defos.tex

\input{preamble}

% OK, start here.
%
\begin{document}

\title{Deformation Theory}


\maketitle

\phantomsection
\label{section-phantom}

\tableofcontents

\section{Introduction}
\label{section-introduction}

\noindent
The goal of this chapter is to give a (relatively) gentle introduction to
deformation theory of modules, morphisms, etc. In this chapter we deal with
those results that can be proven using the naive cotangent complex. In
the chapter on the cotangent complex we will extend these results a little
bit. The advanced reader may wish to consult the treatise by Illusie on this
subject, see \cite{cotangent}.





\section{Deformations of rings and the naive cotangent complex}
\label{section-deformations}

\noindent
In this section we use the naive cotangent complex to do a little bit
of deformation theory. We start with a surjective ring map $A' \to A$
whose kernel is an ideal $I$ of square zero. Moreover we assume
given a ring map $A \to B$, a $B$-module $N$, and an $A$-module map
$c : I \to N$. In this section we ask ourselves whether we can find
the question mark fitting into the following diagram
\begin{equation}
\label{equation-to-solve}
\vcenter{
\xymatrix{
0 \ar[r] & N \ar[r] & {?} \ar[r] & B \ar[r] & 0 \\
0 \ar[r] & I \ar[u]^c \ar[r] & A' \ar[u] \ar[r] & A \ar[u] \ar[r] & 0
}
}
\end{equation}
and moreover how unique the solution is (if it exists). More precisely,
we look for a surjection of $A'$-algebras $B' \to B$ whose kernel is
identified with $N$ such that $A' \to B'$ induces the given map $c$.
We will say $B'$ is a {\it solution} to (\ref{equation-to-solve}).

\begin{lemma}
\label{lemma-huge-diagram}
Given a commutative diagram
$$
\xymatrix{
& 0 \ar[r] & N_2 \ar[r] & B'_2 \ar[r] & B_2 \ar[r] & 0 \\
& 0 \ar[r]|\hole & I_2 \ar[u]_{c_2} \ar[r] &
A'_2 \ar[u] \ar[r]|\hole & A_2 \ar[u] \ar[r] & 0 \\
0 \ar[r] & N_1 \ar[ruu] \ar[r] & B'_1 \ar[r] & B_1 \ar[ruu] \ar[r] & 0 \\
0 \ar[r] & I_1 \ar[ruu]|\hole \ar[u]^{c_1} \ar[r] &
A'_1 \ar[ruu]|\hole \ar[u] \ar[r] & A_1 \ar[ruu]|\hole \ar[u] \ar[r] & 0
}
$$
with front and back solutions to (\ref{equation-to-solve}) we have
\begin{enumerate}
\item There exist a canonical element in
$\text{Ext}^1_{B_1}(\NL_{B_1/A_1}, N_2)$
whose vanishing is a necessary and sufficient condition for the existence
of a ring map $B'_1 \to B'_2$ fitting into the diagram.
\item If there exists a map $B'_1 \to B'_2$ fitting into the diagram
the set of all such maps is a principal homogeneous space under
$\Hom_{B_1}(\Omega_{B_1/A_1}, N_2)$.
\end{enumerate}
\end{lemma}

\begin{proof}
Let $E = B_1$ viewed as a set.
Consider the surjection $A_1[E] \to B_1$ with kernel $J$ used
to define the naive cotangent complex by the formula
$$
\NL_{B_1/A_1} = (J/J^2 \to \Omega_{A_1[E]/A_1} \otimes_{A_1[E]} B_1)
$$
in
Algebra, Section \ref{algebra-section-netherlander}.
Since $\Omega_{A_1[E]/A_1} \otimes B_1$ is a free
$B_1$-module we have
$$
\text{Ext}^1_{B_1}(\NL_{B_1/A_1}, N_2) =
\frac{\Hom_{B_1}(J/J^2, N_2)}
{\Hom_{B_1}(\Omega_{A_1[E]/A_1} \otimes B_1, N_2)}
$$
We will construct an obstruction in the module on the right.
Let $J' = \text{Ker}(A'_1[E] \to B_1)$. Note that there is a surjection
$J' \to J$ whose kernel is $I_1A_1[E]$.
For every $e \in E$ denote $x_e \in A_1[E]$ the corresponding variable.
Choose a lift $y_e \in B'_1$ of the image of $x_e$ in $B_1$ and
a lift $z_e \in B'_2$ of the image of $x_e$ in $B_2$.
These choices determine $A'_1$-algebra maps
$$
A'_1[E] \to B'_1 \quad\text{and}\quad A'_1[E] \to B'_2
$$
The first of these gives a map $J' \to N_1$, $f' \mapsto f'(y_e)$
and the second gives a map $J' \to N_2$, $f' \mapsto f'(z_e)$.
A calculation shows that these maps annihilate $(J')^2$.
Because the left square of the diagram (involving $c_1$ and $c_2$)
commutes we see that these maps agree on $I_1A_1[E]$ as maps into $N_2$.
Observe that $B'_1$ is the pushout of $J' \to A'_1[B_1]$ and $J' \to N_1$. 
Thus, if the maps $J' \to N_1 \to N_2$ and $J' \to N_2$ agree, then we
obtain a map $B'_1 \to B'_2$ fitting into the diagram.
Thus we let the obstruction be the class of the map
$$
J/J^2 \to N_2,\quad f \mapsto f'(z_e) - \nu(f'(y_e))
$$
where $\nu : N_1 \to N_2$ is the given map and where $f' \in J'$
is a lift of $f$. This is well defined by our remarks above.
Note that we have the freedom
to modify our choices of $z_e$ into $z_e + \delta_{2, e}$
and $y_e$ into $y_e + \delta_{1, e}$ for some $\delta_{i, e} \in N_i$.
This will modify the map above into
$$
f \mapsto f'(z_e + \delta_{2, e}) - \nu(f'(y_e + \delta_{1, e})) =
f'(z_e) - \nu(f'(z_e)) +
\sum (\delta_{2, e} - \nu(\delta_{1, e}))\frac{\partial f}{\partial x_e}
$$
This means exactly that we are modifying the map $J/J^2 \to N_2$
by the composition $J/J^2 \to \Omega_{A_1[E]/A_1} \otimes B_1 \to N_2$
where the second map sends $\text{d}x_e$ to
$\delta_{2, e} - \nu(\delta_{1, e})$. Thus our obstruction is well defined
and is zero if and only if a lift exists.

\medskip\noindent
Part (2) comes from the observation that given two maps
$\varphi, \psi : B'_1 \to B'_2$ fitting into the diagram, then
$\varphi - \psi$ factors through a map $D : B_1 \to N_2$ which
is an $A_1$-derivation:
\begin{align*}
D(fg) & = \varphi(f'g') - \psi(f'g') \\
& =
\varphi(f')\varphi(g') - \psi(f')\psi(g') \\
& =
(\varphi(f') - \psi(f'))\varphi(g') + \psi(f')(\varphi(g') - \psi(g')) \\
& =
gD(f) + fD(g)
\end{align*}
Thus $D$ corresponds to a unique $B_1$-linear map
$\Omega_{B_1/A_1} \to N_2$. Conversely, given such a linear map
we get a derivation $D$ and given a ring map $\psi : B'_1 \to B'_2$
fitting into the diagram
the map $\psi + D$ is another ring map fitting into the diagram.
\end{proof}

\noindent
The naive cotangent complex isn't good enough to contain all information
regarding obstructions to finding solutions to (\ref{equation-to-solve}).
However, if the ring map is a local complete intersection, then the
obstruction vanishes. This is a kind of lifting result; observe that
for syntomic ring maps we have proved a rather strong lifting result in
Smoothing Ring Maps, Proposition \ref{smoothing-proposition-lift-smooth}.

\begin{lemma}
\label{lemma-existence-lci}
If $A \to B$ is a local complete intersection ring map, then
there exists a solution to (\ref{equation-to-solve}).
\end{lemma}

\begin{proof}
Write $B = A[x_1, \ldots, x_n]/J$. Let $J' \subset A'[x_1, \ldots, x_n]$
be the inverse image of $J$. Denote $I[x_1, \ldots, x_n]$ the
kernel of $A'[x_1, \ldots, x_n] \to A[x_1, \ldots, x_n]$.
By More on Algebra, Lemma
\ref{more-algebra-lemma-conormal-sequence-H1-regular-ideal} we have
$I[x_1, \ldots, x_n] \cap (J')^2 = J'I[x_1, \ldots, x_n] =
JI[x_1, \ldots, x_n]$. Hence we obtain a short exact sequence
$$
0 \to I \otimes_A B \to J'/(J')^2 \to J/J^2 \to 0
$$
Since $J/J^2$ is projective (More on Algebra, Lemma
\ref{more-algebra-lemma-quasi-regular-ideal-finite-projective})
we can choose a splitting of this sequence
$$
J'/(J')^2 = I \otimes_A B \oplus J/J^2 
$$
Let $(J')^2 \subset J'' \subset J'$ be the elements which map to the
second summand in the decomposition above. Then
$$
0 \to I \otimes_A B \to A'[x_1, \ldots, x_n]/J'' \to B \to 0
$$
is a solution to (\ref{equation-to-solve}) with $N = I \otimes_A B$.
The general case is obtained by doing a pushout along the given
map $I \otimes_A B \to N$.
\end{proof}

\begin{lemma}
\label{lemma-choices}
If there exists a solution to (\ref{equation-to-solve}), then the set of
isomorphism classes of solutions is principal homogeneous under
$\text{Ext}^1_B(\NL_{B/A}, N)$.
\end{lemma}

\begin{proof}
We observe right away that given two solutions $B'_1$ and $B'_2$
to (\ref{equation-to-solve}) we obtain by Lemma \ref{lemma-huge-diagram} an
obstruction element $o(B'_1, B'_2) \in \text{Ext}^1_B(\NL_{B/A}, N)$
to the existence of a map $B'_1 \to B'_2$. Clearly, this element
is the obstruction to the existence of an isomorphism, hence separates
the isomorphism classes. To finish the proof it therefore suffices to
show that given a solution $B'$ and an element
$\xi \in \text{Ext}^1_B(\NL_{B/A}, N)$
we can find a second solution $B'_\xi$ such that
$o(B', B'_\xi) = \xi$.

\medskip\noindent
Let $E = B$ viewed as a set. Consider the surjection $A[E] \to B$ with kernel
$J$ used to define the naive cotangent complex by the formula
$$
\NL_{B/A} = (J/J^2 \to \Omega_{A[E]/A} \otimes_{A[E]} B)
$$
in Algebra, Section \ref{algebra-section-netherlander}.
Since $\Omega_{A[E]/A} \otimes B$ is a free $B$-module we have
$$
\text{Ext}^1_B(\NL_{B/A}, N) =
\frac{\Hom_B(J/J^2, N)}
{\Hom_B(\Omega_{A[E]/A} \otimes B, N)}
$$
Thus we may represent $\xi$ as the class of a morphism $\delta : J/J^2 \to N$.

\medskip\noindent
For every $e \in E$ denote $x_e \in A[E]$ the corresponding variable.
Choose a lift $y_e \in B'$ of the image of $x_e$ in $B$.
These choices determine an $A'$-algebra map $\varphi : A'[E] \to B'$.
Let $J' = \text{Ker}(A'[E] \to B)$. Observe that $\varphi$ induces a map
$\varphi|_{J'} : J' \to N$ and that $B'$ is the pushout, as in the following
diagram
$$
\xymatrix{
0 \ar[r] & N \ar[r] & B' \ar[r] & B \ar[r] & 0 \\
0 \ar[r] & J' \ar[u]^{\varphi|_{J'}} \ar[r] & A'[E] \ar[u] \ar[r] &
B \ar[u]_{=} \ar[r] & 0
}
$$
Let $\psi : J' \to N$ be the sum of the map $\varphi|_{J'}$ and the
composition
$$
J' \to J'/(J')^2 \to J/J^2 \xrightarrow{\delta} N.
$$
Then the pushout along $\psi$ is an other ring extension $B'_\xi$
fitting into a diagram as above. A calculation shows that
$o(B', B'_\xi) = \xi$ as desired.
\end{proof}

\begin{lemma}
\label{lemma-extensions-of-rings}
Let $A$ be a ring and let $I$ be an $A$-module.
\begin{enumerate}
\item The set of extensions of rings $0 \to I \to A' \to A \to 0$
where $I$ is an ideal of square zero is canonically bijective to
$\text{Ext}^1_A(\NL_{A/\mathbf{Z}}, I)$.
\item Given a ring map $A \to B$, a $B$-module $N$, an $A$-module
map $c : I \to N$, and given extensions of rings with square zero kernels:
\begin{enumerate}
\item[(a)] $0 \to I \to A' \to A \to 0$ corresponding to
$\alpha \in \text{Ext}^1_A(\NL_{A/\mathbf{Z}}, I)$, and
\item[(b)] $0 \to N \to B' \to B \to 0$ corresponding to
$\beta \in \text{Ext}^1_B(\NL_{B/\mathbf{Z}}, N)$
\end{enumerate}
then there is a map $A' \to B'$ fitting into a diagram
(\ref{equation-to-solve}) if and only if $\beta$ and $\alpha$
map to the same element of
$\text{Ext}^1_A(\NL_{A/\mathbf{Z}}, N)$.
\end{enumerate}
\end{lemma}

\begin{proof}
To prove this we apply the previous results where we work over
$0 \to 0 \to \mathbf{Z} \to \mathbf{Z} \to 0$, in order words,
we work over the extension of $\mathbf{Z}$ by $0$.
Part (1) follows from Lemma \ref{lemma-choices}
and the fact that there exists a solution, namely $I \oplus A$.
Part (2) follows from Lemma \ref{lemma-huge-diagram}
and a compatibility between the constructions in the proofs
of Lemmas \ref{lemma-choices} and \ref{lemma-huge-diagram}
whose statement and proof we omit.
\end{proof}








\section{Thickenings of ringed spaces}
\label{section-thickenings-spaces}

\noindent
In the following few sections we will use the following notions:
\begin{enumerate}
\item A sheaf of ideals $\mathcal{I} \subset \mathcal{O}_{X'}$ on
a ringed space $(X', \mathcal{O}_{X'})$ is {\it locally nilpotent}
if any local section of $\mathcal{I}$ is locally nilpotent.
Compare with Algebra, Item \ref{algebra-item-ideal-locally-nilpotent}.
\item A {\it thickening} of ringed spaces is a morphism
$i : (X, \mathcal{O}_X) \to (X', \mathcal{O}_{X'})$ of ringed spaces
such that
\begin{enumerate}
\item $i$ induces a homeomorphism $X \to X'$,
\item the map $i^\sharp : \mathcal{O}_{X'} \to i_*\mathcal{O}_X$
is surjective, and
\item the kernel of $i^\sharp$ is a locally nilpotent sheaf of ideals.
\end{enumerate}
\item A {\it first order thickening} of ringed spaces is a thickening
$i : (X, \mathcal{O}_X) \to (X', \mathcal{O}_{X'})$ of ringed spaces
such that $\text{Ker}(i^\sharp)$ has square zero.
\item It is clear how to define {\it morphisms of thickenings},
{\it morphisms of thickenings over a base ringed space}, etc.
\end{enumerate}
If $i : (X, \mathcal{O}_X) \to (X', \mathcal{O}_{X'})$ is a thickening
of ringed spaces then we identify the underlying topological spaces
and think of $\mathcal{O}_X$, $\mathcal{O}_{X'}$, and
$\mathcal{I} = \text{Ker}(i^\sharp)$ as sheaves on $X = X'$. We obtain
a short exact sequence
$$
0 \to \mathcal{I} \to \mathcal{O}_{X'} \to \mathcal{O}_X \to 0
$$
of $\mathcal{O}_{X'}$-modules. By
Modules, Lemma \ref{modules-lemma-i-star-equivalence}
the category of $\mathcal{O}_X$-modules is equivalent to the category
of $\mathcal{O}_{X'}$-modules annihilated by $\mathcal{I}$. In particular,
if $i$ is a first order thickening, then
$\mathcal{I}$ is a $\mathcal{O}_X$-module.

\begin{situation}
\label{situation-morphism-thickenings}
A morphism of thickenings $(f, f')$ is given by a commutative diagram
\begin{equation}
\label{equation-morphism-thickenings}
\vcenter{
\xymatrix{
(X, \mathcal{O}_X) \ar[r]_i \ar[d]_f & (X', \mathcal{O}_{X'}) \ar[d]^{f'} \\
(S, \mathcal{O}_S) \ar[r]^t & (S', \mathcal{O}_{S'})
}
}
\end{equation}
of ringed spaces whose horizontal arrows are thickenings. In this
situation we set
$\mathcal{I} = \text{Ker}(i^\sharp) \subset \mathcal{O}_{X'}$ and
$\mathcal{J} = \text{Ker}(t^\sharp) \subset \mathcal{O}_{S'}$.
As $f = f'$ on underlying topological spaces we will identify
the (topological) pullback functors $f^{-1}$ and $(f')^{-1}$.
Observe that $(f')^\sharp : f^{-1}\mathcal{O}_{S'} \to \mathcal{O}_{X'}$
induces in particular a map $f^{-1}\mathcal{J} \to \mathcal{I}$
and therefore a map of $\mathcal{O}_{X'}$-modules
$$
(f')^*\mathcal{J} \longrightarrow \mathcal{I}
$$
If $i$ and $t$ are first order thickenings, then
$(f')^*\mathcal{J} = f^*\mathcal{J}$ and the map above becomes a
map $f^*\mathcal{J} \to \mathcal{I}$.
\end{situation}

\begin{definition}
\label{definition-strict-morphism-thickenings}
In Situation \ref{situation-morphism-thickenings} we say that $(f, f')$ is a
{\it strict morphism of thickenings}
if the map $(f')^*\mathcal{J} \longrightarrow \mathcal{I}$ is surjective.
\end{definition}

