
%\subsection{Introduction}

As we know, for schemes, the general deformation theory is not as easy as the smooth case. To understand deformation theory of general morphisms, one has to use the full power of the cotangent complex. In log geometry, one can generalize it to get a reasonable theory of logarithmic cotangent complex. 

\zz
This log cotangent complex will be compatible with the usual cotangent complex when the morphism in question is strict and is also compatible with log smooth deformation theory for log smooth morphisms. Basically, this is an application of the deformation theory of representable morphisms to algebraic stacks (\cite{defStacks}) to the classifying morphisms from the underlying scheme of \m X to the stack \m{LOG_Y} (\cite{LogStack}, see also section \ref{LogStacks} of this chapter.)

\begin{conv}
We will focus on the category of fine log schemes, and the functors will be always in the derived sense. For a log scheme \m X, \underline{X} means the underlying scheme of \m X.
\end{conv}

\begin{rmk}We will work with the category \m{D'(\underline{X}_{\acute{e}t})} and similar categories, and one can talk about distinguished triangles and Ext's in these categories. For relevant definitions, see \cite{LogCot} and \cite{defStacks}.
\end{rmk}

Our presentation here follows \cite{LogCot}.

\subsection{The Log Cotangent Complex}
In \cite{LogStack}, an Artin stack \m{LOG_Y} is defined so that to give a morphism of log schemes \m{X\ra Y} is equivalent to give a representable morphism \m{\underline{X}\ra LOG_Y}. Thus one may think deformations of the morphism of log schemes \m{X\ra Y} as deformations of the representable morphism from \underline{X} to the stack \m{LOG_Y}. In \cite{defStacks}, the deformation theory of such morphisms was studied in detail. As an application of this theory, one makes the following definition:

\begin{defn}
For a morphism of log schemes \m{f:X\ra Y}, the logarithemic cotangent complex of \m{f} is defined by \m{L_f=L_{\underline{X}/LOG_Y}}, where the right hand side is the cotangent complex of the morphism \m{\underline{X}\ra LOG_Y} defined in \cite{defStacks}.
\end{defn}

\begin{rmk}
One should think about \m{L_f} as an object of the category \m{D'_{qcoh}(\underline{X}_{\acute{e}t})}. In the above definition, the right hand side is an object of the category \m{D'_{qcoh}(\underline{X}_{lis-\acute{e}t})}. As the restriction functor
\m{
D'_{qcoh}(\underline{X}_{lis-\acute{e}t})\ra D'_{qcoh}(\underline{X}_{\acute{e}t})
}
is an equivalence of categories, no information of the cotangent complex would lost.
\end{rmk}

\subsection{Basic Properties}

For every morphism of fine log schemes \m{f:X\ra Y} the log cotangent complex is a projective system
\[
L_f=(\cdots\ra L_f^{\geq -n-1}\ra L_f^{\geq -n}\ra\cdots\ra L_f^{\geq0})
\]
where each \m{L_f^{\geq -n}} is an essentially constant ind-object in \m{D^{[-n,0]}(\ox)} (The derived category of \m{\ox}-modules supported in  \m{[-n,0]}).

The log cotangent complex \m{L_f} has the following properties:
\begin{enumerate}
\item For any \m{n\geq 0}, the natural map \m{\tau_{\geq -n}L_f^{\geq -n-1}\ra L_f^{\geq -n}} is an isomorphism.
\item If \m f is strict, then the system \m{(\tau_{\geq -n}L_{f'})} represents \m{L_f}, where \m{L_{f'}} is the usual cotangent complex of the underlying morphism of schemes \m{f'}.
\item If \m{f:X\ra Y} is log smooth, then the sheaf of log differentials \m{\Omega^1_{X/Y}} represents \m{L_f}.\label{logsm}
\item If \[\squarediagram{X'}{X}{Y'}{Y}{a}{b}{g}{f}\] is a commutative diagram of fine log schemes, then there is a natural map \[
a^*L_f\ra L_g
\]
which is an isomorphism if the square above is cartesian and \m f is log flat. Furthermore, if the composite \m{X'\ra Y'\ra Y} satisfies the condition (T) below, then the map
\[
g^*L_b\oplus a^*L_f\ra L_{bg}
\]
is also an isomorphism.
\item Given a composite
\[
\xymatrix{X\ar[r]^{f}&Y\ar[r]^{g}&Z}
\]
satisfying condition (T) below, there is a natural map
\[
L_f\ra f^*L_g[1]
\]
making the resulting triangle
\begin{equation}
f^*L_g\ra L_{gf}\ra L_f \ra f^*L_g[1]
\label{tri}
\end{equation}
distinguished.
\end{enumerate}

\begin{rmk}
One might hope a theory of log cotangent complex in which every triangle (\ref{tri}) is distinguished. However, such a theory is too good to exist.
\end{rmk}

On the other hand, Gabber has shown in \cite{LogCot} that if one loosen the requirement \ref{logsm}, then one can obtain a theory of log cotangent complexes for which one has a distinguished triangle (\ref{tri}) for all composites \xymatrix{X\ar[r]^{f}&Y\ar[r]^{g}&Z}.

