $U_i$ has allows a chart $U_i\rightarrow\mathop{\mathrm{S}}\nolimits(P_i)$
and $\mathcal{K}|_{U_i}$ is generated by
global sections.
Then from what we have seen above the local log blow-up's
$\mathop{\mathrm{B}}\nolimits_{\mathcal{K}|_{U_i}}(U_i)\rightarrow U_i$ glue to a morphism of log schemes
$\mathop{\mathrm{B}}\nolimits_{\mathcal{K}}(X)\rightarrow X$ which is nothing but the log
blow-up of $X$ along the coherent log-ideal $\mathcal{K}$.
Note that log blow-up's are log \'etale.

The following lemmas are easy to see by the universality of log blow-up's:
\begin{lem}\label{lem-logblowupselfint}
{\it Let $X$ be a locally noetherian fs log scheme and $\mathcal{K}$ a log-ideal
of $X$. Then we have}
$\mathop{\mathrm{B}}\nolimits_{\mathcal{K}}(X)\times_X\mathop{\mathrm{B}}\nolimits_{\mathcal{K}}(X)\cong\mathop{\mathrm{B}}\nolimits_{\mathcal{K}}(X)$.
\end{lem}
\begin{lem}\label{lem-logblowupcommutative}
{\it Let $\mathcal{K}$ and $\mathcal{K}'$ be coherent log-ideals of $X$. Then we have 
a canonical isomorphisms} 
$$
\mathop{\mathrm{B}}\nolimits_{\mathcal{K}\ast\mathcal{K}'}(X)\cong\mathop{\mathrm{B}}\nolimits_{\rho_{\mathcal{K}}^{\bullet}\mathcal{K}'}
(\mathop{\mathrm{B}}\nolimits_{\mathcal{K}}(X))\cong\mathop{\mathrm{B}}\nolimits_{\rho_{\mathcal{K}'}^{\bullet}\mathcal{K}}
(\mathop{\mathrm{B}}\nolimits_{\mathcal{K}'}(X))
$$
{\it over $X$, where $\mathcal{K}\ast\mathcal{K}'$ denote the log-ideal of $X$
generated by $\mathcal{K}$ and $\mathcal{K}'$.}
\end{lem}
\begin{exa}\label{exa-logblowuptoric}
It is worth while describing log blow-up's in the toric case:
Let $\Sigma$ be a fan of $N\cong\Z^r$ and $T_{\Sigma}$ the assiciated 
toric variety over a ring $A$.
For any cone $\sigma\in\Sigma$, the affine patch $T_{\sigma}=
\Spec A[\sigma^{\vee}\cap M]$ (where $M=\Hom_{\Z}(N,\Z)$) has the canonical
log structure which gives rise to $\mathop{\mathrm{S}}\nolimits(\sigma^{\vee}\cap M)_A$.
These patches glue to an fs log scheme which is also denoted by $T_{\Sigma}$.
Let $T=\Spec A[M]$ be the torus included in $T_{\Sigma}$ as a Zariski open
set and $i\colon T\hookrightarrow T_{\Sigma}$ the open immersion.
Then $T$ is exactly the locus on which the log structure is trivial.
Furthermore, the log structure $\mathcal{M}_{T_{\Sigma}}$ on $T_{\Sigma}$ is 
isomorphic to
$$
\O_{T_{\Sigma}}\cap i_{\ast}\Oi_T\lhook\joinrel\longrightarrow\O_{T_{\Sigma}}.
$$

Log blow-up's of toric varieties coincides with those described
in \cite[I, \S 2]{KKM}:
For a log-ideal $\mathcal{K}$ of $T_{\Sigma}$, the ideal
$\mathcal{I}(\mathcal{K})$ of 
$\O_{T_{\Sigma}}$ generated by $\mathcal{K}$ is invariant under the
action of $T$. 
Conversely, to a $T$-invariant ideal of $\O_{T_{\Sigma}}$ we can associate
the log ideal of $T_{\Sigma}$ by taking the intersection with
$\mathcal{M}_{T_{\Sigma}}$.
This correspondence gives rise to the bijection between the set of all 
log-ideals of $T_{\Sigma}$ and that of all $T$-invariant ideals of
$\O_{T_{\Sigma}}$.
If $\mathcal{I}$ is a $T$-invariant ideal of $\O_{T_{\Sigma}}$, the log blow-up
along $\mathcal{I}$ is given by a certain subdivision
$\til{\Sigma}\rightarrow\Sigma$ 
of the fan $\Sigma$; the fan $\til{\Sigma}$ is the subdivision consisting of
cones on which the associated piecewise linear function
 $\mathrm{ord}\mathcal{I}$. 
\end{exa}