\noindent
The following lemma in particular shows that a morphism
$(f, f') : (X \subset X') \to (S \subset S')$ of
thickenings of schemes is strict if and only if $X = S \times_{S'} X'$.

\begin{lemma}
\label{lemma-strict-morphism-thickenings}
In Situation \ref{situation-morphism-thickenings} the morphism $(f, f')$
is a strict morphism of thickenings if and only if
(\ref{equation-morphism-thickenings}) is cartesian in the category
of ringed spaces.
\end{lemma}

\begin{proof}
Omitted.
\end{proof}




\section{Modules on first order thickenings of ringed spaces}
\label{section-modules-thickenings}

\noindent
In this section we discuss some preliminaries to the deformation theory
of modules. Let $i : (X, \mathcal{O}_X) \to (X', \mathcal{O}_{X'})$
be a first order thickening of ringed spaces. We will freely use the notation
introduced in Section \ref{section-thickenings-spaces}, in particular we will
identify the underlying topological spaces.
In this section we consider short exact sequences
\begin{equation}
\label{equation-extension}
0 \to \mathcal{K} \to \mathcal{F}' \to \mathcal{F} \to 0
\end{equation}
of $\mathcal{O}_{X'}$-modules, where $\mathcal{F}$, $\mathcal{K}$ are
$\mathcal{O}_X$-modules and $\mathcal{F}'$ is an $\mathcal{O}_{X'}$-module.
In this situation we have a canonical $\mathcal{O}_X$-module map
$$
c_{\mathcal{F}'} :
\mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{F}
\longrightarrow
\mathcal{K}
$$
where $\mathcal{I} = \text{Ker}(i^\sharp)$.
Namely, given local sections $f$ of $\mathcal{I}$ and $s$
of $\mathcal{F}$ we set $c_{\mathcal{F}'}(f \otimes s) = fs'$
where $s'$ is a local section of $\mathcal{F}'$ lifting $s$.

\begin{lemma}
\label{lemma-inf-map}
Let $i : (X, \mathcal{O}_X) \to (X', \mathcal{O}_{X'})$
be a first order thickening of ringed spaces. Assume given
extensions
$$
0 \to \mathcal{K} \to \mathcal{F}' \to \mathcal{F} \to 0
\quad\text{and}\quad
0 \to \mathcal{L} \to \mathcal{G}' \to \mathcal{G} \to 0
$$
as in (\ref{equation-extension})
and maps $\varphi : \mathcal{F} \to \mathcal{G}$ and
$\psi : \mathcal{K} \to \mathcal{L}$.
\begin{enumerate}
\item If there exists an $\mathcal{O}_{X'}$-module
map $\varphi' : \mathcal{F}' \to \mathcal{G}'$ compatible with $\varphi$
and $\psi$, then the diagram
$$
\xymatrix{
\mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{F}
\ar[r]_-{c_{\mathcal{F}'}} \ar[d]_{1 \otimes \varphi} &
\mathcal{K} \ar[d]^\psi \\
\mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{G}
\ar[r]^-{c_{\mathcal{G}'}} &
\mathcal{L}
}
$$
is commutative.
\item The set of $\mathcal{O}_{X'}$-module
maps $\varphi' : \mathcal{F}' \to \mathcal{G}'$ compatible with $\varphi$
and $\psi$ is, if nonempty, a principal homogeneous space under
$\Hom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{L})$.
\end{enumerate}
\end{lemma}

\begin{proof}
Part (1) is immediate from the description of the maps.
For (2), if $\varphi'$ and $\varphi''$ are two maps
$\mathcal{F}' \to \mathcal{G}'$ compatible with $\varphi$
and $\psi$, then $\varphi' - \varphi''$ factors as
$$
\mathcal{F}' \to \mathcal{F} \to \mathcal{L} \to \mathcal{G}'
$$
The map in the middle comes from a unique element of
$\Hom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{L})$ by
Modules, Lemma \ref{modules-lemma-i-star-equivalence}.
Conversely, given an element $\alpha$ of this group we can add the
composition (as displayed above with $\alpha$ in the middle)
to $\varphi'$. Some details omitted.
\end{proof}

\begin{lemma}
\label{lemma-inf-obs-map}
Let $i : (X, \mathcal{O}_X) \to (X', \mathcal{O}_{X'})$
be a first order thickening of ringed spaces. Assume given
extensions
$$
0 \to \mathcal{K} \to \mathcal{F}' \to \mathcal{F} \to 0
\quad\text{and}\quad
0 \to \mathcal{L} \to \mathcal{G}' \to \mathcal{G} \to 0
$$
as in (\ref{equation-extension})
and maps $\varphi : \mathcal{F} \to \mathcal{G}$ and
$\psi : \mathcal{K} \to \mathcal{L}$. Assume the diagram
$$
\xymatrix{
\mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{F}
\ar[r]_-{c_{\mathcal{F}'}} \ar[d]_{1 \otimes \varphi} &
\mathcal{K} \ar[d]^\psi \\
\mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{G}
\ar[r]^-{c_{\mathcal{G}'}} &
\mathcal{L}
}
$$
is commutative. Then there exists an element
$$
o(\varphi, \psi) \in
\text{Ext}^1_{\mathcal{O}_X}(\mathcal{F}, \mathcal{L})
$$
whose vanishing is a necessary and sufficient condition for the existence
of a map $\varphi' : \mathcal{F}' \to \mathcal{G}'$ compatible with
$\varphi$ and $\psi$.
\end{lemma}

\begin{proof}
We can construct explicitly an extension
$$
0 \to \mathcal{L} \to \mathcal{H} \to \mathcal{F} \to 0
$$
by taking $\mathcal{H}$ to be the cohomology of the complex
$$
\mathcal{K}
\xrightarrow{1, - \psi}
\mathcal{F}' \oplus \mathcal{G}' \xrightarrow{\varphi, 1}
\mathcal{G}
$$
in the middle (with obvious notation). A calculation with local sections
using the assumption that the diagram of the lemma commutes
shows that $\mathcal{H}$ is annihilated by $\mathcal{I}$. Hence
$\mathcal{H}$ defines a class in
$$
\text{Ext}^1_{\mathcal{O}_X}(\mathcal{F}, \mathcal{L})
\subset
\text{Ext}^1_{\mathcal{O}_{X'}}(\mathcal{F}, \mathcal{L})
$$
Finally, the class of $\mathcal{H}$ is the difference of the pushout
of the extension $\mathcal{F}'$ via $\psi$ and the pullback
of the extension $\mathcal{G}'$ via $\varphi$ (calculations omitted).
Thus the vanishing of the class of $\mathcal{H}$ is equivalent to the
existence of a commutative diagram
$$
\xymatrix{
0 \ar[r] &
\mathcal{K} \ar[r] \ar[d]_{\psi} &
\mathcal{F}' \ar[r] \ar[d]_{\varphi'} &
\mathcal{F} \ar[r] \ar[d]_\varphi & 0\\
0 \ar[r] &
\mathcal{L} \ar[r] &
\mathcal{G}' \ar[r] &
\mathcal{G} \ar[r] & 0
}
$$
as desired.
\end{proof}

\begin{lemma}
\label{lemma-inf-ext}
Let $i : (X, \mathcal{O}_X) \to (X', \mathcal{O}_{X'})$ be a first order
thickening of ringed spaces.
Assume given $\mathcal{O}_X$-modules $\mathcal{F}$, $\mathcal{K}$
and an $\mathcal{O}_X$-linear map
$c : \mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{F} \to \mathcal{K}$.
If there exists a sequence (\ref{equation-extension}) with
$c_{\mathcal{F}'} = c$ then the set of isomorphism classes of these
extensions is principal homogeneous under
$\text{Ext}^1_{\mathcal{O}_X}(\mathcal{F}, \mathcal{K})$.
\end{lemma}

\begin{proof}
Assume given extensions
$$
0 \to \mathcal{K} \to \mathcal{F}'_1 \to \mathcal{F} \to 0
\quad\text{and}\quad
0 \to \mathcal{K} \to \mathcal{F}'_2 \to \mathcal{F} \to 0
$$
with $c_{\mathcal{F}'_1} = c_{\mathcal{F}'_2} = c$. Then the difference
(in the extension group, see
Homology, Section \ref{homology-section-extensions})
is an extension
$$
0 \to \mathcal{K} \to \mathcal{E} \to \mathcal{F} \to 0
$$
where $\mathcal{E}$ is annihilated by $\mathcal{I}$ (local computation
omitted). Hence the sequence is an extension of $\mathcal{O}_X$-modules,
see Modules, Lemma \ref{modules-lemma-i-star-equivalence}.
Conversely, given such an extension $\mathcal{E}$ we can add the extension
$\mathcal{E}$ to the $\mathcal{O}_{X'}$-extension $\mathcal{F}'$ without
affecting the map $c_{\mathcal{F}'}$. Some details omitted.
\end{proof}

\begin{lemma}
\label{lemma-inf-obs-ext}
Let $i : (X, \mathcal{O}_X) \to (X', \mathcal{O}_{X'})$
be a first order thickening of ringed spaces. Assume given
$\mathcal{O}_X$-modules $\mathcal{F}$, $\mathcal{K}$
and an $\mathcal{O}_X$-linear map
$c : \mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{F} \to \mathcal{K}$.
Then there exists an element
$$
o(\mathcal{F}, \mathcal{K}, c) \in
\text{Ext}^2_{\mathcal{O}_X}(\mathcal{F}, \mathcal{K})
$$
whose vanishing is a necessary and sufficient condition for the existence
of a sequence (\ref{equation-extension}) with $c_{\mathcal{F}'} = c$.
\end{lemma}

\begin{proof}
We first show that if $\mathcal{K}$ is an injective $\mathcal{O}_X$-module,
then there does exist a sequence (\ref{equation-extension}) with
$c_{\mathcal{F}'} = c$. To do this, choose a flat
$\mathcal{O}_{X'}$-module $\mathcal{H}'$ and a surjection
$\mathcal{H}' \to \mathcal{F}$
(Modules, Lemma \ref{modules-lemma-module-quotient-flat}).
Let $\mathcal{J} \subset \mathcal{H}'$ be the kernel. Since $\mathcal{H}'$
is flat we have
$$
\mathcal{I} \otimes_{\mathcal{O}_{X'}} \mathcal{H}' =
\mathcal{I}\mathcal{H}'
\subset \mathcal{J} \subset \mathcal{H}'
$$
Observe that the map
$$
\mathcal{I}\mathcal{H}' =
\mathcal{I} \otimes_{\mathcal{O}_{X'}} \mathcal{H}'
\longrightarrow
\mathcal{I} \otimes_{\mathcal{O}_{X'}} \mathcal{F} =
\mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{F}
$$
annihilates $\mathcal{I}\mathcal{J}$. Namely, if $f$ is a local section
of $\mathcal{I}$ and $s$ is a local section of $\mathcal{H}$, then
$fs$ is mapped to $f \otimes \overline{s}$ where $\overline{s}$ is
the image of $s$ in $\mathcal{F}$. Thus we obtain
$$
\xymatrix{
\mathcal{I}\mathcal{H}'/\mathcal{I}\mathcal{J}
\ar@{^{(}->}[r] \ar[d] &
\mathcal{J}/\mathcal{I}\mathcal{J} \ar@{..>}[d]_\gamma \\
\mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{F} \ar[r]^-c &
\mathcal{K}
}
$$
a diagram of $\mathcal{O}_X$-modules. If $\mathcal{K}$ is injective
as an $\mathcal{O}_X$-module, then we obtain the dotted arrow.
Denote $\gamma' : \mathcal{J} \to \mathcal{K}$ the composition
of $\gamma$ with $\mathcal{J} \to \mathcal{J}/\mathcal{I}\mathcal{J}$.
A local calculation shows the pushout
$$
\xymatrix{
0 \ar[r] &
\mathcal{J} \ar[r] \ar[d]_{\gamma'} &
\mathcal{H}' \ar[r] \ar[d] &
\mathcal{F} \ar[r] \ar@{=}[d] &
0 \\
0 \ar[r] &
\mathcal{K} \ar[r] &
\mathcal{F}' \ar[r] &
\mathcal{F} \ar[r] &
0
}
$$
is a solution to the problem posed by the lemma.

\medskip\noindent
General case. Choose an embedding $\mathcal{K} \subset \mathcal{K}'$
with $\mathcal{K}'$ an injective $\mathcal{O}_X$-module. Let $\mathcal{Q}$
be the quotient, so that we have an exact sequence
$$
0 \to \mathcal{K} \to \mathcal{K}' \to \mathcal{Q} \to 0
$$
Denote
$c' : \mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{F} \to \mathcal{K}'$
be the composition. By the paragraph above there exists a sequence
$$
0 \to \mathcal{K}' \to \mathcal{E}' \to \mathcal{F} \to 0
$$
as in (\ref{equation-extension}) with $c_{\mathcal{E}'} = c'$.
Note that $c'$ composed with the map $\mathcal{K}' \to \mathcal{Q}$
is zero, hence the pushout of $\mathcal{E}'$ by
$\mathcal{K}' \to \mathcal{Q}$ is an extension
$$
0 \to \mathcal{Q} \to \mathcal{D}' \to \mathcal{F} \to 0
$$
as in (\ref{equation-extension}) with $c_{\mathcal{D}'} = 0$.
This means exactly that $\mathcal{D}'$ is annihilated by
$\mathcal{I}$, in other words, the $\mathcal{D}'$ is an extension
of $\mathcal{O}_X$-modules, i.e., defines an element
$$
o(\mathcal{F}, \mathcal{K}, c) \in
\text{Ext}^1_{\mathcal{O}_X}(\mathcal{F}, \mathcal{Q}) =
\text{Ext}^2_{\mathcal{O}_X}(\mathcal{F}, \mathcal{K})
$$
(the equality holds by the long exact cohomology sequence associated
to the exact sequence above and the vanishing of higher ext groups
into the injective module $\mathcal{K}'$). If
$o(\mathcal{F}, \mathcal{K}, c) = 0$, then we can choose a splitting
$s : \mathcal{F} \to \mathcal{D}'$ and we can set
$$
\mathcal{F}' = \text{Ker}(\mathcal{E}' \to \mathcal{D}'/s(\mathcal{F}))
$$
so that we obtain the following diagram
$$
\xymatrix{
0 \ar[r] &
\mathcal{K} \ar[r] \ar[d] &
\mathcal{F}' \ar[r] \ar[d] &
\mathcal{F} \ar[r] \ar@{=}[d] &
0 \\
0 \ar[r] &
\mathcal{K}' \ar[r] &
\mathcal{E}' \ar[r] &
\mathcal{F} \ar[r] & 0
}
$$
with exact rows which shows that $c_{\mathcal{F}'} = c$. Conversely, if
$\mathcal{F}'$ exists, then the pushout of $\mathcal{F}'$ by the map
$\mathcal{K} \to \mathcal{K}'$ is isomorphic to $\mathcal{E}'$ by
Lemma \ref{lemma-inf-ext} and the vanishing of higher ext groups
into the injective module $\mathcal{K}'$. This gives a diagram
as above, which implies that $\mathcal{D}'$ is split as an extension, i.e.,
the class $o(\mathcal{F}, \mathcal{K}, c)$ is zero.
\end{proof}

\begin{remark}
\label{remark-trivial-thickening}
Let $(X, \mathcal{O}_X)$ be a ringed space. A first order thickening
$i : (X, \mathcal{O}_X) \to (X', \mathcal{O}_{X'})$ is said
to be {\it trivial} if there exists a morphism of ringed spaces
$\pi : (X', \mathcal{O}_{X'}) \to (X, \mathcal{O}_X)$ which is a
left inverse to $i$. The choice of such a morphism
$\pi$ is called a {\it trivialization} of the first order thickening.
Given $\pi$ we obtain a splitting
\begin{equation}
\label{equation-splitting}
\mathcal{O}_{X'} = \mathcal{O}_X \oplus \mathcal{I}
\end{equation}
as sheaves of algebras on $X$ by using $\pi^\sharp$ to split the surjection
$\mathcal{O}_{X'} \to \mathcal{O}_X$. Conversely, such a splitting determines
a morphism $\pi$. The category of trivialized first order thickenings of
$(X, \mathcal{O}_X)$ is equivalent to the category of 
$\mathcal{O}_X$-modules.
\end{remark}

\begin{remark}
\label{remark-trivial-extension}
Let $i : (X, \mathcal{O}_X) \to (X', \mathcal{O}_{X'})$
be a trivial first order thickening of ringed spaces
and let $\pi : (X', \mathcal{O}_{X'}) \to (X, \mathcal{O}_X)$
be a trivialization. Then given any triple
$(\mathcal{F}, \mathcal{K}, c)$ consisting of a pair of
$\mathcal{O}_X$-modules and a map
$c : \mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{F} \to \mathcal{K}$
we may set
$$
\mathcal{F}'_{c, triv} = \mathcal{F} \oplus \mathcal{K}
$$
and use the splitting (\ref{equation-splitting}) associated to $\pi$
and the map $c$ to define the $\mathcal{O}_{X'}$-module structure
and obtain an extension (\ref{equation-extension}). We will call
$\mathcal{F}'_{c, triv}$ the {\it trivial extension} of $\mathcal{F}$
by $\mathcal{K}$ corresponding
to $c$ and the trivialization $\pi$. Given any extension
$\mathcal{F}'$ as in (\ref{equation-extension}) we can use
$\pi^\sharp : \mathcal{O}_X \to \mathcal{O}_{X'}$ to think of $\mathcal{F}'$
as an $\mathcal{O}_X$-module extension, hence a class $\xi_{\mathcal{F}'}$
in $\text{Ext}^1_{\mathcal{O}_X}(\mathcal{F}, \mathcal{K})$.
Lemma \ref{lemma-inf-ext} assures that
$\mathcal{F}' \mapsto \xi_{\mathcal{F}'}$
induces a bijection
$$
\left\{
\begin{matrix}
\text{isomorphism classes of extensions}\\
\mathcal{F}'\text{ as in (\ref{equation-extension}) with }c = c_{\mathcal{F}'}
\end{matrix}
\right\}
\longrightarrow
\text{Ext}^1_{\mathcal{O}_X}(\mathcal{F}, \mathcal{K})
$$
Moreover, the trivial extension $\mathcal{F}'_{c, triv}$ maps to the zero class.
\end{remark}