\zz
The Condition (T) mentioned above is the following:

There exists a family of commutative diagrams
\[
\xymatrix
{
X_i \ar[r]^-{\pi_{X_i}}\ar[dr] &X\times_Y Y_i\ar[d]\ar[r]&X\times_Z Z_i\ar[d]\ar[r]&X\ar[d]_f\\
& Y_i \ar[r]^-{\pi_{Y_i}}\ar[dr] &Y\times_Z Z_i\ar[r]\ar[d]&Y\ar[d]_g\\
&&Z_i\ar[r]^-{\pi_{Z_i}}& Z
}
\]
such that
\begin{enumerate}
\item The underlying schemes of $X_i, Y_i, Z_i$ are all affine.
\item The \m{\pi}'s are all strict, and their underlying morphisms are flat and locally of finite presentation.
\item The underlying family of morphisms of schemes of \{\m{X_i\ra X}\} is jointly surjective.
\item There exists charts 
\[\beta_{X_i}:Q_{X_i}\ra \mM_{X_i},\beta_{Y_i}:Q_{Y_i}\ra \mM_{Y_i},\beta_{Z_i}:Q_{Z_i}\ra \mM_{Z_i}\]
and injective maps
\[
Q_{Z_i}\ra Q_{Y_i}\ra Q_{X_i}
\]
compatible with the morphisms \m{f_i, g_i} and
\[
\Tor^j_{\mO_{Z_i}\otimes_{\ZZ[Q_{Z_i}]}\ZZ[Q_{Y_i}]}(\mO_{Z_i}\otimes_{\ZZ[Q_{Z_i}]}\ZZ[Q_{X_i}], \mO_{Y_i}[G])=0 \text{ for all } j>0.
\]
Here \m{G:= \Coker(Q_{Z_i}^{gp}\ra Q_{Y_i}^{gp})} and \m{\mO_{Y_i}[G]} is viewed as an \m{\mO_{Z_i}\otimes_{\ZZ[Q_{Z_i}]}\ZZ[Q_{Y_i}]}-algebra via the map
\[
\mO_{Z_i}\otimes_{\ZZ[Q_{Z_i}]}\ZZ[Q_{Y_i}]\ra\mO_{Y_i}[G], t\otimes e_q\mapsto g_i^*(t)\beta_{Q_{Y_i}}(q)\cdot \bar{q}
\]
where \m{\bar{q}} denotes the image of \m q in \m G.
\end{enumerate}

\subsection{Deformation Theory of Log Schemes in General}

In this section, we explain the relation between the log cotangent complex and deformation theory of log schemes.
\zz
Let \m{f:X\ra Y} be a morphism of fine log schemes and let \m I be a quasi-coherent sheaf on \underline{\m X}. Define a \m{Y}-extension of \m{X} by \m{I} to be a commutative diagram of log schemes
\[
\xymatrix
{
X\ar[r]^{j}\ar[d]_f&X'\ar[dl]^{f'}\\
Y
}
\]
where \m{j} is an strict closed immersion (meaning that the log structure on \m{X} is isomorphic to the pull-back of that of \m{X'}'s) defined by a square-zero ideal, together with an isomorphism \m{\epsilon_j: I\cong \Ker(\mO_{X'}\ra\mO_X)}. The set of \m Y extensions by \m I forms in a natural way a category \m{\underline{\Exal}_Y(X,I)}. Let \m{\Exal_Y(X,I)} be the set of isomorphism classes of this category.

There is a tautological equivalence of categories (see \cite{defStacks} for the meaning of the right hand side):
\[
\underline{\Exal}_Y(X,I)\cong\underline{\Exal}_{LOG_Y}(\underline{X},I).
\]
Hence by (\cite{defStacks}, 1.1) and our definition of \m{L_f} we obtain the following result:

\begin{thm}
There is a natural bijection
\[
\Exal_Y(X,I)\cong\Ext^1(L_f,I).
\]
\end{thm}

It is precisely the theorem above that guarantees that general deformation theory is controlled by our logarithmic cotangent complex.

\begin{defn}
Let \m{j_0:Y_0\into Y} be an strict closed immersion of fine log schemes defined by a square-zero ideal \m{I\subset\mO_Y}, and let \m{f_0:X_0\ra Y_0} be a log flat morphism. A log flat deformation of \m{X_0} to \m Y is a cartesian square
\[
\squarediagram{X_0}{X}{Y_0}{Y}{j}{j_0}{f_0}{f}
\]
with \m{f} log flat.
\end{defn}

To give a log flat deformation as above is equivalent to give a 2-commutative diagram
\[
\squarediagram{\underline{X_0}}{\underline{X}}{LOG_{Y_0}}{LOG_Y}{j}{j_0}{\mL_{f_0}}{\mL_{f}}
\]
with \m{\mL_f} flat. Thus from (\cite{defStacks}, 1.4) we obtain the following:

\begin{thm}
Let \m J denote the ideal of \m{LOG_{Y_0}} in \m{LOG_{Y}}. Then
\begin{enumerate}
\item{There exists a canonical class \m{o\in\Ext^2(L_{f_0},\mL_{f_0}^*J)} whose vanishing is equivalent to the existence of a log flat deformation of \m{X_0} to \m Y.}
\item{If \m{o=0}, then the set of isomorphism classes of log flat deformations of \m{X_0} to \m Y is naturally a torsor under \m{\Ext^1(L_{f_0},\mL^*_{f_0}J)}.}
\item{The automorphism group of any log flat deformation of \m{X_0} to \m{Y} is canonically isomorphic to \m{\Ext^0(L_{f_0},\mL^*_{f_0}J))}.}
\end{enumerate}
\end{thm}
This theorem gives a satisfactory answer to the question of general deformation theory of log schemes.

% 2. check if D'(A) is triangulated.