Any log blow-up of a toric variety therefore comes from subdivision of
the fan. But a morphism of toric varieties by subdivision is not
necessarily a log blow-up.
\begin{lem}\label{lem-blowupbasechange1}
{\it Let $Q\rightarrow P$ be a homomorphism of fs monoids such that}
$\gp{Q}\cong\gp{P}$.
{\it Then there exists an ideal $K\subseteq Q$ and a strict open immersion}
$\mathop{\mathrm{S}}\nolimits(P)\hookrightarrow\mathop{\mathrm{B}}\nolimits_K(Q)$ {\it such that}
$$
\begin{array}{ccc}
\mathop{\mathrm{S}}\nolimits(P)&=&\mathop{\mathrm{S}}\nolimits(P)\\
\vphantom{\bigg|}\Big\downarrow&&\vphantom{\bigg|}\Big\downarrow\\
\mathop{\mathrm{S}}\nolimits(Q)&\longleftarrow&\mathop{\mathrm{B}}\nolimits_K(Q)
\end{array}
$$
{\it is a Cartesian diagram of fs log schemes.}
\end{lem}

\noindent{\bf Proof.}\hspace{2mm}
Write $T=\mathop{\mathrm{S}}\nolimits(P)$ and $S=\mathop{\mathrm{S}}\nolimits(Q)$.
We can set $P=Q\langle\frac{a_1}{b},\ldots,\frac{a_r}{b}\rangle$ in
$\gp{Q}\cong\gp{P}$, where $b,a_1,\ldots,a_r\in Q$.
Let $K=(b,a_1,\ldots,a_r)$ be the ideal of $Q$.
Then the log blow-up $\mathop{\mathrm{B}}\nolimits_K(Q)$ has the covering
$$
\mathop{\mathrm{B}}\nolimits_K(Q)=\bigcup_{i=0,\ldots,r}V_i,
$$
where 
$$
V_0=\mathop{\mathrm{S}}\nolimits(Q\langle\frac{a_1}{b},\ldots,\frac{a_r}{b}\rangle),
$$
which is isomorphic to $T$, and
$$
V_j=\mathop{\mathrm{S}}\nolimits(Q\langle\frac{b}{a_j},\frac{a_1}{a_j},\ldots,\frac{a_r}{a_j}
\rangle)
$$
for $j>0$.
By Lemma \ref{lem-pushoutfs} we have $T\times_SV_0\cong T$ and 
$T\times_SV_j\cong\mathop{\mathrm{S}}\nolimits(P\langle\frac{b}{a_j}\rangle)$ for $j>0$.
Since we can check $T\times_S(V_0\cap V_j)=T\times_SV_j$, we have
$T\times_S\mathop{\mathrm{B}}\nolimits_K(Q)\cong T$.
\hfill$\square$
\begin{thm}\label{thm-blowupbasechange2}
{\it Let $f\colon X\rightarrow Y$ be a morphism of locally noetherian fs log 
schemes.
Suppose $\under{f}$ and $\under{Y}$ are quasi-compact.
Then there exists a log-ideal $\mathcal{K}$ of $Y$
such that the base change} $X_{\mathop{\mathrm{B}}\nolimits_{\mathcal{K}}(Y)}\rightarrow\mathop{\mathrm{B}}\nolimits_{\mathcal{K}}(Y)$
{\it by the log blow-up of $Y$ along $\mathcal{K}$ is exact.}
\end{thm}