\begin{remark}
\label{remark-extension-functorial}
Let $(X, \mathcal{O}_X)$ be a ringed space. Let
$(X, \mathcal{O}_X) \to (X'_i, \mathcal{O}_{X'_i})$, $i = 1, 2$
be first order thickenings with ideal sheaves $\mathcal{I}_i$.
Let $h : (X'_1, \mathcal{O}_{X'_1}) \to (X'_2, \mathcal{O}_{X'_2})$
be a morphism of first order thickenings of $(X, \mathcal{O}_X)$.
Picture
$$
\xymatrix{
& (X, \mathcal{O}_X) \ar[ld] \ar[rd] & \\
(X'_1, \mathcal{O}_{X'_1}) \ar[rr]^h & & 
(X'_2, \mathcal{O}_{X'_2})
}
$$
Observe that $h^\sharp : \mathcal{O}_{X'_2} \to \mathcal{O}_{X'_1}$
in particular induces an $\mathcal{O}_X$-module map
$\mathcal{I}_2 \to \mathcal{I}_1$.
Let $\mathcal{F}$ be an
$\mathcal{O}_X$-module. Let $(\mathcal{K}_i, c_i)$, $i = 1, 2$ be a pair
consisting of an $\mathcal{O}_X$-module $\mathcal{K}_i$ and a map
$c_i : \mathcal{I}_i \otimes_{\mathcal{O}_X} \mathcal{F} \to
\mathcal{K}_i$. Assume furthermore given a map
of $\mathcal{O}_X$-modules $\mathcal{K}_2 \to \mathcal{K}_1$
such that
$$
\xymatrix{
\mathcal{I}_2 \otimes_{\mathcal{O}_X} \mathcal{F}
\ar[r]_-{c_2} \ar[d] &
\mathcal{K}_2 \ar[d] \\
\mathcal{I}_1 \otimes_{\mathcal{O}_X} \mathcal{F}
\ar[r]^-{c_1} &
\mathcal{K}_1
}
$$
is commutative. Then there is a canonical functoriality
$$
\left\{
\begin{matrix}
\mathcal{F}'_2\text{ as in (\ref{equation-extension}) with }\\
c_2 = c_{\mathcal{F}'_2}\text{ and }\mathcal{K} = \mathcal{K}_2
\end{matrix}
\right\}
\longrightarrow
\left\{
\begin{matrix}
\mathcal{F}'_1\text{ as in (\ref{equation-extension}) with }\\
c_1 = c_{\mathcal{F}'_1}\text{ and }\mathcal{K} = \mathcal{K}_1
\end{matrix}
\right\}
$$
Namely, thinking of all sheaves $\mathcal{O}_X$, $\mathcal{O}_{X'_i}$,
$\mathcal{F}$, $\mathcal{K}_i$, etc as sheaves on $X$, we set
given $\mathcal{F}'_2$ the sheaf $\mathcal{F}'_1$ equal to the
pushout, i.e., fitting into the following diagram of extensions
$$
\xymatrix{
0 \ar[r] &
\mathcal{K}_2 \ar[r] \ar[d] &
\mathcal{F}'_2 \ar[r] \ar[d] &
\mathcal{F} \ar@{=}[d] \ar[r] & 0 \\
0 \ar[r] &
\mathcal{K}_1 \ar[r] &
\mathcal{F}'_1 \ar[r] &
\mathcal{F} \ar[r] & 0
}
$$
We omit the construction of the $\mathcal{O}_{X'_1}$-module structure
on the pushout (this uses the commutativity of the diagram
involving $c_1$ and $c_2$).
\end{remark}

\begin{remark}
\label{remark-trivial-extension-functorial}
Let $(X, \mathcal{O}_X)$, $(X, \mathcal{O}_X) \to (X'_i, \mathcal{O}_{X'_i})$,
$\mathcal{I}_i$, and
$h : (X'_1, \mathcal{O}_{X'_1}) \to (X'_2, \mathcal{O}_{X'_2})$
be as in Remark \ref{remark-extension-functorial}. Assume that we are
given given trivializations $\pi_i : X'_i \to X$ such that
$\pi_1 = h \circ \pi_2$. In other words, assume $h$ is a morphism
of trivialized first order thickening of $(X, \mathcal{O}_X)$. Let
$(\mathcal{K}_i, c_i)$, $i = 1, 2$ be a pair consisting of an
$\mathcal{O}_X$-module $\mathcal{K}_i$ and a map
$c_i : \mathcal{I}_i \otimes_{\mathcal{O}_X} \mathcal{F} \to
\mathcal{K}_i$. Assume furthermore given a map
of $\mathcal{O}_X$-modules $\mathcal{K}_2 \to \mathcal{K}_1$
such that
$$
\xymatrix{
\mathcal{I}_2 \otimes_{\mathcal{O}_X} \mathcal{F}
\ar[r]_-{c_2} \ar[d] &
\mathcal{K}_2 \ar[d] \\
\mathcal{I}_1 \otimes_{\mathcal{O}_X} \mathcal{F}
\ar[r]^-{c_1} &
\mathcal{K}_1
}
$$
is commutative. In this situation the construction of
Remark \ref{remark-trivial-extension} induces
a commutative diagram
$$
\xymatrix{
\{\mathcal{F}'_2\text{ as in (\ref{equation-extension}) with }
c_2 = c_{\mathcal{F}'_2}\text{ and }\mathcal{K} = \mathcal{K}_2\}
\ar[d] \ar[rr] & &
\text{Ext}^1_{\mathcal{O}_X}(\mathcal{F}, \mathcal{K}_2) \ar[d] \\
\{\mathcal{F}'_1\text{ as in (\ref{equation-extension}) with }
c_1 = c_{\mathcal{F}'_1}\text{ and }\mathcal{K} = \mathcal{K}_1\}
\ar[rr] & &
\text{Ext}^1_{\mathcal{O}_X}(\mathcal{F}, \mathcal{K}_1)
}
$$
where the vertical map on the right is given by functoriality of $\text{Ext}$
and the map $\mathcal{K}_2 \to \mathcal{K}_1$ and the vertical map on the left
is the one from Remark \ref{remark-extension-functorial}.
\end{remark}

\begin{remark}
\label{remark-short-exact-sequence-thickenings}
Let $(X, \mathcal{O}_X)$ be a ringed space. We define a sequence of morphisms
of first order thickenings
$$
(X'_1, \mathcal{O}_{X'_1}) \to
(X'_2, \mathcal{O}_{X'_2}) \to
(X'_3, \mathcal{O}_{X'_3})
$$
of $(X, \mathcal{O}_X)$ to be a {\it complex}
if the corresponding maps between
the ideal sheaves $\mathcal{I}_i$
give a complex of $\mathcal{O}_X$-modules
$\mathcal{I}_3 \to \mathcal{I}_2 \to \mathcal{I}_1$
(i.e., the composition is zero). In this case the composition
$(X'_1, \mathcal{O}_{X'_1}) \to (X_3', \mathcal{O}_{X'_3})$ factors through
$(X, \mathcal{O}_X) \to (X'_3, \mathcal{O}_{X'_3})$, i.e.,
the first order thickening $(X'_1, \mathcal{O}_{X'_1})$ of
$(X, \mathcal{O}_X)$ is trivial and comes with
a canonical trivialization
$\pi : (X'_1, \mathcal{O}_{X'_1}) \to (X, \mathcal{O}_X)$.

\medskip\noindent
We say a sequence of morphisms of first order thickenings
$$
(X'_1, \mathcal{O}_{X'_1}) \to
(X'_2, \mathcal{O}_{X'_2}) \to
(X'_3, \mathcal{O}_{X'_3})
$$
of $(X, \mathcal{O}_X)$ is {\it a short exact sequence} if the
corresponding maps between ideal sheaves is a short exact sequence
$$
0 \to \mathcal{I}_3 \to \mathcal{I}_2 \to \mathcal{I}_1 \to 0
$$
of $\mathcal{O}_X$-modules.
\end{remark}

\begin{remark}
\label{remark-complex-thickenings-and-ses-modules}
Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{F}$ be an
$\mathcal{O}_X$-module. Let
$$
(X'_1, \mathcal{O}_{X'_1}) \to
(X'_2, \mathcal{O}_{X'_2}) \to
(X'_3, \mathcal{O}_{X'_3})
$$
be a complex first order thickenings of $(X, \mathcal{O}_X)$, see
Remark \ref{remark-short-exact-sequence-thickenings}.
Let $(\mathcal{K}_i, c_i)$, $i = 1, 2, 3$ be pairs consisting of
an $\mathcal{O}_X$-module $\mathcal{K}_i$ and a map
$c_i : \mathcal{I}_i \otimes_{\mathcal{O}_X} \mathcal{F} \to
\mathcal{K}_i$. Assume given a short exact sequence
of $\mathcal{O}_X$-modules
$$
0 \to \mathcal{K}_3 \to \mathcal{K}_2 \to \mathcal{K}_1 \to 0
$$
such that
$$
\vcenter{
\xymatrix{
\mathcal{I}_2 \otimes_{\mathcal{O}_X} \mathcal{F}
\ar[r]_-{c_2} \ar[d] &
\mathcal{K}_2 \ar[d] \\
\mathcal{I}_1 \otimes_{\mathcal{O}_X} \mathcal{F}
\ar[r]^-{c_1} &
\mathcal{K}_1
}
}
\quad\text{and}\quad
\vcenter{
\xymatrix{
\mathcal{I}_3 \otimes_{\mathcal{O}_X} \mathcal{F}
\ar[r]_-{c_3} \ar[d] &
\mathcal{K}_3 \ar[d] \\
\mathcal{I}_2 \otimes_{\mathcal{O}_X} \mathcal{F}
\ar[r]^-{c_2} &
\mathcal{K}_2
}
}
$$
are commutative. Finally, assume given an extension
$$
0 \to \mathcal{K}_2 \to \mathcal{F}'_2 \to \mathcal{F} \to 0
$$
as in (\ref{equation-extension}) with $\mathcal{K} = \mathcal{K}_2$
of $\mathcal{O}_{X'_2}$-modules with $c_{\mathcal{F}'_2} = c_2$.
In this situation we can apply the functoriality of
Remark \ref{remark-extension-functorial} to obtain an extension
$\mathcal{F}'_1$ on $X'_1$ (we'll describe $\mathcal{F}'_1$
in this special case below). By
Remark \ref{remark-trivial-extension}
using the canonical splitting
$\pi : (X'_1, \mathcal{O}_{X'_1}) \to (X, \mathcal{O}_X)$ of
Remark \ref{remark-short-exact-sequence-thickenings}
we obtain
$\xi_{\mathcal{F}'_1} \in
\text{Ext}^1_{\mathcal{O}_X}(\mathcal{F}, \mathcal{K}_1)$.
Finally, we have the obstruction
$$
o(\mathcal{F}, \mathcal{K}_3, c_3) \in
\text{Ext}^2_{\mathcal{O}_X}(\mathcal{F}, \mathcal{K}_3)
$$
see Lemma \ref{lemma-inf-obs-ext}.
In this situation we {\bf claim} that the canonical map
$$
\partial :
\text{Ext}^1_{\mathcal{O}_X}(\mathcal{F}, \mathcal{K}_1)
\longrightarrow
\text{Ext}^2_{\mathcal{O}_X}(\mathcal{F}, \mathcal{K}_3)
$$
coming from the short exact sequence
$0 \to \mathcal{K}_3 \to \mathcal{K}_2 \to \mathcal{K}_1 \to 0$
sends $\xi_{\mathcal{F}'_1}$
to the obstruction class $o(\mathcal{F}, \mathcal{K}_3, c_3)$.

\medskip\noindent
To prove this claim choose an embedding $j : \mathcal{K}_3 \to \mathcal{K}$
where $\mathcal{K}$ is an injective $\mathcal{O}_X$-module.
We can lift $j$ to a map $j' : \mathcal{K}_2 \to \mathcal{K}$.
Set $\mathcal{E}'_2 = j'_*\mathcal{F}'_2$ equal to the pushout
of $\mathcal{F}'_2$ by $j'$ so that $c_{\mathcal{E}'_2} = j' \circ c_2$.
Picture:
$$
\xymatrix{
0 \ar[r] &
\mathcal{K}_2 \ar[r] \ar[d]_{j'} &
\mathcal{F}'_2 \ar[r] \ar[d] &
\mathcal{F} \ar[r] \ar[d] & 0 \\
0 \ar[r] &
\mathcal{K} \ar[r] &
\mathcal{E}'_2 \ar[r] &
\mathcal{F} \ar[r] & 0
}
$$
Set $\mathcal{E}'_3 = \mathcal{E}'_2$ but viewed as an
$\mathcal{O}_{X'_3}$-module via $\mathcal{O}_{X'_3} \to \mathcal{O}_{X'_2}$.
Then $c_{\mathcal{E}'_3} = j \circ c_3$.
The proof of Lemma \ref{lemma-inf-obs-ext} constructs
$o(\mathcal{F}, \mathcal{K}_3, c_3)$
as the boundary of the class of the extension of $\mathcal{O}_X$-modules
$$
0 \to
\mathcal{K}/\mathcal{K}_3 \to
\mathcal{E}'_3/\mathcal{K}_3 \to
\mathcal{F} \to 0
$$
On the other hand, note that $\mathcal{F}'_1 = \mathcal{F}'_2/\mathcal{K}_3$
hence the class $\xi_{\mathcal{F}'_1}$ is the class
of the extension
$$
0 \to \mathcal{K}_2/\mathcal{K}_3 \to \mathcal{F}'_2/\mathcal{K}_3
\to \mathcal{F} \to 0
$$
seen as a sequence of $\mathcal{O}_X$-modules using $\pi^\sharp$
where $\pi : (X'_1, \mathcal{O}_{X'_1}) \to (X, \mathcal{O}_X)$
is the canonical splitting.
Thus finally, the claim follows from the fact that we have
a commutative diagram
$$
\xymatrix{
0 \ar[r] &
\mathcal{K}_2/\mathcal{K}_3 \ar[r] \ar[d] &
\mathcal{F}'_2/\mathcal{K}_3 \ar[r] \ar[d] &
\mathcal{F} \ar[r] \ar[d] & 0 \\
0 \ar[r] &
\mathcal{K}/\mathcal{K}_3 \ar[r] &
\mathcal{E}'_3/\mathcal{K}_3 \ar[r] &
\mathcal{F} \ar[r] & 0
}
$$
which is $\mathcal{O}_X$-linear (with the $\mathcal{O}_X$-module
structures given above).
\end{remark}








\section{Infinitesimal deformations of modules on ringed spaces}
\label{section-deformation-modules}

\noindent
Let $i : (X, \mathcal{O}_X) \to (X', \mathcal{O}_{X'})$ be a first
order thickening of ringed spaces. We freely use the notation introduced in
Section \ref{section-thickenings-spaces}.
Let $\mathcal{F}'$ be an $\mathcal{O}_{X'}$-module
and set $\mathcal{F} = i^*\mathcal{F}'$.
In this situation we have a short exact sequence
$$
0 \to \mathcal{I}\mathcal{F}' \to \mathcal{F}' \to \mathcal{F} \to 0
$$
of $\mathcal{O}_{X'}$-modules. Since $\mathcal{I}^2 = 0$ the
$\mathcal{O}_{X'}$-module structure on $\mathcal{I}\mathcal{F}'$
comes from a unique $\mathcal{O}_X$-module structure.
Thus the sequence above is an extension as in (\ref{equation-extension}).
As a special case, if $\mathcal{F}' = \mathcal{O}_{X'}$ we have
$i^*\mathcal{O}_{X'} = \mathcal{O}_X$ and
$\mathcal{I}\mathcal{O}_{X'} = \mathcal{I}$ and we recover the
sequence of structure sheaves
$$
0 \to \mathcal{I} \to \mathcal{O}_{X'} \to \mathcal{O}_X \to 0
$$

\begin{lemma}
\label{lemma-inf-map-special}
Let $i : (X, \mathcal{O}_X) \to (X', \mathcal{O}_{X'})$
be a first order thickening of ringed spaces.
Let $\mathcal{F}'$, $\mathcal{G}'$ be $\mathcal{O}_{X'}$-modules.
Set $\mathcal{F} = i^*\mathcal{F}'$ and $\mathcal{G} = i^*\mathcal{G}'$.
Let $\varphi : \mathcal{F} \to \mathcal{G}$ be an $\mathcal{O}_X$-linear map.
The set of lifts of $\varphi$ to an $\mathcal{O}_{X'}$-linear map
$\varphi' : \mathcal{F}' \to \mathcal{G}'$ is, if nonempty, a principal
homogeneous space under
$\Hom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{I}\mathcal{G}')$.
\end{lemma}

\begin{proof}
This is a special case of Lemma \ref{lemma-inf-map} but we also
give a direct proof. We have short exact sequences of modules
$$
0 \to \mathcal{I} \to \mathcal{O}_{X'} \to \mathcal{O}_X \to 0
\quad\text{and}\quad
0 \to \mathcal{I}\mathcal{G}' \to \mathcal{G}' \to \mathcal{G} \to 0
$$
and similarly for $\mathcal{F}'$.
Since $\mathcal{I}$ has square zero the $\mathcal{O}_{X'}$-module
structure on $\mathcal{I}$ and $\mathcal{I}\mathcal{G}'$ comes from
a unique $\mathcal{O}_X$-module structure. It follows that
$$
\Hom_{\mathcal{O}_{X'}}(\mathcal{F}', \mathcal{I}\mathcal{G}') =
\Hom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{I}\mathcal{G}')
\quad\text{and}\quad
\Hom_{\mathcal{O}_{X'}}(\mathcal{F}', \mathcal{G}) =
\Hom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})
$$
The lemma now follows from the exact sequence
$$
0 \to \Hom_{\mathcal{O}_{X'}}(\mathcal{F}', \mathcal{I}\mathcal{G}') \to
\Hom_{\mathcal{O}_{X'}}(\mathcal{F}', \mathcal{G}') \to
\Hom_{\mathcal{O}_{X'}}(\mathcal{F}', \mathcal{G})
$$
see Homology, Lemma \ref{homology-lemma-check-exactness}.
\end{proof}

\begin{lemma}
\label{lemma-deform-module}
Let $(f, f')$ be a morphism of first order thickenings of ringed spaces
as in Situation \ref{situation-morphism-thickenings}.
Let $\mathcal{F}'$ be an $\mathcal{O}_{X'}$-module
and set $\mathcal{F} = i^*\mathcal{F}'$.
Assume that $\mathcal{F}$ is flat over $S$
and that $(f, f')$ is a strict morphism of thickenings
(Definition \ref{definition-strict-morphism-thickenings}).
Then the following are equivalent
\begin{enumerate}
\item $\mathcal{F}'$ is flat over $S'$, and
\item the canonical map
$f^*\mathcal{J} \otimes_{\mathcal{O}_X} \mathcal{F} \to
\mathcal{I}\mathcal{F}'$
is an isomorphism.
\end{enumerate}
Moreover, in this case the maps
$$
f^*\mathcal{J} \otimes_{\mathcal{O}_X} \mathcal{F} \to
\mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{F} \to
\mathcal{I}\mathcal{F}'
$$
are isomorphisms.
\end{lemma}

\begin{proof}
The map $f^*\mathcal{J} \to \mathcal{I}$ is surjective
as $(f, f')$ is a strict morphism of thickenings.
Hence the final statement is a consequence of (2).

\medskip\noindent
Proof of the equivalence of (1) and (2). We may check these conditions
at stalks. Let $x \in X \subset X'$
be a point with image $s = f(x) \in S \subset S'$.
Set $A' = \mathcal{O}_{S', s}$, $B' = \mathcal{O}_{X', x}$,
$A = \mathcal{O}_{S, s}$, and $B = \mathcal{O}_{X, x}$.
Then $A = A'/J$ and $B = B'/I$ for some square zero ideals.
Since $(f, f')$ is a strict morphism of thickenings we have $I = JB'$.
Let $M' = \mathcal{F}'_x$ and $M = \mathcal{F}_x$.
Then $M'$ is a $B'$-module and $M$ is a $B$-module.
Since $\mathcal{F} = i^*\mathcal{F}'$ we see that the kernel of the
surjection $M' \to M$ is $IM' = JM'$. Thus we have a short exact
sequence
$$
0 \to JM' \to M' \to M \to 0
$$
Using
Sheaves, Lemma \ref{sheaves-lemma-stalk-pullback-modules}
and
Modules, Lemma \ref{modules-lemma-stalk-tensor-product}
to identify stalks of pullbacks and tensor products we see
that the stalk at $x$ of the canonical map of the lemma is the map
$$
(J \otimes_A B) \otimes_B M = J \otimes_A M = J \otimes_{A'} M'
\longrightarrow JM'
$$
The assumption that $\mathcal{F}$ is flat over $S$ signifies that
$M$ is a flat $A$-module.

\medskip\noindent
Assume (1). Flatness implies $\text{Tor}_1^{A'}(M', A) = 0$ by
Algebra, Lemma \ref{algebra-lemma-characterize-flat}.
This means $J \otimes_{A'} M' \to M'$ is injective by
Algebra, Remark \ref{algebra-remark-Tor-ring-mod-ideal}.
Hence $J \otimes_A M \to JM'$ is an isomorphism.

\medskip\noindent
Assume (2). Then $J \otimes_{A'} M' \to M'$ is injective. Hence
$\text{Tor}_1^{A'}(M', A) = 0$ by
Algebra, Remark \ref{algebra-remark-Tor-ring-mod-ideal}.
Hence $M'$ is flat over $A'$ by
Algebra, Lemma \ref{algebra-lemma-what-does-it-mean}.
\end{proof}

\begin{lemma}
\label{lemma-inf-map-rel}
Let $(f, f')$ be a morphism of first order thickenings as in
Situation \ref{situation-morphism-thickenings}.
Let $\mathcal{F}'$, $\mathcal{G}'$ be $\mathcal{O}_{X'}$-modules and set
$\mathcal{F} = i^*\mathcal{F}'$ and $\mathcal{G} = i^*\mathcal{G}'$.
Let $\varphi : \mathcal{F} \to \mathcal{G}$ be an $\mathcal{O}_X$-linear map.
Assume that $\mathcal{G}'$ is flat over $S'$ and that
$(f, f')$ is a strict morphism of thickenings.
The set of lifts of $\varphi$ to an $\mathcal{O}_{X'}$-linear map
$\varphi' : \mathcal{F}' \to \mathcal{G}'$ is, if nonempty, a principal
homogeneous space under
$$
\Hom_{\mathcal{O}_X}(\mathcal{F},
\mathcal{G} \otimes_{\mathcal{O}_X} f^*\mathcal{J})
$$
\end{lemma}

\begin{proof}
Combine Lemmas \ref{lemma-inf-map-special} and \ref{lemma-deform-module}.
\end{proof}

\begin{lemma}
\label{lemma-inf-obs-map-special}
Let $i : (X, \mathcal{O}_X) \to (X', \mathcal{O}_{X'})$
be a first order thickening of ringed spaces.
Let $\mathcal{F}'$, $\mathcal{G}'$ be $\mathcal{O}_{X'}$-modules and set
$\mathcal{F} = i^*\mathcal{F}'$ and $\mathcal{G} = i^*\mathcal{G}'$.
Let $\varphi : \mathcal{F} \to \mathcal{G}$ be an $\mathcal{O}_X$-linear map.
There exists an element
$$
o(\varphi) \in
\text{Ext}^1_{\mathcal{O}_X}(Li^*\mathcal{F}',
\mathcal{I}\mathcal{G}')
$$
whose vanishing is a necessary and sufficient condition for the
existence of a lift of $\varphi$ to an $\mathcal{O}_{X'}$-linear map
$\varphi' : \mathcal{F}' \to \mathcal{G}'$.
\end{lemma}

\begin{proof}
It is clear from the proof of Lemma \ref{lemma-inf-map-special} that the
vanishing of the boundary of $\varphi$ via the map
$$
\Hom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G}) =
\Hom_{\mathcal{O}_{X'}}(\mathcal{F}', \mathcal{G}) \longrightarrow
\text{Ext}^1_{\mathcal{O}_{X'}}(\mathcal{F}', \mathcal{I}\mathcal{G}')
$$
is a necessary and sufficient condition for the existence of a lift. We
conclude as
$$
\text{Ext}^1_{\mathcal{O}_{X'}}(\mathcal{F}', \mathcal{I}\mathcal{G}') =
\text{Ext}^1_{\mathcal{O}_X}(Li^*\mathcal{F}', \mathcal{I}\mathcal{G}')
$$
the adjointness of $i_* = Ri_*$ and $Li^*$ on the derived category
(Cohomology, Lemma \ref{cohomology-lemma-adjoint}).
\end{proof}

\begin{lemma}
\label{lemma-inf-obs-map-rel}
Let $(f, f')$ be a morphism of first
order thickenings as in Situation \ref{situation-morphism-thickenings}.
Let $\mathcal{F}'$, $\mathcal{G}'$ be $\mathcal{O}_{X'}$-modules and set
$\mathcal{F} = i^*\mathcal{F}'$ and $\mathcal{G} = i^*\mathcal{G}'$.
Let $\varphi : \mathcal{F} \to \mathcal{G}$ be an $\mathcal{O}_X$-linear map.
Assume that $\mathcal{F}'$ and $\mathcal{G}'$ are flat over $S'$ and
that $(f, f')$ is a strict morphism of thickenings. There exists an element
$$
o(\varphi) \in  \text{Ext}^1_{\mathcal{O}_X}(\mathcal{F},
\mathcal{G} \otimes_{\mathcal{O}_X} f^*\mathcal{J})
$$
whose vanishing is a necessary and sufficient condition for the
existence of a lift of $\varphi$ to an $\mathcal{O}_{X'}$-linear map
$\varphi' : \mathcal{F}' \to \mathcal{G}'$.
\end{lemma}

\begin{proof}[First proof]
This follows from Lemma \ref{lemma-inf-obs-map-special}
as we claim that under the assumptions of the lemma we have
$$
\text{Ext}^1_{\mathcal{O}_X}(Li^*\mathcal{F}',
\mathcal{I}\mathcal{G}') =
\text{Ext}^1_{\mathcal{O}_X}(\mathcal{F},
\mathcal{G} \otimes_{\mathcal{O}_X} f^*\mathcal{J})
$$
Namely, we have
$\mathcal{I}\mathcal{G}' =
\mathcal{G} \otimes_{\mathcal{O}_X} f^*\mathcal{J}$
by Lemma \ref{lemma-deform-module}.
On the other hand, observe that
$$
H^{-1}(Li^*\mathcal{F}') =
\text{Tor}_1^{\mathcal{O}_{X'}}(\mathcal{F}', \mathcal{O}_X)
$$
(local computation omitted). Using the short exact sequence
$$
0 \to \mathcal{I} \to \mathcal{O}_{X'} \to \mathcal{O}_X \to 0
$$
we see that this $\text{Tor}_1$ is computed by the kernel of the map
$\mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{F} \to \mathcal{I}\mathcal{F}'$
which is zero by the final assertion of Lemma \ref{lemma-deform-module}.
Thus $\tau_{\geq -1}Li^*\mathcal{F}' = \mathcal{F}$.
On the other hand, we have
$$
\text{Ext}^1_{\mathcal{O}_X}(Li^*\mathcal{F}',
\mathcal{I}\mathcal{G}') =
\text{Ext}^1_{\mathcal{O}_X}(\tau_{\geq -1}Li^*\mathcal{F}',
\mathcal{I}\mathcal{G}')
$$
by the dual of
Derived Categories, Lemma \ref{derived-lemma-negative-vanishing}.
\end{proof}

\begin{proof}[Second proof]
We can apply Lemma \ref{lemma-inf-obs-map} as follows. Note that
$\mathcal{K} = \mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{F}$ and
$\mathcal{L} = \mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{G}$
by Lemma \ref{lemma-deform-module}, that
$c_{\mathcal{F}'} = 1 \otimes 1$ and $c_{\mathcal{G}'} = 1 \otimes 1$
and taking $\psi = 1 \otimes \varphi$ the diagram of the lemma
commutes. Thus $o(\varphi) = o(\varphi, 1 \otimes \varphi)$
works.
\end{proof}

\begin{lemma}
\label{lemma-inf-ext-rel}
Let $(f, f')$ be a morphism of first order thickenings as in
Situation \ref{situation-morphism-thickenings}.
Let $\mathcal{F}$ be an $\mathcal{O}_X$-module.
Assume $(f, f')$ is a strict morphism of thickenings and
$\mathcal{F}$ flat over $S$. If there exists a pair
$(\mathcal{F}', \alpha)$ consisting of an
$\mathcal{O}_{X'}$-module $\mathcal{F}'$ flat over $S'$ and an isomorphism
$\alpha : i^*\mathcal{F}' \to \mathcal{F}$, then the set of
isomorphism classes of such pairs is principal homogeneous
under
$\text{Ext}^1_{\mathcal{O}_X}(
\mathcal{F}, \mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{F})$.
\end{lemma}

\begin{proof}
If we assume there exists one such module, then the canonical map
$$
f^*\mathcal{J} \otimes_{\mathcal{O}_X} \mathcal{F} \to
\mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{F}
$$
is an isomorphism by Lemma \ref{lemma-deform-module}. Apply
Lemma \ref{lemma-inf-ext} with $\mathcal{K} = 
\mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{F}$
and $c = 1$. By Lemma \ref{lemma-deform-module} the corresponding extensions
$\mathcal{F}'$ are all flat over $S'$.
\end{proof}

\begin{lemma}
\label{lemma-inf-obs-ext-rel}
Let $(f, f')$ be a morphism of first order thickenings as in
Situation \ref{situation-morphism-thickenings}.
Let $\mathcal{F}$ be an $\mathcal{O}_X$-module. Assume
$(f, f')$ is a strict morphism of thickenings
and $\mathcal{F}$ flat over $S$. There exists an
$\mathcal{O}_{X'}$-module $\mathcal{F}'$ flat over $S'$ with
$i^*\mathcal{F}' \cong \mathcal{F}$, if and only if
\begin{enumerate}
\item the canonical map $
f^*\mathcal{J} \otimes_{\mathcal{O}_X} \mathcal{F} \to
\mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{F}$
is an isomorphism, and
\item the class
$o(\mathcal{F}, \mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{F}, 1)
\in \text{Ext}^2_{\mathcal{O}_X}(
\mathcal{F}, \mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{F})$
of Lemma \ref{lemma-inf-obs-ext} is zero.
\end{enumerate}
\end{lemma}

\begin{proof}
This follows immediately from the characterization of
$\mathcal{O}_{X'}$-modules flat over $S'$ of 
Lemma \ref{lemma-deform-module} and
Lemma \ref{lemma-inf-obs-ext}.
\end{proof}






\section{Application to flat modules on flat thickenings of ringed spaces}
\label{section-flat}

\noindent
Consider a commutative diagram
$$
\xymatrix{
(X, \mathcal{O}_X) \ar[r]_i \ar[d]_f & (X', \mathcal{O}_{X'}) \ar[d]^{f'} \\
(S, \mathcal{O}_S) \ar[r]^t & (S', \mathcal{O}_{S'})
}
$$
of ringed spaces whose horizontal arrows are first order thickenings as in
Situation \ref{situation-morphism-thickenings}. Set
$\mathcal{I} = \text{Ker}(i^\sharp) \subset \mathcal{O}_{X'}$ and
$\mathcal{J} = \text{Ker}(t^\sharp) \subset \mathcal{O}_{S'}$.
Let $\mathcal{F}$ be an $\mathcal{O}_X$-module. Assume that
\begin{enumerate}
\item $(f, f')$ is a strict morphism of thickenings,
\item $f'$ is flat, and
\item $\mathcal{F}$ is flat over $S$.
\end{enumerate}
Note that (1) $+$ (2) imply that $\mathcal{I} = f^*\mathcal{J}$
(apply Lemma \ref{lemma-deform-module} to $\mathcal{O}_{X'}$).
The theory of the preceding section is especially nice
under these assumptions. We summarize the results already obtained
in the following lemma.