\noindent{\bf Proof.}\hspace{2mm}
First we assume that $\gp{\ovl{\mathcal{M}}}_{X/Y}$ is constant and that there
exists a chart $Y\rightarrow\mathop{\mathrm{S}}\nolimits(Q)$.
Then by Lemma \ref{lem-neatchart}
there exists an fppf covering $\{X_i\rightarrow X\}$ of $X$ consisting of
finitely many members such that each $X_i\rightarrow Y$ allows a chart
$(\la_i\colon X_i\rightarrow\mathop{\mathrm{S}}\nolimits(P_i),\mu\colon Y\rightarrow\mathop{\mathrm{S}}\nolimits(Q),
h_i\colon Q\rightarrow P_i)$
such that $h_i$ is injective and $\gp{\ovl{\mathcal{M}}}_{X/Y}\cong\gp{P}_i/\gp{Q}$.
Set $Q_i=P_i\cap\gp{Q}$. Then $Q\rightarrow Q_i$ yields $\gp{Q}\cong\gp{Q}_i$.
Set $Y_i=Y\times_{\mathop{\mathrm{S}}\nolimits(Q)}\mathop{\mathrm{S}}\nolimits(Q_i)$ for each $i$.
By Lemma \ref{lem-blowupbasechange1} we have an ideal $K_i$ of $Q$ such 
that $\mathop{\mathrm{S}}\nolimits(Q_i)\hookrightarrow\mathop{\mathrm{B}}\nolimits_{K_i}(Q)$ gives the base change.
Set $\mathcal{K}_i=\mu^{\bullet}\til{K}_i$ and consider the log blow-up
$\mathop{\mathrm{B}}\nolimits_{\mathcal{K}_i}(Y)\rightarrow Y$.
Then we have $X_i\times_Y\mathop{\mathrm{B}}\nolimits_{\mathcal{K}_i}(Y)\cong X_i$, since
$Y_i\times_Y\mathop{\mathrm{B}}\nolimits_{\mathcal{K}_i}(Y)\cong Y_i$.
Since $Y_i\rightarrow\mathop{\mathrm{B}}\nolimits_{\mathcal{K}_i}(Y)$ is strict, we have
$\gp{\ovl{\mathcal{M}}}_{X_i/\mathop{\mathrm{B}}\nolimits_{\mathcal{K}_i}(Y)}\cong\gp{\ovl{\mathcal{M}}}_{X_i/Y}$,
which is constant and is isomorphic to $\gp{P}_i/\gp{Q}_i$.
Hence it follows from the exactness of $Q_i\rightarrow P_i$ that 
$X_i\rightarrow Y_i$ is exact, and hence $X_i\rightarrow\mathop{\mathrm{B}}\nolimits_{\mathcal{K}_i}(Y)$ is
exact.
Let $\mathcal{K}$ be the log-ideal of $Y$ generated by $\mathcal{K}_i$'s.
Since the exactness is stable under base changes, the base change 
$X_{\mathop{\mathrm{B}}\nolimits_{\mathcal{K}}(Y)}\rightarrow\mathop{\mathrm{B}}\nolimits_{\mathcal{K}}(Y)$ restricted on each $X_i$
is exact, and hence the whole
$X_{\mathop{\mathrm{B}}\nolimits_{\mathcal{K}}(Y)}\rightarrow\mathop{\mathrm{B}}\nolimits_{\mathcal{K}}(Y)$ is exact.
Hence we have proved the theorem in this particular case.

In general we may still assume $Y$ allows a chart $Y\rightarrow\mathop{\mathrm{S}}\nolimits(Q)$.
The sheaf $\gp{\ovl{\mathcal{M}}}_{X/Y}$ is constructible.
We look at each locally closed set of $\under{X}$ on which 
$\gp{\ovl{\mathcal{M}}}_{X/Y}$ is constant, and take a log blow-up to make $f$ exact 
on this locus.
Finally, we gather up the resulting log-ideal of $Y$ and then take 
sufficiently large log blow-up.
\hfill$\square$
\begin{dfn}\label{dfn-logmodification}
A morphism $\rho\colon\til{X}\rightarrow X$ of locally noetherian 
fs log schemes is called a {\it log modification} if for any point 
$x\in X$ there exists an \'etale neighborhood $U\rightarrow X$ of 
$x$, and a log blow-up $Z\rightarrow\til{X}_U$ such that the composite 
$Z\rightarrow\til{X}_U\rightarrow U$ is also a log blow-up.
\end{dfn}

Note that if $X$ is quasi-compact the morphism 
$\rho\colon\til{X}\rightarrow X$ is a log modification if and only if 
there exists a log blow-up $Z\rightarrow\til{X}$ such that the composite 
$Z\rightarrow\til{X}\rightarrow X$ is also a log blow-up.