\begin{lemma}
\label{lemma-flat}
In the situation above.
\begin{enumerate}
\item There exists an $\mathcal{O}_{X'}$-module $\mathcal{F}'$ flat over
$S'$ with $i^*\mathcal{F}' \cong \mathcal{F}$, if and only if
the class
$o(\mathcal{F}, f^*\mathcal{J} \otimes_{\mathcal{O}_X} \mathcal{F}, 1)
\in \text{Ext}^2_{\mathcal{O}_X}(
\mathcal{F}, f^*\mathcal{J} \otimes_{\mathcal{O}_X} \mathcal{F})$
of Lemma \ref{lemma-inf-obs-ext} is zero.
\item If such a module exists, then the set of isomorphism classes
of lifts is principal homogeneous under
$\text{Ext}^1_{\mathcal{O}_X}(
\mathcal{F}, f^*\mathcal{J} \otimes_{\mathcal{O}_X} \mathcal{F})$.
\item Given a lift $\mathcal{F}'$, the set of automorphisms of
$\mathcal{F}'$ which pull back to $\text{id}_\mathcal{F}$ is canonically
isomorphic to $\text{Ext}^0_{\mathcal{O}_X}(
\mathcal{F}, f^*\mathcal{J} \otimes_{\mathcal{O}_X} \mathcal{F})$.
\end{enumerate}
\end{lemma}

\begin{proof}
Part (1) follows from Lemma \ref{lemma-inf-obs-ext-rel}
as we have seen above that $\mathcal{I} = f^*\mathcal{J}$.
Part (2) follows from Lemma \ref{lemma-inf-ext-rel}.
Part (3) follows from Lemma \ref{lemma-inf-map-rel}.
\end{proof}

\begin{situation}
\label{situation-ses-flat-thickenings}
Let $f : (X, \mathcal{O}_X) \to (S, \mathcal{O}_S)$ be a morphism of
ringed spaces. Consider a commutative diagram
$$
\xymatrix{
(X'_1, \mathcal{O}'_1) \ar[r]_h \ar[d]_{f'_1} &
(X'_2, \mathcal{O}'_2) \ar[r] \ar[d]_{f'_2} &
(X'_3, \mathcal{O}'_3) \ar[d]_{f'_3} \\
(S'_1, \mathcal{O}_{S'_1}) \ar[r] &
(S'_2, \mathcal{O}_{S'_2}) \ar[r] &
(S'_3, \mathcal{O}_{S'_3})
}
$$
where (a) the top row is a short exact sequence of first order thickenings
of $X$, (b) the lower row is a short exact sequence of first order
thickenings of $S$, (c) each $f'_i$ restricts to $f$, (d) each pair
$(f, f_i')$ is a strict morphism of thickenings, and (e) each $f'_i$ is flat.
Finally, let $\mathcal{F}'_2$ be an $\mathcal{O}'_2$-module flat over
$S'_2$ and set $\mathcal{F} = \mathcal{F}'_2|_X$. Let $\pi : X'_1 \to X$
be the canonical splitting
(Remark \ref{remark-short-exact-sequence-thickenings}).
\end{situation}

\begin{lemma}
\label{lemma-verify-iv}
In Situation \ref{situation-ses-flat-thickenings} the modules
$\pi^*\mathcal{F}$ and $h^*\mathcal{F}'_2$ are $\mathcal{O}'_1$-modules
flat over $S'_1$ restricting to $\mathcal{F}$ on $X$.
Their difference (Lemma \ref{lemma-flat}) is an element
$\theta$ of $\text{Ext}^1_{\mathcal{O}_X}(
\mathcal{F}, f^*\mathcal{J}_1 \otimes_{\mathcal{O}_X} \mathcal{F})$
whose boundary in
$\text{Ext}^2_{\mathcal{O}_X}(
\mathcal{F}, f^*\mathcal{J}_3 \otimes_{\mathcal{O}_X} \mathcal{F})$
equals the obstruction (Lemma \ref{lemma-flat})
to lifting $\mathcal{F}$ to an $\mathcal{O}'_3$-module flat over $S'_3$.
\end{lemma}

\begin{proof}
Note that both $\pi^*\mathcal{F}$ and $h^*\mathcal{F}'_2$
restrict to $\mathcal{F}$ on $X$ and that the kernels of
$\pi^*\mathcal{F} \to \mathcal{F}$ and $h^*\mathcal{F}'_2 \to \mathcal{F}$
are given by $f^*\mathcal{J}_1 \otimes_{\mathcal{O}_X} \mathcal{F}$.
Hence flatness by Lemma \ref{lemma-deform-module}.
Taking the boundary makes sense as the sequence of modules
$$
0 \to f^*\mathcal{J}_3 \otimes_{\mathcal{O}_X} \mathcal{F} \to
f^*\mathcal{J}_2 \otimes_{\mathcal{O}_X} \mathcal{F} \to
f^*\mathcal{J}_1 \otimes_{\mathcal{O}_X} \mathcal{F} \to 0
$$
is short exact due to the assumptions in
Situation \ref{situation-ses-flat-thickenings}
and the fact that $\mathcal{F}$ is flat over $S$.
The statement on the obstruction class is a direct translation
of the result of
Remark \ref{remark-complex-thickenings-and-ses-modules}
to this particular situation.
\end{proof}






\section{Deformations of ringed spaces and the naive cotangent complex}
\label{section-deformations-ringed-spaces}

\noindent
In this section we use the naive cotangent complex to do a little bit
of deformation theory. We start with a first order thickening
$t : (S, \mathcal{O}_S) \to (S', \mathcal{O}_{S'})$ of ringed spaces.
We denote $\mathcal{J} = \text{Ker}(t^\sharp)$ and we
identify the underlying topological spaces of $S$ and $S'$.
Moreover we assume given a morphism of ringed spaces
$f : (X, \mathcal{O}_X) \to (S, \mathcal{O}_S)$, an $\mathcal{O}_X$-module
$\mathcal{G}$, and an $f$-map $c : \mathcal{J} \to \mathcal{G}$
of sheaves of modules (Sheaves, Definition \ref{sheaves-definition-f-map}
and Section \ref{sheaves-section-ringed-spaces-functoriality-modules}).
In this section we ask ourselves whether we can find
the question mark fitting into the following diagram
\begin{equation}
\label{equation-to-solve-ringed-spaces}
\vcenter{
\xymatrix{
0 \ar[r] & \mathcal{G} \ar[r] & {?} \ar[r] & \mathcal{O}_X \ar[r] & 0 \\
0 \ar[r] & \mathcal{J} \ar[u]^c \ar[r] & \mathcal{O}_{S'} \ar[u] \ar[r] &
\mathcal{O}_S \ar[u] \ar[r] & 0
}
}
\end{equation}
(where the vertical arrows are $f$-maps)
and moreover how unique the solution is (if it exists). More precisely,
we look for a first order thickening
$i : (X, \mathcal{O}_X) \to (X', \mathcal{O}_{X'})$
and a morphism of thickenings $(f, f')$ as in
(\ref{equation-morphism-thickenings})
where $\text{Ker}(i^\sharp)$ is identified with $\mathcal{G}$
such that $(f')^\sharp$ induces the given map $c$.
We will say $X'$ is a {\it solution} to
(\ref{equation-to-solve-ringed-spaces}).

\begin{lemma}
\label{lemma-huge-diagram-ringed-spaces}
Assume given a commutative diagram of morphisms ringed spaces
\begin{equation}
\label{equation-huge-1}
\vcenter{
\xymatrix{
& (X_2, \mathcal{O}_{X_2}) \ar[r]_{i_2} \ar[d]_{f_2} \ar[ddl]_g &
(X'_2, \mathcal{O}_{X'_2}) \ar[d]^{f'_2} \\
& (S_2, \mathcal{O}_{S_2}) \ar[r]^{t_2} \ar[ddl]|\hole &
(S'_2, \mathcal{O}_{S'_2}) \ar[ddl] \\
(X_1, \mathcal{O}_{X_1}) \ar[r]_{i_1} \ar[d]_{f_1} &
(X'_1, \mathcal{O}_{X'_1}) \ar[d]^{f'_1} \\
(S_1, \mathcal{O}_{S_1}) \ar[r]^{t_1} &
(S'_1, \mathcal{O}_{S'_1})
}
}
\end{equation}
whose horizontal arrows are first order thickenings. Set
$\mathcal{G}_j = \text{Ker}(i_j^\sharp)$ and assume given
a $g$-map $\nu : \mathcal{G}_1 \to \mathcal{G}_2$ of modules
giving rise to the commutative diagram
\begin{equation}
\label{equation-huge-2}
\vcenter{
\xymatrix{
& 0 \ar[r] & \mathcal{G}_2 \ar[r] &
\mathcal{O}_{X'_2} \ar[r] &
\mathcal{O}_{X_2} \ar[r] & 0 \\
& 0 \ar[r]|\hole &
\mathcal{J}_2 \ar[u]_{c_2} \ar[r] &
\mathcal{O}_{S'_2} \ar[u] \ar[r]|\hole &
\mathcal{O}_{S_2} \ar[u] \ar[r] & 0 \\
0 \ar[r] & \mathcal{G}_1 \ar[ruu] \ar[r] &
\mathcal{O}_{X'_1} \ar[r] &
\mathcal{O}_{X_1} \ar[ruu] \ar[r] & 0 \\
0 \ar[r] & \mathcal{J}_1 \ar[ruu]|\hole \ar[u]^{c_1} \ar[r] &
\mathcal{O}_{S'_1} \ar[ruu]|\hole \ar[u] \ar[r] &
\mathcal{O}_{S_1} \ar[ruu]|\hole \ar[u] \ar[r] & 0
}
}
\end{equation}
with front and back solutions to (\ref{equation-to-solve-ringed-spaces}).
\begin{enumerate}
\item There exist a canonical element in
$\text{Ext}^1_{\mathcal{O}_{X_2}}(Lg^*\NL_{X_1/S_1}, \mathcal{G}_2)$
whose vanishing is a necessary and sufficient condition for the existence
of a morphism of ringed spaces $X'_2 \to X'_1$ fitting into
(\ref{equation-huge-1}) compatibly with $\nu$.
\item If there exists a morphism $X'_2 \to X'_1$ fitting into
(\ref{equation-huge-1}) compatibly with $\nu$ the set of all such morphisms
is a principal homogeneous space under
$$
\Hom_{\mathcal{O}_{X_1}}(\Omega_{X_1/S_1}, g_*\mathcal{G}_2) =
\Hom_{\mathcal{O}_{X_2}}(g^*\Omega_{X_1/S_1}, \mathcal{G}_2) =
\text{Ext}^0_{\mathcal{O}_{X_2}}(Lg^*\NL_{X_1/S_1}, \mathcal{G}_2).
$$
\end{enumerate}
\end{lemma}

\begin{proof}
The equalities in the last statement of the lemma are immediate from
the definitions. Thus we will work with the groups
$\text{Ext}^k_{\mathcal{O}_{X_2}}(Lg^*\NL_{X_1/S_1}, \mathcal{G}_2)$,
$k = 0, 1$ in the rest of the proof. We first argue that we can reduce
to the case where the underlying topological spaces of all ringed
spaces in the lemma is the same.

\medskip\noindent
To do this, observe that $g^{-1}\NL_{X_1/S_1}$ is equal to the naive
cotangent complex of the homomorphism of sheaves of rings
$g^{-1}f_1^{-1}\mathcal{O}_{S_1} \to g^{-1}\mathcal{O}_{X_1}$, see
Modules, Lemma \ref{modules-lemma-pullback-differentials}.
Moreover, the degree $0$ term of $\NL_{X_1/S_1}$ is a flat
$\mathcal{O}_{X_1}$-module, hence the canonical map
$$
Lg^*\NL_{X_1/S_1}
\longrightarrow
g^{-1}\NL_{X_1/S_1} \otimes_{g^{-1}\mathcal{O}_{X_1}} \mathcal{O}_{X_2}
$$
induces an isomorphism on cohomology sheaves in degrees $0$ and $-1$.
Thus we may replace the Ext groups of the lemma with
$$
\text{Ext}^k_{g^{-1}\mathcal{O}_{X_1}}(g^{-1}\NL_{X_1/S_1}, \mathcal{G}_2) =
\text{Ext}^k_{g^{-1}\mathcal{O}_{X_1}}(
\NL_{g^{-1}\mathcal{O}_{X_1}/g^{-1}f_1^{-1}\mathcal{O}_{S_1}}, \mathcal{G}_2)
$$
The set of morphism of ringed spaces $X'_2 \to X'_1$ fitting into
(\ref{equation-huge-1}) compatibly with $\nu$
is in one-to-one bijection with
the set of homomorphisms of $g^{-1}f_1^{-1}\mathcal{O}_{S'_1}$-algebras
$g^{-1}\mathcal{O}_{X'_1} \to \mathcal{O}_{X'_2}$ which are compatible with
$f^\sharp$ and $\nu$. In this way we see that we may assume we have a
diagram (\ref{equation-huge-2}) of sheaves on $X$ and we are looking to
find a homomorphism of sheaves of rings
$\mathcal{O}_{X'_1} \to \mathcal{O}_{X'_2}$ fitting into it.

\medskip\noindent
In the rest of the proof of the lemma we assume
all underlying topological spaces are the
same, i.e., we have a diagram (\ref{equation-huge-2}) of sheaves on
a space $X$ and we are looking for homomorphisms of sheaves of rings
$\mathcal{O}_{X'_1} \to \mathcal{O}_{X'_2}$ fitting into it.
As ext groups we will use
$\text{Ext}^k_{\mathcal{O}_{X_1}}(
\NL_{\mathcal{O}_{X_1}/\mathcal{O}_{S_1}}, \mathcal{G}_2)$, $k = 0, 1$.

\medskip\noindent
Step 1. Construction of the obstruction class. Consider the sheaf
of sets
$$
\mathcal{E} = \mathcal{O}_{X'_1} \times_{\mathcal{O}_{X_2}} \mathcal{O}_{X'_2}
$$
This comes with a surjective map $\alpha : \mathcal{E} \to \mathcal{O}_{X_1}$
and hence we can use $\NL(\alpha)$ instead of
$\NL_{\mathcal{O}_{X_1}/\mathcal{O}_{S_1}}$, see
Modules, Lemma \ref{modules-lemma-NL-up-to-qis}.
Set
$$
\mathcal{I}' =
\text{Ker}(\mathcal{O}_{S'_1}[\mathcal{E}] \to \mathcal{O}_{X_1})
\quad\text{and}\quad
\mathcal{I} =
\text{Ker}(\mathcal{O}_{S_1}[\mathcal{E}] \to \mathcal{O}_{X_1})
$$
There is a surjection $\mathcal{I}' \to \mathcal{I}$ whose kernel
is $\mathcal{J}_1\mathcal{O}_{S'_1}[\mathcal{E}]$.
We obtain two homomorphisms of $\mathcal{O}_{S'_2}$-algebras
$$
a : \mathcal{O}_{S'_1}[\mathcal{E}] \to \mathcal{O}_{X'_1}
\quad\text{and}\quad
b : \mathcal{O}_{S'_1}[\mathcal{E}] \to \mathcal{O}_{X'_2}
$$
which induce maps $a|_{\mathcal{I}'} : \mathcal{I}' \to \mathcal{G}_1$ and
$b|_{\mathcal{I}'} : \mathcal{I}' \to \mathcal{G}_2$. Both $a$ and $b$
annihilate $(\mathcal{I}')^2$. Moreover $a$ and $b$ agree on
$\mathcal{J}_1\mathcal{O}_{S'_1}[\mathcal{E}]$ as maps into $\mathcal{G}_2$
because the left hand square of (\ref{equation-huge-2}) is commutative.
Thus the difference
$b|_{\mathcal{I}'} - \nu \circ a|_{\mathcal{I}'}$
induces a well defined $\mathcal{O}_{X_1}$-linear map
$$
\xi : \mathcal{I}/\mathcal{I}^2 \longrightarrow \mathcal{G}_2
$$
which sends the class of a local section $f$ of $\mathcal{I}$ to
$a(f') - \nu(b(f'))$ where $f'$ is a lift of $f$ to a local
section of $\mathcal{I}'$. We let
$[\xi] \in \text{Ext}^1_{\mathcal{O}_{X_1}}(\NL(\alpha), \mathcal{G}_2)$
be the image (see below).

\medskip\noindent
Step 2. Vanishing of $[\xi]$ is necessary. Let us write
$\Omega = \Omega_{\mathcal{O}_{S_1}[\mathcal{E}]/\mathcal{O}_{S_1}}
\otimes_{\mathcal{O}_{S_1}[\mathcal{E}]} \mathcal{O}_{X_1}$.
Observe that $\NL(\alpha) = (\mathcal{I}/\mathcal{I}^2 \to \Omega)$
fits into a distinguished triangle
$$
\Omega[0] \to
\NL(\alpha) \to
\mathcal{I}/\mathcal{I}^2[1] \to
\Omega[1]
$$
Thus we see that $[\xi]$ is zero if and only if $\xi$
is a composition $\mathcal{I}/\mathcal{I}^2 \to \Omega \to \mathcal{G}_2$
for some map $\Omega \to \mathcal{G}_2$. Suppose there exists a
homomorphisms of sheaves of rings
$\varphi : \mathcal{O}_{X'_1} \to \mathcal{O}_{X'_2}$ fitting into
(\ref{equation-huge-2}). In this case consider the map
$\mathcal{O}_{S'_1}[\mathcal{E}] \to \mathcal{G}_2$,
$f' \mapsto b(f') - \varphi(a(f'))$. A calculation
shows this annihilates $\mathcal{J}_1\mathcal{O}_{S'_1}[\mathcal{E}]$
and induces a derivation $\mathcal{O}_{S_1}[\mathcal{E}] \to \mathcal{G}_2$.
The resulting linear map $\Omega \to \mathcal{G}_2$ witnesses the
fact that $[\xi] = 0$ in this case.

\medskip\noindent
Step 3. Vanishing of $[\xi]$ is sufficient. Let
$\theta : \Omega \to \mathcal{G}_2$ be a $\mathcal{O}_{X_1}$-linear map
such that $\xi$ is equal to
$\theta \circ (\mathcal{I}/\mathcal{I}^2 \to \Omega)$.
Then a calculation shows that
$$
b + \theta \circ d : \mathcal{O}_{S'_1}[\mathcal{E}] \to \mathcal{O}_{X'_2}
$$
annihilates $\mathcal{I}'$ and hence defines a map
$\mathcal{O}_{X'_1} \to \mathcal{O}_{X'_2}$ fitting into
(\ref{equation-huge-2}).

\medskip\noindent
Proof of (2) in the special case above. Omitted. Hint:
This is exactly the same as the proof of (2) of Lemma \ref{lemma-huge-diagram}.
\end{proof}

\begin{lemma}
\label{lemma-NL-represent-ext-class}
Let $X$ be a topological space. Let $\mathcal{A} \to \mathcal{B}$ be a
homomorphism of sheaves of rings. Let $\mathcal{G}$ be a $\mathcal{B}$-module.
Let
$\xi \in \text{Ext}^1_\mathcal{B}(\NL_{\mathcal{B}/\mathcal{A}}, \mathcal{G})$. 
There exists a map of sheaves of sets $\alpha : \mathcal{E} \to \mathcal{B}$
such that $\xi \in \text{Ext}^1_\mathcal{B}(\NL(\alpha), \mathcal{G})$
is the class of a map $\mathcal{I}/\mathcal{I}^2 \to \mathcal{G}$
(see proof for notation).
\end{lemma}

\begin{proof}
Recall that given $\alpha : \mathcal{E} \to \mathcal{B}$
such that $\mathcal{A}[\mathcal{E}] \to \mathcal{B}$ is surjective
with kernel $\mathcal{I}$ the complex
$\NL(\alpha) = (\mathcal{I}/\mathcal{I}^2 \to 
\Omega_{\mathcal{A}[\mathcal{E}]/\mathcal{A}}
\otimes_{\mathcal{A}[\mathcal{E}]} \mathcal{B})$ is canonically
isomorphic to $\NL_{\mathcal{B}/\mathcal{A}}$, see
Modules, Lemma \ref{modules-lemma-NL-up-to-qis}.
Observe moreover, that
$\Omega = \Omega_{\mathcal{A}[\mathcal{E}]/\mathcal{A}}
\otimes_{\mathcal{A}[\mathcal{E}]} \mathcal{B}$ is the sheaf
associated to the presheaf
$U \mapsto \bigoplus_{e \in \mathcal{E}(U)} \mathcal{B}(U)$.
In other words, $\Omega$ is the free $\mathcal{B}$-module on the
sheaf of sets $\mathcal{E}$ and in particular there is a canonical
map $\mathcal{E} \to \Omega$.