The above definition of log modifications is motivated by the following
facts:
\begin{itemize}\vspace{-1ex}
\item A morphism $\rho\colon\til{X}\rightarrow X$ of quasi-projective 
varieties over a field is a modification (i.e.\ proper and birational
morphism) if and only if there exists a blow-up $Z\rightarrow\til{X}$
such that the composite $Z\rightarrow X$ is a blow-up; this follows
easily from \cite[Theorem 7.17]{Ha1} and Chow's lemma.
\vspace{-1.7ex}
\item Any morphism of toric varieties coming from subdivision of fans
are log modifications due to De Concini-Procesi's Theorem (cf.\ \cite[\S
      1.7]{TO1}). 
\vspace{-1ex}
\end{itemize}

Note that log modifications are log \'etale.

\begin{lem}\label{lem-logmodificationcomposite}
\begin{enumerate}\vspace{-1ex}
\item {\it Log modifications are universally surjective. $($A morphism
      $X\rightarrow Y$ of fs log schemes is said to be universally
      surjective if for any 
morphism $Y'\rightarrow Y$ of fs log schemes the underlying morphism of
the base change $X_{Y'}\rightarrow Y'$ is surjective.$)$}
\vspace{-1.7ex}
\item {\it If $X\rightarrow Y$ and $Y\rightarrow Z$ are log modifications,
then the composite $X\rightarrow Z$ is a log modification.}
\vspace{-1.7ex}
\item {\it If $\til{X}\rightarrow X$ is a log modification and $Y\rightarrow X$
is a morphism, then the base change $\til{X}_Y\rightarrow Y$ is a log
modification.}
\vspace{-1ex}
\end{enumerate}
\end{lem}

\noindent{\bf Proof.}\hspace{2mm}
First we prove (1).
It is enough to prove that log blow-up's are universally surjective.
By the universality of log blow-up's it suffices to show that any blow-up
is surjective.
By an easy observation we reduce to verify whether $\mathop{\mathrm{B}}\nolimits_{K}(P)\rightarrow
\mathop{\mathrm{S}}\nolimits(P)$ is surjective for a monoid $P$ and an ideal $K$.
But this is trivial since the underlying morphism of $\mathop{\mathrm{B}}\nolimits_{K}(P)\rightarrow
\mathop{\mathrm{S}}\nolimits(P)$ is a normalized blow-up in the usual sense.
To prove (2) we may assume there exists a log blow-up $\mathop{\mathrm{B}}\nolimits(Z)\rightarrow Z$
which factors through $Y$ by a log blow-up.
We can take a log blow-up $\mathop{\mathrm{B}}\nolimits(Y)\rightarrow Y$ which factors through
$X$ by a log blow-up, since $Y$ is easily seen to be quasi-compact.
Take the log blow-up $\mathop{\mathrm{B}}\nolimits(\mathop{\mathrm{B}}\nolimits(Z))\rightarrow\mathop{\mathrm{B}}\nolimits(Z)$ along the pull-back of
the blown-up center of $\mathop{\mathrm{B}}\nolimits(Y)\rightarrow Y$.
Then we have the unique morphism $\mathop{\mathrm{B}}\nolimits(\mathop{\mathrm{B}}\nolimits(Z))\rightarrow\mathop{\mathrm{B}}\nolimits(Y)$, and then
(2) is proved.
(3) is straightforward.
\hfill$\square$

\vspace{1ex}
\begin{thm}\label{thm-blowupbasechange3}
{\it Let $f\colon X\rightarrow Y$ be a morphism of locally noetherian fs log 
schemes.
Suppose $\under{f}$ and $\under{Y}$ are quasi-compact.
Then there exists a commutative diagram}
$$
\begin{array}{ccc}
\til{X}&\longrightarrow&X\\
\llap{$\scriptstyle{\til{f}}$}\vphantom{\bigg|}\Big\downarrow&&\vphantom{\bigg|}\Big\downarrow
\rlap{$\scriptstyle{f}$}\\
\til{Y}&\longrightarrow&Y
\end{array}
$$
{\it of fs log schemes such that the horizontal arrows are log modifications
and $\til{f}$ is integral.}
\end{thm}