\medskip\noindent
Having said this, pick some $\mathcal{E}$ (for example
$\mathcal{E} = \mathcal{B}$ as in the definition of the naive cotangent
complex). The obstruction to writing $\xi$ as the class of a map
$\mathcal{I}/\mathcal{I}^2 \to \mathcal{G}$ is an element in
$\text{Ext}^1_\mathcal{B}(\Omega, \mathcal{G})$. Say this is represented
by the extension $0 \to \mathcal{G} \to \mathcal{H} \to \Omega \to 0$
of $\mathcal{B}$-modules. Consider the sheaf of sets
$\mathcal{E}' = \mathcal{E} \times_\Omega \mathcal{H}$
which comes with an induced map $\alpha' : \mathcal{E}' \to \mathcal{B}$.
Let $\mathcal{I}' = \text{Ker}(\mathcal{A}[\mathcal{E}'] \to \mathcal{B})$
and $\Omega' = \Omega_{\mathcal{A}[\mathcal{E}']/\mathcal{A}}
\otimes_{\mathcal{A}[\mathcal{E}']} \mathcal{B}$.
The pullback of $\xi$ under the quasi-isomorphism
$\NL(\alpha') \to \NL(\alpha)$ maps to zero in
$\text{Ext}^1_\mathcal{B}(\Omega', \mathcal{G})$
because the pullback of the extension $\mathcal{H}$
by the map $\Omega' \to \Omega$ is split as $\Omega'$ is the free
$\mathcal{B}$-module on the sheaf of sets $\mathcal{E}'$ and since
by construction there is a commutative diagram
$$
\xymatrix{
\mathcal{E}' \ar[r] \ar[d] & \mathcal{E} \ar[d] \\
\mathcal{H} \ar[r] & \Omega
}
$$
This finishes the proof.
\end{proof}

\begin{lemma}
\label{lemma-choices-ringed-spaces}
If there exists a solution to (\ref{equation-to-solve-ringed-spaces}),
then the set of isomorphism classes of solutions is principal homogeneous
under $\text{Ext}^1_{\mathcal{O}_X}(\NL_{X/S}, \mathcal{G})$.
\end{lemma}

\begin{proof}
We observe right away that given two solutions $X'_1$ and $X'_2$
to (\ref{equation-to-solve-ringed-spaces}) we obtain by
Lemma \ref{lemma-huge-diagram-ringed-spaces} an obstruction element
$o(X'_1, X'_2) \in \text{Ext}^1_{\mathcal{O}_X}(\NL_{X/S}, \mathcal{G})$
to the existence of a map $X'_1 \to X'_2$. Clearly, this element
is the obstruction to the existence of an isomorphism, hence separates
the isomorphism classes. To finish the proof it therefore suffices to
show that given a solution $X'$ and an element
$\xi \in \text{Ext}^1_{\mathcal{O}_X}(\NL_{X/S}, \mathcal{G})$
we can find a second solution $X'_\xi$ such that
$o(X', X'_\xi) = \xi$.

\medskip\noindent
Pick $\alpha : \mathcal{E} \to \mathcal{O}_X$ as in
Lemma \ref{lemma-NL-represent-ext-class}
for the class $\xi$. Consider the surjection
$f^{-1}\mathcal{O}_S[\mathcal{E}] \to \mathcal{O}_X$
with kernel $\mathcal{I}$ and corresponding naive cotangent complex
$\NL(\alpha) = (\mathcal{I}/\mathcal{I}^2 \to
\Omega_{f^{-1}\mathcal{O}_S[\mathcal{E}]/f^{-1}\mathcal{O}_S}
\otimes_{f^{-1}\mathcal{O}_S[\mathcal{E}]} \mathcal{O}_X)$.
By the lemma $\xi$ is the class of a morphism
$\delta : \mathcal{I}/\mathcal{I}^2 \to \mathcal{G}$.
After replacing $\mathcal{E}$ by
$\mathcal{E} \times_{\mathcal{O}_X} \mathcal{O}_{X'}$ we may also assume
that $\alpha$ factors through a map
$\alpha' : \mathcal{E} \to \mathcal{O}_{X'}$.

\medskip\noindent
These choices determine an $f^{-1}\mathcal{O}_{S'}$-algebra map
$\varphi : \mathcal{O}_{S'}[\mathcal{E}] \to \mathcal{O}_{X'}$.
Let $\mathcal{I}' = \text{Ker}(\varphi)$.
Observe that $\varphi$ induces a map
$\varphi|_{\mathcal{I}'} : \mathcal{I}' \to \mathcal{G}$
and that $\mathcal{O}_{X'}$ is the pushout, as in the following
diagram
$$
\xymatrix{
0 \ar[r] & \mathcal{G} \ar[r] & \mathcal{O}_{X'} \ar[r] &
\mathcal{O}_X \ar[r] & 0 \\
0 \ar[r] & \mathcal{I}' \ar[u]^{\varphi|_{\mathcal{I}'}} \ar[r] &
f^{-1}\mathcal{O}_{S'}[\mathcal{E}] \ar[u] \ar[r] &
\mathcal{O}_X \ar[u]_{=} \ar[r] & 0
}
$$
Let $\psi : \mathcal{I}' \to \mathcal{G}$ be the sum of the map
$\varphi|_{\mathcal{I}'}$ and the composition
$$
\mathcal{I}' \to \mathcal{I}'/(\mathcal{I}')^2 \to
\mathcal{I}/\mathcal{I}^2 \xrightarrow{\delta} \mathcal{G}.
$$
Then the pushout along $\psi$ is an other ring extension
$\mathcal{O}_{X'_\xi}$ fitting into a diagram as above.
A calculation (omitted) shows that $o(X', X'_\xi) = \xi$ as desired.
\end{proof}

\begin{lemma}
\label{lemma-extensions-of-ringed-spaces}
Let $(S, \mathcal{O}_S)$ be a ringed space and let $\mathcal{J}$
be an $\mathcal{O}_S$-module.
\begin{enumerate}
\item The set of extensions of sheaves of rings
$0 \to \mathcal{J} \to \mathcal{O}_{S'} \to \mathcal{O}_S \to 0$
where $\mathcal{J}$ is an ideal of square zero is canonically bijective to
$\text{Ext}^1_{\mathcal{O}_S}(\NL_{S/\mathbf{Z}}, \mathcal{J})$.
\item Given a morphism of ringed spaces
$f : (X, \mathcal{O}_X) \to (S, \mathcal{O}_S)$, an $\mathcal{O}_X$-module
$\mathcal{G}$, an $f$-map $c : \mathcal{J} \to \mathcal{G}$, and
given extensions of sheaves of rings with square zero kernels:
\begin{enumerate}
\item[(a)] $0 \to \mathcal{J} \to \mathcal{O}_{S'} \to \mathcal{O}_S \to 0$
corresponding to
$\alpha \in \text{Ext}^1_{\mathcal{O}_S}(\NL_{S/\mathbf{Z}}, \mathcal{J})$,
\item[(b)] $0 \to \mathcal{G} \to \mathcal{O}_{X'} \to \mathcal{O}_X \to 0$
corresponding to
$\beta \in \text{Ext}^1_{\mathcal{O}_X}(\NL_{X/\mathbf{Z}}, \mathcal{G})$
\end{enumerate}
then there is a morphism $X' \to S'$ fitting into a diagram
(\ref{equation-to-solve-ringed-spaces}) if and only if $\beta$ and $\alpha$
map to the same element of
$\text{Ext}^1_{\mathcal{O}_X}(Lf^*\NL_{S/\mathbf{Z}}, \mathcal{G})$.
\end{enumerate}
\end{lemma}

\begin{proof}
To prove this we apply the previous results where we work over
the base ringed space $(*, \mathbf{Z})$ with trivial thickening.
Part (1) follows from Lemma \ref{lemma-choices-ringed-spaces}
and the fact that there exists a solution, namely
$\mathcal{J} \oplus \mathcal{O}_S$.
Part (2) follows from Lemma \ref{lemma-huge-diagram-ringed-spaces}
and a compatibility between the constructions in the proofs
of Lemmas \ref{lemma-choices-ringed-spaces} and
\ref{lemma-huge-diagram-ringed-spaces}
whose statement and proof we omit.
\end{proof}









\section{Thickenings of ringed topoi}
\label{section-thickenings-ringed-topoi}

\noindent
This section is the analogue of Section \ref{section-thickenings-spaces}
for ringed topoi.
In the following few sections we will use the following notions:
\begin{enumerate}
\item A sheaf of ideals $\mathcal{I} \subset \mathcal{O}'$ on
a ringed topos $(\Sh(\mathcal{D}), \mathcal{O}')$ is {\it locally nilpotent}
if any local section of $\mathcal{I}$ is locally nilpotent.
\item A {\it thickening} of ringed topoi is a morphism
$i : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{D}), \mathcal{O}')$
of ringed topoi such that
\begin{enumerate}
\item $i_*$ is an equivalence $\Sh(\mathcal{C}) \to \Sh(\mathcal{D})$,
\item the map $i^\sharp : \mathcal{O}' \to i_*\mathcal{O}$
is surjective, and
\item the kernel of $i^\sharp$ is a locally nilpotent sheaf of ideals.
\end{enumerate}
\item A {\it first order thickening} of ringed topoi is a thickening
$i : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{D}), \mathcal{O}')$
of ringed topoi such that $\text{Ker}(i^\sharp)$ has square zero.
\item It is clear how to define
{\it morphisms of thickenings of ringed topoi},
{\it morphisms of thickenings of ringed topoi over a base ringed topos}, etc.
\end{enumerate}
If
$i : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{D}), \mathcal{O}')$
is a thickening of ringed topoi then we identify the underlying topoi
and think of $\mathcal{O}$, $\mathcal{O}'$, and
$\mathcal{I} = \text{Ker}(i^\sharp)$ as sheaves on $\mathcal{C}$.
We obtain a short exact sequence
$$
0 \to \mathcal{I} \to \mathcal{O}' \to \mathcal{O} \to 0
$$
of $\mathcal{O}'$-modules. By
Modules on Sites, Lemma \ref{sites-modules-lemma-i-star-equivalence}
the category of $\mathcal{O}$-modules is equivalent to the category
of $\mathcal{O}'$-modules annihilated by $\mathcal{I}$. In particular,
if $i$ is a first order thickening, then
$\mathcal{I}$ is a $\mathcal{O}$-module.

\begin{situation}
\label{situation-morphism-thickenings-ringed-topoi}
A morphism of thickenings of ringed topoi $(f, f')$
is given by a commutative diagram
\begin{equation}
\label{equation-morphism-thickenings-ringed-topoi}
\vcenter{
\xymatrix{
(\Sh(\mathcal{C}), \mathcal{O}) \ar[r]_i \ar[d]_f &
(\Sh(\mathcal{D}), \mathcal{O}') \ar[d]^{f'} \\
(\Sh(\mathcal{B}), \mathcal{O}_\mathcal{B}) \ar[r]^t &
(\Sh(\mathcal{B}'), \mathcal{O}_{\mathcal{B}'})
}
}
\end{equation}
of ringed topoi whose horizontal arrows are thickenings. In this
situation we set
$\mathcal{I} = \text{Ker}(i^\sharp) \subset \mathcal{O}'$ and
$\mathcal{J} = \text{Ker}(t^\sharp) \subset \mathcal{O}_{\mathcal{B}'}$.
As $f = f'$ on underlying topoi we will identify
the pullback functors $f^{-1}$ and $(f')^{-1}$.
Observe that
$(f')^\sharp : f^{-1}\mathcal{O}_{\mathcal{B}'} \to \mathcal{O}'$
induces in particular a map $f^{-1}\mathcal{J} \to \mathcal{I}$
and therefore a map of $\mathcal{O}'$-modules
$$
(f')^*\mathcal{J} \longrightarrow \mathcal{I}
$$
If $i$ and $t$ are first order thickenings, then
$(f')^*\mathcal{J} = f^*\mathcal{J}$ and the map above becomes a
map $f^*\mathcal{J} \to \mathcal{I}$.
\end{situation}

\begin{definition}
\label{definition-strict-morphism-thickenings-ringed-topoi}
In Situation \ref{situation-morphism-thickenings-ringed-topoi}
we say that $(f, f')$ is a {\it strict morphism of thickenings}
if the map $(f')^*\mathcal{J} \longrightarrow \mathcal{I}$ is surjective.
\end{definition}








\section{Modules on first order thickenings of ringed topoi}
\label{section-modules-thickenings-ringed-topoi}

\noindent
In this section we discuss some preliminaries to the deformation theory
of modules. Let
$i : (\Sh(\mathcal{C}, \mathcal{O}) \to (\Sh(\mathcal{D}), \mathcal{O}')$
be a first order thickening of ringed topoi. We will freely use the notation
introduced in Section \ref{section-thickenings-ringed-topoi},
in particular we will identify the underlying topological topoi.
In this section we consider short exact sequences
\begin{equation}
\label{equation-extension-ringed-topoi}
0 \to \mathcal{K} \to \mathcal{F}' \to \mathcal{F} \to 0
\end{equation}
of $\mathcal{O}'$-modules, where $\mathcal{F}$, $\mathcal{K}$ are
$\mathcal{O}$-modules and $\mathcal{F}'$ is an $\mathcal{O}'$-module.
In this situation we have a canonical $\mathcal{O}$-module map
$$
c_{\mathcal{F}'} :
\mathcal{I} \otimes_\mathcal{O} \mathcal{F}
\longrightarrow
\mathcal{K}
$$
where $\mathcal{I} = \text{Ker}(i^\sharp)$.
Namely, given local sections $f$ of $\mathcal{I}$ and $s$
of $\mathcal{F}$ we set $c_{\mathcal{F}'}(f \otimes s) = fs'$
where $s'$ is a local section of $\mathcal{F}'$ lifting $s$.

\begin{lemma}
\label{lemma-inf-map-ringed-topoi}
Let $i : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{D}), \mathcal{O}')$
be a first order thickening of ringed topoi. Assume given
extensions
$$
0 \to \mathcal{K} \to \mathcal{F}' \to \mathcal{F} \to 0
\quad\text{and}\quad
0 \to \mathcal{L} \to \mathcal{G}' \to \mathcal{G} \to 0
$$
as in (\ref{equation-extension-ringed-topoi})
and maps $\varphi : \mathcal{F} \to \mathcal{G}$ and
$\psi : \mathcal{K} \to \mathcal{L}$.
\begin{enumerate}
\item If there exists an $\mathcal{O}'$-module
map $\varphi' : \mathcal{F}' \to \mathcal{G}'$ compatible with $\varphi$
and $\psi$, then the diagram
$$
\xymatrix{
\mathcal{I} \otimes_\mathcal{O} \mathcal{F}
\ar[r]_-{c_{\mathcal{F}'}} \ar[d]_{1 \otimes \varphi} &
\mathcal{K} \ar[d]^\psi \\
\mathcal{I} \otimes_\mathcal{O} \mathcal{G}
\ar[r]^-{c_{\mathcal{G}'}} &
\mathcal{L}
}
$$
is commutative.
\item The set of $\mathcal{O}'$-module
maps $\varphi' : \mathcal{F}' \to \mathcal{G}'$ compatible with $\varphi$
and $\psi$ is, if nonempty, a principal homogeneous space under
$\Hom_\mathcal{O}(\mathcal{F}, \mathcal{L})$.
\end{enumerate}
\end{lemma}

\begin{proof}
Part (1) is immediate from the description of the maps.
For (2), if $\varphi'$ and $\varphi''$ are two maps
$\mathcal{F}' \to \mathcal{G}'$ compatible with $\varphi$
and $\psi$, then $\varphi' - \varphi''$ factors as
$$
\mathcal{F}' \to \mathcal{F} \to \mathcal{L} \to \mathcal{G}'
$$
The map in the middle comes from a unique element of
$\Hom_\mathcal{O}(\mathcal{F}, \mathcal{L})$ by
Modules on Sites, Lemma \ref{sites-modules-lemma-i-star-equivalence}.
Conversely, given an element $\alpha$ of this group we can add the
composition (as displayed above with $\alpha$ in the middle)
to $\varphi'$. Some details omitted.
\end{proof}

\begin{lemma}
\label{lemma-inf-obs-map-ringed-topoi}
Let $i : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{D}), \mathcal{O}')$
be a first order thickening of ringed topoi. Assume given extensions
$$
0 \to \mathcal{K} \to \mathcal{F}' \to \mathcal{F} \to 0
\quad\text{and}\quad
0 \to \mathcal{L} \to \mathcal{G}' \to \mathcal{G} \to 0
$$
as in (\ref{equation-extension-ringed-topoi})
and maps $\varphi : \mathcal{F} \to \mathcal{G}$ and
$\psi : \mathcal{K} \to \mathcal{L}$. Assume the diagram
$$
\xymatrix{
\mathcal{I} \otimes_\mathcal{O} \mathcal{F}
\ar[r]_-{c_{\mathcal{F}'}} \ar[d]_{1 \otimes \varphi} &
\mathcal{K} \ar[d]^\psi \\
\mathcal{I} \otimes_\mathcal{O} \mathcal{G}
\ar[r]^-{c_{\mathcal{G}'}} &
\mathcal{L}
}
$$
is commutative. Then there exists an element
$$
o(\varphi, \psi) \in
\text{Ext}^1_\mathcal{O}(\mathcal{F}, \mathcal{L})
$$
whose vanishing is a necessary and sufficient condition for the existence
of a map $\varphi' : \mathcal{F}' \to \mathcal{G}'$ compatible with
$\varphi$ and $\psi$.
\end{lemma}

\begin{proof}
We can construct explicitly an extension
$$
0 \to \mathcal{L} \to \mathcal{H} \to \mathcal{F} \to 0
$$
by taking $\mathcal{H}$ to be the cohomology of the complex
$$
\mathcal{K}
\xrightarrow{1, - \psi}
\mathcal{F}' \oplus \mathcal{G}' \xrightarrow{\varphi, 1}
\mathcal{G}
$$
in the middle (with obvious notation). A calculation with local sections
using the assumption that the diagram of the lemma commutes
shows that $\mathcal{H}$ is annihilated by $\mathcal{I}$. Hence
$\mathcal{H}$ defines a class in
$$
\text{Ext}^1_\mathcal{O}(\mathcal{F}, \mathcal{L})
\subset
\text{Ext}^1_{\mathcal{O}'}(\mathcal{F}, \mathcal{L})
$$
Finally, the class of $\mathcal{H}$ is the difference of the pushout
of the extension $\mathcal{F}'$ via $\psi$ and the pullback
of the extension $\mathcal{G}'$ via $\varphi$ (calculations omitted).
Thus the vanishing of the class of $\mathcal{H}$ is equivalent to the
existence of a commutative diagram
$$
\xymatrix{
0 \ar[r] &
\mathcal{K} \ar[r] \ar[d]_{\psi} &
\mathcal{F}' \ar[r] \ar[d]_{\varphi'} &
\mathcal{F} \ar[r] \ar[d]_\varphi & 0\\
0 \ar[r] &
\mathcal{L} \ar[r] &
\mathcal{G}' \ar[r] &
\mathcal{G} \ar[r] & 0
}
$$
as desired.
\end{proof}

\begin{lemma}
\label{lemma-inf-ext-ringed-topoi}
Let $i : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{D}), \mathcal{O}')$
be a first order thickening of ringed topoi. Assume given
$\mathcal{O}$-modules $\mathcal{F}$, $\mathcal{K}$
and an $\mathcal{O}$-linear map
$c : \mathcal{I} \otimes_\mathcal{O} \mathcal{F} \to \mathcal{K}$.
If there exists a sequence (\ref{equation-extension-ringed-topoi}) with
$c_{\mathcal{F}'} = c$ then the set of isomorphism classes of these
extensions is principal homogeneous under
$\text{Ext}^1_\mathcal{O}(\mathcal{F}, \mathcal{K})$.
\end{lemma}

\begin{proof}
Assume given extensions
$$
0 \to \mathcal{K} \to \mathcal{F}'_1 \to \mathcal{F} \to 0
\quad\text{and}\quad
0 \to \mathcal{K} \to \mathcal{F}'_2 \to \mathcal{F} \to 0
$$
with $c_{\mathcal{F}'_1} = c_{\mathcal{F}'_2} = c$. Then the difference
(in the extension group, see
Homology, Section \ref{homology-section-extensions})
is an extension
$$
0 \to \mathcal{K} \to \mathcal{E} \to \mathcal{F} \to 0
$$
where $\mathcal{E}$ is annihilated by $\mathcal{I}$ (local computation
omitted). Hence the sequence is an extension of $\mathcal{O}$-modules,
see Modules on Sites, Lemma \ref{sites-modules-lemma-i-star-equivalence}.
Conversely, given such an extension $\mathcal{E}$ we can add the extension
$\mathcal{E}$ to the $\mathcal{O}'$-extension $\mathcal{F}'$ without
affecting the map $c_{\mathcal{F}'}$. Some details omitted.
\end{proof}

\begin{lemma}
\label{lemma-inf-obs-ext-ringed-topoi}
Let $i : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{D}), \mathcal{O}')$
be a first order thickening of ringed topoi. Assume given
$\mathcal{O}$-modules $\mathcal{F}$, $\mathcal{K}$
and an $\mathcal{O}$-linear map
$c : \mathcal{I} \otimes_\mathcal{O} \mathcal{F} \to \mathcal{K}$.
Then there exists an element
$$
o(\mathcal{F}, \mathcal{K}, c) \in
\text{Ext}^2_\mathcal{O}(\mathcal{F}, \mathcal{K})
$$
whose vanishing is a necessary and sufficient condition for the existence
of a sequence (\ref{equation-extension-ringed-topoi})
with $c_{\mathcal{F}'} = c$.
\end{lemma}

\begin{proof}
We first show that if $\mathcal{K}$ is an injective $\mathcal{O}$-module,
then there does exist a sequence (\ref{equation-extension-ringed-topoi}) with
$c_{\mathcal{F}'} = c$. To do this, choose a flat
$\mathcal{O}'$-module $\mathcal{H}'$ and a surjection
$\mathcal{H}' \to \mathcal{F}$
(Modules on Sites, Lemma \ref{sites-modules-lemma-module-quotient-flat}).
Let $\mathcal{J} \subset \mathcal{H}'$ be the kernel. Since $\mathcal{H}'$
is flat we have
$$
\mathcal{I} \otimes_{\mathcal{O}'} \mathcal{H}' =
\mathcal{I}\mathcal{H}'
\subset \mathcal{J} \subset \mathcal{H}'
$$
Observe that the map
$$
\mathcal{I}\mathcal{H}' =
\mathcal{I} \otimes_{\mathcal{O}'} \mathcal{H}'
\longrightarrow
\mathcal{I} \otimes_{\mathcal{O}'} \mathcal{F} =
\mathcal{I} \otimes_\mathcal{O} \mathcal{F}
$$
annihilates $\mathcal{I}\mathcal{J}$. Namely, if $f$ is a local section
of $\mathcal{I}$ and $s$ is a local section of $\mathcal{H}$, then
$fs$ is mapped to $f \otimes \overline{s}$ where $\overline{s}$ is
the image of $s$ in $\mathcal{F}$. Thus we obtain
$$
\xymatrix{
\mathcal{I}\mathcal{H}'/\mathcal{I}\mathcal{J}
\ar@{^{(}->}[r] \ar[d] &
\mathcal{J}/\mathcal{I}\mathcal{J} \ar@{..>}[d]_\gamma \\
\mathcal{I} \otimes_\mathcal{O} \mathcal{F} \ar[r]^-c &
\mathcal{K}
}
$$
a diagram of $\mathcal{O}$-modules. If $\mathcal{K}$ is injective
as an $\mathcal{O}$-module, then we obtain the dotted arrow.
Denote $\gamma' : \mathcal{J} \to \mathcal{K}$ the composition
of $\gamma$ with $\mathcal{J} \to \mathcal{J}/\mathcal{I}\mathcal{J}$.
A local calculation shows the pushout
$$
\xymatrix{
0 \ar[r] &
\mathcal{J} \ar[r] \ar[d]_{\gamma'} &
\mathcal{H}' \ar[r] \ar[d] &
\mathcal{F} \ar[r] \ar@{=}[d] &
0 \\
0 \ar[r] &
\mathcal{K} \ar[r] &
\mathcal{F}' \ar[r] &
\mathcal{F} \ar[r] &
0
}
$$
is a solution to the problem posed by the lemma.

\medskip\noindent
General case. Choose an embedding $\mathcal{K} \subset \mathcal{K}'$
with $\mathcal{K}'$ an injective $\mathcal{O}$-module. Let $\mathcal{Q}$
be the quotient, so that we have an exact sequence
$$
0 \to \mathcal{K} \to \mathcal{K}' \to \mathcal{Q} \to 0
$$
Denote
$c' : \mathcal{I} \otimes_\mathcal{O} \mathcal{F} \to \mathcal{K}'$
be the composition. By the paragraph above there exists a sequence
$$
0 \to \mathcal{K}' \to \mathcal{E}' \to \mathcal{F} \to 0
$$
as in (\ref{equation-extension-ringed-topoi}) with $c_{\mathcal{E}'} = c'$.
Note that $c'$ composed with the map $\mathcal{K}' \to \mathcal{Q}$
is zero, hence the pushout of $\mathcal{E}'$ by
$\mathcal{K}' \to \mathcal{Q}$ is an extension
$$
0 \to \mathcal{Q} \to \mathcal{D}' \to \mathcal{F} \to 0
$$
as in (\ref{equation-extension-ringed-topoi}) with $c_{\mathcal{D}'} = 0$.
This means exactly that $\mathcal{D}'$ is annihilated by
$\mathcal{I}$, in other words, the $\mathcal{D}'$ is an extension
of $\mathcal{O}$-modules, i.e., defines an element
$$
o(\mathcal{F}, \mathcal{K}, c) \in
\text{Ext}^1_\mathcal{O}(\mathcal{F}, \mathcal{Q}) =
\text{Ext}^2_\mathcal{O}(\mathcal{F}, \mathcal{K})
$$
(the equality holds by the long exact cohomology sequence associated
to the exact sequence above and the vanishing of higher ext groups
into the injective module $\mathcal{K}'$). If
$o(\mathcal{F}, \mathcal{K}, c) = 0$, then we can choose a splitting
$s : \mathcal{F} \to \mathcal{D}'$ and we can set
$$
\mathcal{F}' = \text{Ker}(\mathcal{E}' \to \mathcal{D}'/s(\mathcal{F}))
$$
so that we obtain the following diagram
$$
\xymatrix{
0 \ar[r] &
\mathcal{K} \ar[r] \ar[d] &
\mathcal{F}' \ar[r] \ar[d] &
\mathcal{F} \ar[r] \ar@{=}[d] &
0 \\
0 \ar[r] &
\mathcal{K}' \ar[r] &
\mathcal{E}' \ar[r] &
\mathcal{F} \ar[r] & 0
}
$$
with exact rows which shows that $c_{\mathcal{F}'} = c$. Conversely, if
$\mathcal{F}'$ exists, then the pushout of $\mathcal{F}'$ by the map
$\mathcal{K} \to \mathcal{K}'$ is isomorphic to $\mathcal{E}'$ by
Lemma \ref{lemma-inf-ext-ringed-topoi} and the vanishing of higher ext groups
into the injective module $\mathcal{K}'$. This gives a diagram
as above, which implies that $\mathcal{D}'$ is split as an extension, i.e.,
the class $o(\mathcal{F}, \mathcal{K}, c)$ is zero.
\end{proof}

\begin{remark}
\label{remark-trivial-thickening-ringed-topoi}
Let $(\Sh(\mathcal{C}), \mathcal{O})$ be a ringed topos. A first order
thickening $i : (\Sh(\mathcal{C}), \mathcal{O}) \to
(\Sh(\mathcal{D}), \mathcal{O}')$ is said
to be {\it trivial} if there exists a morphism of ringed topoi
$\pi : (\Sh(\mathcal{D}), \mathcal{O}') \to (\Sh(\mathcal{C}), \mathcal{O})$
which is a left inverse to $i$. The choice of such a morphism
$\pi$ is called a {\it trivialization} of the first order thickening.
Given $\pi$ we obtain a splitting
\begin{equation}
\label{equation-splitting-ringed-topoi}
\mathcal{O}' = \mathcal{O} \oplus \mathcal{I}
\end{equation}
as sheaves of algebras on $\mathcal{C}$ by using $\pi^\sharp$
to split the surjection $\mathcal{O}' \to \mathcal{O}$.
Conversely, such a splitting determines
a morphism $\pi$. The category of trivialized first order thickenings of
$(\Sh(\mathcal{C}), \mathcal{O})$ is equivalent to the category of 
$\mathcal{O}$-modules.
\end{remark}

\begin{remark}
\label{remark-trivial-extension-ringed-topoi}
Let $i : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{D}), \mathcal{O}')$
be a trivial first order thickening of ringed topoi
and let $\pi : (\Sh(\mathcal{D}), \mathcal{O}') \to
(\Sh(\mathcal{C}), \mathcal{O})$ be a trivialization. Then given any triple
$(\mathcal{F}, \mathcal{K}, c)$ consisting of a pair of
$\mathcal{O}$-modules and a map
$c : \mathcal{I} \otimes_\mathcal{O} \mathcal{F} \to \mathcal{K}$
we may set
$$
\mathcal{F}'_{c, triv} = \mathcal{F} \oplus \mathcal{K}
$$
and use the splitting (\ref{equation-splitting-ringed-topoi})
associated to $\pi$ and the map $c$ to define the $\mathcal{O}'$-module
structure and obtain an extension (\ref{equation-extension-ringed-topoi}).
We will call $\mathcal{F}'_{c, triv}$ the {\it trivial extension} of
$\mathcal{F}$ by $\mathcal{K}$ corresponding
to $c$ and the trivialization $\pi$. Given any extension
$\mathcal{F}'$ as in (\ref{equation-extension-ringed-topoi}) we can use
$\pi^\sharp : \mathcal{O} \to \mathcal{O}'$ to think of $\mathcal{F}'$
as an $\mathcal{O}$-module extension, hence a class $\xi_{\mathcal{F}'}$
in $\text{Ext}^1_\mathcal{O}(\mathcal{F}, \mathcal{K})$.
Lemma \ref{lemma-inf-ext-ringed-topoi} assures that
$\mathcal{F}' \mapsto \xi_{\mathcal{F}'}$
induces a bijection
$$
\left\{
\begin{matrix}
\text{isomorphism classes of extensions}\\
\mathcal{F}'\text{ as in (\ref{equation-extension-ringed-topoi}) with }
c = c_{\mathcal{F}'}
\end{matrix}
\right\}
\longrightarrow
\text{Ext}^1_\mathcal{O}(\mathcal{F}, \mathcal{K})
$$
Moreover, the trivial extension $\mathcal{F}'_{c, triv}$ maps to the zero class.
\end{remark}

\begin{remark}
\label{remark-extension-functorial-ringed-topoi}
Let $(\Sh(\mathcal{C}), \mathcal{O})$ be a ringed topos. Let
$(\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{D}_i), \mathcal{O}'_i)$,
$i = 1, 2$ be first order thickenings with ideal sheaves $\mathcal{I}_i$.
Let $h : (\Sh(\mathcal{D}_1), \mathcal{O}'_1) \to
(\Sh(\mathcal{D}_2), \mathcal{O}'_2)$
be a morphism of first order thickenings of $(\Sh(\mathcal{C}), \mathcal{O})$.
Picture
$$
\xymatrix{
& (\Sh(\mathcal{C}), \mathcal{O}) \ar[ld] \ar[rd] & \\
(\Sh(\mathcal{D}_1), \mathcal{O}'_1) \ar[rr]^h & & 
(\Sh(\mathcal{D}_2), \mathcal{O}'_2)
}
$$
Observe that $h^\sharp : \mathcal{O}'_2 \to \mathcal{O}'_1$
in particular induces an $\mathcal{O}$-module map
$\mathcal{I}_2 \to \mathcal{I}_1$.
Let $\mathcal{F}$ be an $\mathcal{O}$-module.
Let $(\mathcal{K}_i, c_i)$, $i = 1, 2$ be a pair
consisting of an $\mathcal{O}$-module $\mathcal{K}_i$ and a map
$c_i : \mathcal{I}_i \otimes_\mathcal{O} \mathcal{F} \to
\mathcal{K}_i$. Assume furthermore given a map
of $\mathcal{O}$-modules $\mathcal{K}_2 \to \mathcal{K}_1$
such that
$$
\xymatrix{
\mathcal{I}_2 \otimes_\mathcal{O} \mathcal{F}
\ar[r]_-{c_2} \ar[d] &
\mathcal{K}_2 \ar[d] \\
\mathcal{I}_1 \otimes_\mathcal{O} \mathcal{F}
\ar[r]^-{c_1} &
\mathcal{K}_1
}
$$
is commutative. Then there is a canonical functoriality
$$
\left\{
\begin{matrix}
\mathcal{F}'_2\text{ as in (\ref{equation-extension-ringed-topoi}) with }\\
c_2 = c_{\mathcal{F}'_2}\text{ and }\mathcal{K} = \mathcal{K}_2
\end{matrix}
\right\}
\longrightarrow
\left\{
\begin{matrix}
\mathcal{F}'_1\text{ as in (\ref{equation-extension-ringed-topoi}) with }\\
c_1 = c_{\mathcal{F}'_1}\text{ and }\mathcal{K} = \mathcal{K}_1
\end{matrix}
\right\}
$$
Namely, thinking of all sheaves $\mathcal{O}$, $\mathcal{O}'_i$,
$\mathcal{F}$, $\mathcal{K}_i$, etc as sheaves on $\mathcal{C}$, we set
given $\mathcal{F}'_2$ the sheaf $\mathcal{F}'_1$ equal to the
pushout, i.e., fitting into the following diagram of extensions
$$
\xymatrix{
0 \ar[r] &
\mathcal{K}_2 \ar[r] \ar[d] &
\mathcal{F}'_2 \ar[r] \ar[d] &
\mathcal{F} \ar@{=}[d] \ar[r] & 0 \\
0 \ar[r] &
\mathcal{K}_1 \ar[r] &
\mathcal{F}'_1 \ar[r] &
\mathcal{F} \ar[r] & 0
}
$$
We omit the construction of the $\mathcal{O}'_1$-module structure
on the pushout (this uses the commutativity of the diagram
involving $c_1$ and $c_2$).
\end{remark}