\noindent{\bf Proof.}\hspace{2mm}
By Theorem \ref{thm-blowupbasechange2} we may assume $f$ is exact, since
base changes by log blow-up's are log modifications.
Since we have assumed $\under{Y}$ is quasi-compact, we may assume $Y$ allows
a chart $\mu\colon Y\rightarrow\mathop{\mathrm{S}}\nolimits(Q)$ which is good at a point.
By Corollary \ref{cor-goodchart} we can take fppf locally on $X$ a chart
$(\la\colon X\rightarrow\mathop{\mathrm{S}}\nolimits(P),\mu\colon Y\rightarrow\mathop{\mathrm{S}}\nolimits(Q),h\colon
Q\rightarrow P)$ of $f$ which is good at a point.
Then $\mathop{\mathrm{S}}\nolimits(h)\colon\mathop{\mathrm{S}}\nolimits(P)\rightarrow\mathop{\mathrm{S}}\nolimits(Q)$ is a morphism of toric varieties
which comes from a morphism of fans $\Sigma'\rightarrow\Sigma$ (both of which
are defined by single cones) underlain by $\varphi\colon
N'\rightarrow N$.

For any ray $\rho\in\Sigma'_1$ the image of $\rho$ in $N$ is either a
ray or a point.
If it is a ray, then take the primitive base $n_1\in N$ of $\varphi(\rho)$, 
and extend it to a $\Z$-basis $n_1,\ldots,n_r$ of $N$.
The $r+1$-rays spanned by $n_1,\ldots,n_r,-(n_1+\cdots+n_r)$ give
the projective space
$\mathbf{P}^{r}_{\Z}$, and the very ample divisor $\O(1)$ gives rise to
the support function, which we denote by $s_{\rho}$.
In case $\rho$ is mapped to the point we set $s_{\rho}=0$.
Then 
$$
s=\sum_{\rho\in\Sigma'_1}s_{\rho}
$$
gives a piecewise linear function on $N\times_{\Z}\R$ which yields a 
log-ideal $\mathcal{K}$ on $\mathop{\mathrm{S}}\nolimits(Q)$
(cf.\ \cite[I, Theorem 9]{KKM}).
Set $\mathcal{K}'=\mathop{\mathrm{S}}\nolimits(h)^{\bullet}\mathcal{K}$.
The induced morphism $\mathop{\mathrm{B}}\nolimits_{K'}(P)\rightarrow\mathop{\mathrm{B}}\nolimits_K(Q)$ is also an
equivariant morphism of toric varieties which comes from $\til{\Sigma}'
\rightarrow\til{\Sigma}$, where $\til{\Sigma}$ (resp.\ $\til{\Sigma}'$) is
the subdivision of $\Sigma$ (resp.\ $\Sigma'$) consisting of biggest cones
on which $s$ (resp.\ $\varphi_{\R}^{\ast}s$) is linear.
Since any cones in $\til{\Sigma}'$ is mapped surjectively onto some cone in
$\til{\Sigma}$, that the morphism of associated toric varieties
$\mathop{\mathrm{B}}\nolimits_{K'}(P)\rightarrow\mathop{\mathrm{B}}\nolimits_K(Q)$ has constant fiber dimension.
Taking more auxiliary log blow-up's of $\mathop{\mathrm{B}}\nolimits_K(Q)$ and $\mathop{\mathrm{B}}\nolimits_{K'}(P)$, we may
assume $\mathop{\mathrm{B}}\nolimits_K(Q)$ is non-singular.
In this case the morphism $\mathop{\mathrm{B}}\nolimits_{K'}(P)\rightarrow\mathop{\mathrm{B}}\nolimits_K(Q)$ is flat (since
toric varieties are Cohen-Macaulay), or what amounts to the same, integral
(as a morphism of fs log schemes) by \cite[(4,1)]{KK1}.

We gather up these resulting local log-ideals on $Y$ to make the log-ideal
$\mathcal{K}$.
We set $\mathcal{K}'=f^{\bullet}\mathcal{K}$ and then consider $\mathop{\mathrm{B}}\nolimits_{\mathcal{K}'}(X)
\rightarrow\mathop{\mathrm{B}}\nolimits_{\mathcal{K}}(Y)$.
Then by what we have seen above it is fppf locally integral, and hence
is integral.
\hfill$\square$
\begin{small}