\begin{remark}
\label{remark-trivial-extension-functorial-ringed-topoi}
Let $(\Sh(\mathcal{C}), \mathcal{O})$,
$(\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{D}_i), \mathcal{O}'_i)$,
$\mathcal{I}_i$, and $h : (\Sh(\mathcal{D}_1), \mathcal{O}'_1) \to
(\Sh(\mathcal{D}_2), \mathcal{O}'_2)$ be as in
Remark \ref{remark-extension-functorial-ringed-topoi}.
Assume that we are given given trivializations
$\pi_i : (\Sh(\mathcal{D}_i), \mathcal{O}'_i) \to
(\Sh(\mathcal{C}), \mathcal{O})$ such that
$\pi_1 = h \circ \pi_2$. In other words, assume $h$ is a morphism
of trivialized first order thickenings of $(\Sh(\mathcal{C}), \mathcal{O})$.
Let $(\mathcal{K}_i, c_i)$, $i = 1, 2$ be a pair consisting of an
$\mathcal{O}$-module $\mathcal{K}_i$ and a map
$c_i : \mathcal{I}_i \otimes_\mathcal{O} \mathcal{F} \to
\mathcal{K}_i$. Assume furthermore given a map
of $\mathcal{O}$-modules $\mathcal{K}_2 \to \mathcal{K}_1$
such that
$$
\xymatrix{
\mathcal{I}_2 \otimes_\mathcal{O} \mathcal{F}
\ar[r]_-{c_2} \ar[d] &
\mathcal{K}_2 \ar[d] \\
\mathcal{I}_1 \otimes_\mathcal{O} \mathcal{F}
\ar[r]^-{c_1} &
\mathcal{K}_1
}
$$
is commutative. In this situation the construction of
Remark \ref{remark-trivial-extension-ringed-topoi} induces
a commutative diagram
$$
\xymatrix{
\{\mathcal{F}'_2\text{ as in (\ref{equation-extension-ringed-topoi}) with }
c_2 = c_{\mathcal{F}'_2}\text{ and }\mathcal{K} = \mathcal{K}_2\}
\ar[d] \ar[rr] & &
\text{Ext}^1_\mathcal{O}(\mathcal{F}, \mathcal{K}_2) \ar[d] \\
\{\mathcal{F}'_1\text{ as in (\ref{equation-extension-ringed-topoi}) with }
c_1 = c_{\mathcal{F}'_1}\text{ and }\mathcal{K} = \mathcal{K}_1\}
\ar[rr] & &
\text{Ext}^1_\mathcal{O}(\mathcal{F}, \mathcal{K}_1)
}
$$
where the vertical map on the right is given by functoriality of $\text{Ext}$
and the map $\mathcal{K}_2 \to \mathcal{K}_1$ and the vertical map on the left
is the one from Remark \ref{remark-extension-functorial-ringed-topoi}.
\end{remark}

\begin{remark}
\label{remark-short-exact-sequence-thickenings-ringed-topoi}
Let $(\Sh(\mathcal{C}), \mathcal{O})$ be a ringed topos.
We define a sequence of morphisms of first order thickenings
$$
(\Sh(\mathcal{D}_1), \mathcal{O}'_1) \to
(\Sh(\mathcal{D}_2), \mathcal{O}'_2) \to
(\Sh(\mathcal{D}_3), \mathcal{O}'_3)
$$
of $(\Sh(\mathcal{C}), \mathcal{O})$ to be a {\it complex}
if the corresponding maps between
the ideal sheaves $\mathcal{I}_i$
give a complex of $\mathcal{O}$-modules
$\mathcal{I}_3 \to \mathcal{I}_2 \to \mathcal{I}_1$
(i.e., the composition is zero). In this case the composition
$(\Sh(\mathcal{D}_1), \mathcal{O}'_1) \to
(\Sh(\mathcal{D}_3), \mathcal{O}'_3)$ factors through
$(\Sh(\mathcal{C}), \mathcal{O}) \to
(\Sh(\mathcal{D}_3), \mathcal{O}'_3)$, i.e.,
the first order thickening
$(\Sh(\mathcal{D}_1), \mathcal{O}'_1)$ of
$(\Sh(\mathcal{C}), \mathcal{O})$ is trivial and comes with
a canonical trivialization
$\pi : (\Sh(\mathcal{D}_1), \mathcal{O}'_1) \to
(\Sh(\mathcal{C}), \mathcal{O})$.

\medskip\noindent
We say a sequence of morphisms of first order thickenings
$$
(\Sh(\mathcal{D}_1), \mathcal{O}'_1) \to
(\Sh(\mathcal{D}_2), \mathcal{O}'_2) \to
(\Sh(\mathcal{D}_3), \mathcal{O}'_3)
$$
of $(\Sh(\mathcal{C}), \mathcal{O})$ is {\it a short exact sequence} if the
corresponding maps between ideal sheaves is a short exact sequence
$$
0 \to \mathcal{I}_3 \to \mathcal{I}_2 \to \mathcal{I}_1 \to 0
$$
of $\mathcal{O}$-modules.
\end{remark}

\begin{remark}
\label{remark-complex-thickenings-and-ses-modules-ringed-topoi}
Let $(\Sh(\mathcal{C}), \mathcal{O})$ be a ringed topos.
Let $\mathcal{F}$ be an $\mathcal{O}$-module. Let
$$
(\Sh(\mathcal{D}_1), \mathcal{O}'_1) \to
(\Sh(\mathcal{D}_2), \mathcal{O}'_2) \to
(\Sh(\mathcal{D}_3), \mathcal{O}'_3)
$$
be a complex first order thickenings of $(\Sh(\mathcal{C}), \mathcal{O})$, see
Remark \ref{remark-short-exact-sequence-thickenings-ringed-topoi}.
Let $(\mathcal{K}_i, c_i)$, $i = 1, 2, 3$ be pairs consisting of
an $\mathcal{O}$-module $\mathcal{K}_i$ and a map
$c_i : \mathcal{I}_i \otimes_\mathcal{O} \mathcal{F} \to
\mathcal{K}_i$. Assume given a short exact sequence
of $\mathcal{O}$-modules
$$
0 \to \mathcal{K}_3 \to \mathcal{K}_2 \to \mathcal{K}_1 \to 0
$$
such that
$$
\vcenter{
\xymatrix{
\mathcal{I}_2 \otimes_\mathcal{O} \mathcal{F}
\ar[r]_-{c_2} \ar[d] &
\mathcal{K}_2 \ar[d] \\
\mathcal{I}_1 \otimes_\mathcal{O} \mathcal{F}
\ar[r]^-{c_1} &
\mathcal{K}_1
}
}
\quad\text{and}\quad
\vcenter{
\xymatrix{
\mathcal{I}_3 \otimes_\mathcal{O} \mathcal{F}
\ar[r]_-{c_3} \ar[d] &
\mathcal{K}_3 \ar[d] \\
\mathcal{I}_2 \otimes_\mathcal{O} \mathcal{F}
\ar[r]^-{c_2} &
\mathcal{K}_2
}
}
$$
are commutative. Finally, assume given an extension
$$
0 \to \mathcal{K}_2 \to \mathcal{F}'_2 \to \mathcal{F} \to 0
$$
as in (\ref{equation-extension-ringed-topoi})
with $\mathcal{K} = \mathcal{K}_2$
of $\mathcal{O}'_2$-modules with $c_{\mathcal{F}'_2} = c_2$.
In this situation we can apply the functoriality of
Remark \ref{remark-extension-functorial-ringed-topoi}
to obtain an extension $\mathcal{F}'_1$ of $\mathcal{O}'_1$-modules
(we'll describe $\mathcal{F}'_1$ in this special case below). By
Remark \ref{remark-trivial-extension-ringed-topoi}
using the canonical splitting
$\pi : (\Sh(\mathcal{D}_1), \mathcal{O}'_1) \to
(\Sh(\mathcal{C}), \mathcal{O})$ of
Remark \ref{remark-short-exact-sequence-thickenings-ringed-topoi}
we obtain
$\xi_{\mathcal{F}'_1} \in
\text{Ext}^1_\mathcal{O}(\mathcal{F}, \mathcal{K}_1)$.
Finally, we have the obstruction
$$
o(\mathcal{F}, \mathcal{K}_3, c_3) \in
\text{Ext}^2_\mathcal{O}(\mathcal{F}, \mathcal{K}_3)
$$
see Lemma \ref{lemma-inf-obs-ext-ringed-topoi}.
In this situation we {\bf claim} that the canonical map
$$
\partial :
\text{Ext}^1_\mathcal{O}(\mathcal{F}, \mathcal{K}_1)
\longrightarrow
\text{Ext}^2_\mathcal{O}(\mathcal{F}, \mathcal{K}_3)
$$
coming from the short exact sequence
$0 \to \mathcal{K}_3 \to \mathcal{K}_2 \to \mathcal{K}_1 \to 0$
sends $\xi_{\mathcal{F}'_1}$
to the obstruction class $o(\mathcal{F}, \mathcal{K}_3, c_3)$.

\medskip\noindent
To prove this claim choose an embedding $j : \mathcal{K}_3 \to \mathcal{K}$
where $\mathcal{K}$ is an injective $\mathcal{O}$-module.
We can lift $j$ to a map $j' : \mathcal{K}_2 \to \mathcal{K}$.
Set $\mathcal{E}'_2 = j'_*\mathcal{F}'_2$ equal to the pushout
of $\mathcal{F}'_2$ by $j'$ so that $c_{\mathcal{E}'_2} = j' \circ c_2$.
Picture:
$$
\xymatrix{
0 \ar[r] &
\mathcal{K}_2 \ar[r] \ar[d]_{j'} &
\mathcal{F}'_2 \ar[r] \ar[d] &
\mathcal{F} \ar[r] \ar[d] & 0 \\
0 \ar[r] &
\mathcal{K} \ar[r] &
\mathcal{E}'_2 \ar[r] &
\mathcal{F} \ar[r] & 0
}
$$
Set $\mathcal{E}'_3 = \mathcal{E}'_2$ but viewed as an
$\mathcal{O}'_3$-module via $\mathcal{O}'_3 \to \mathcal{O}'_2$.
Then $c_{\mathcal{E}'_3} = j \circ c_3$.
The proof of Lemma \ref{lemma-inf-obs-ext-ringed-topoi} constructs
$o(\mathcal{F}, \mathcal{K}_3, c_3)$
as the boundary of the class of the extension of $\mathcal{O}$-modules
$$
0 \to
\mathcal{K}/\mathcal{K}_3 \to
\mathcal{E}'_3/\mathcal{K}_3 \to
\mathcal{F} \to 0
$$
On the other hand, note that $\mathcal{F}'_1 = \mathcal{F}'_2/\mathcal{K}_3$
hence the class $\xi_{\mathcal{F}'_1}$ is the class
of the extension
$$
0 \to \mathcal{K}_2/\mathcal{K}_3 \to \mathcal{F}'_2/\mathcal{K}_3
\to \mathcal{F} \to 0
$$
seen as a sequence of $\mathcal{O}$-modules using $\pi^\sharp$
where $\pi : (\Sh(\mathcal{D}_1), \mathcal{O}'_1) \to
(\Sh(\mathcal{C}), \mathcal{O})$ is the canonical splitting.
Thus finally, the claim follows from the fact that we have
a commutative diagram
$$
\xymatrix{
0 \ar[r] &
\mathcal{K}_2/\mathcal{K}_3 \ar[r] \ar[d] &
\mathcal{F}'_2/\mathcal{K}_3 \ar[r] \ar[d] &
\mathcal{F} \ar[r] \ar[d] & 0 \\
0 \ar[r] &
\mathcal{K}/\mathcal{K}_3 \ar[r] &
\mathcal{E}'_3/\mathcal{K}_3 \ar[r] &
\mathcal{F} \ar[r] & 0
}
$$
which is $\mathcal{O}$-linear (with the $\mathcal{O}$-module
structures given above).
\end{remark}







\section{Infinitesimal deformations of modules on ringed topi}
\label{section-deformation-modules-ringed-topoi}

\noindent
Let $i : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{D}), \mathcal{O}')$
be a first order thickening of ringed topoi. We freely use the notation
introduced in Section \ref{section-thickenings-ringed-topoi}.
Let $\mathcal{F}'$ be an $\mathcal{O}'$-module
and set $\mathcal{F} = i^*\mathcal{F}'$.
In this situation we have a short exact sequence
$$
0 \to \mathcal{I}\mathcal{F}' \to \mathcal{F}' \to \mathcal{F} \to 0
$$
of $\mathcal{O}'$-modules. Since $\mathcal{I}^2 = 0$ the
$\mathcal{O}'$-module structure on $\mathcal{I}\mathcal{F}'$
comes from a unique $\mathcal{O}$-module structure.
Thus the sequence above is an extension as in
(\ref{equation-extension-ringed-topoi}).
As a special case, if $\mathcal{F}' = \mathcal{O}'$ we have
$i^*\mathcal{O}' = \mathcal{O}$ and
$\mathcal{I}\mathcal{O}' = \mathcal{I}$ and we recover the
sequence of structure sheaves
$$
0 \to \mathcal{I} \to \mathcal{O}' \to \mathcal{O} \to 0
$$

\begin{lemma}
\label{lemma-inf-map-special-ringed-topoi}
Let $i : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{D}), \mathcal{O}')$
be a first order thickening of ringed topoi.
Let $\mathcal{F}'$, $\mathcal{G}'$ be $\mathcal{O}'$-modules.
Set $\mathcal{F} = i^*\mathcal{F}'$ and $\mathcal{G} = i^*\mathcal{G}'$.
Let $\varphi : \mathcal{F} \to \mathcal{G}$ be an $\mathcal{O}$-linear map.
The set of lifts of $\varphi$ to an $\mathcal{O}'$-linear map
$\varphi' : \mathcal{F}' \to \mathcal{G}'$ is, if nonempty, a principal
homogeneous space under
$\Hom_\mathcal{O}(\mathcal{F}, \mathcal{I}\mathcal{G}')$.
\end{lemma}

\begin{proof}
This is a special case of Lemma \ref{lemma-inf-map-ringed-topoi} but we also
give a direct proof. We have short exact sequences of modules
$$
0 \to \mathcal{I} \to \mathcal{O}' \to \mathcal{O} \to 0
\quad\text{and}\quad
0 \to \mathcal{I}\mathcal{G}' \to \mathcal{G}' \to \mathcal{G} \to 0
$$
and similarly for $\mathcal{F}'$.
Since $\mathcal{I}$ has square zero the $\mathcal{O}'$-module
structure on $\mathcal{I}$ and $\mathcal{I}\mathcal{G}'$ comes from
a unique $\mathcal{O}$-module structure. It follows that
$$
\Hom_{\mathcal{O}'}(\mathcal{F}', \mathcal{I}\mathcal{G}') =
\Hom_\mathcal{O}(\mathcal{F}, \mathcal{I}\mathcal{G}')
\quad\text{and}\quad
\Hom_{\mathcal{O}'}(\mathcal{F}', \mathcal{G}) =
\Hom_\mathcal{O}(\mathcal{F}, \mathcal{G})
$$
The lemma now follows from the exact sequence
$$
0 \to \Hom_{\mathcal{O}'}(\mathcal{F}', \mathcal{I}\mathcal{G}') \to
\Hom_{\mathcal{O}'}(\mathcal{F}', \mathcal{G}') \to
\Hom_{\mathcal{O}'}(\mathcal{F}', \mathcal{G})
$$
see Homology, Lemma \ref{homology-lemma-check-exactness}.
\end{proof}

\begin{lemma}
\label{lemma-deform-module-ringed-topoi}
Let $(f, f')$ be a morphism of first order thickenings of ringed topoi
as in Situation \ref{situation-morphism-thickenings-ringed-topoi}.
Let $\mathcal{F}'$ be an $\mathcal{O}'$-module
and set $\mathcal{F} = i^*\mathcal{F}'$.
Assume that $\mathcal{F}$ is flat over $\mathcal{O}_\mathcal{B}$
and that $(f, f')$ is a strict morphism of thickenings
(Definition \ref{definition-strict-morphism-thickenings-ringed-topoi}).
Then the following are equivalent
\begin{enumerate}
\item $\mathcal{F}'$ is flat over $\mathcal{O}_{\mathcal{B}'}$, and
\item the canonical map
$f^*\mathcal{J} \otimes_\mathcal{O} \mathcal{F} \to
\mathcal{I}\mathcal{F}'$
is an isomorphism.
\end{enumerate}
Moreover, in this case the maps
$$
f^*\mathcal{J} \otimes_\mathcal{O} \mathcal{F} \to
\mathcal{I} \otimes_\mathcal{O} \mathcal{F} \to
\mathcal{I}\mathcal{F}'
$$
are isomorphisms.
\end{lemma}

\begin{proof}
The map $f^*\mathcal{J} \to \mathcal{I}$ is surjective
as $(f, f')$ is a strict morphism of thickenings.
Hence the final statement is a consequence of (2).

\medskip\noindent
Proof of the equivalence of (1) and (2). By definition flatness over
$\mathcal{O}_\mathcal{B}$ means flatness over $f^{-1}\mathcal{O}_\mathcal{B}$.
Similarly for flatness over $f^{-1}\mathcal{O}_{\mathcal{B}'}$.
Note that the strictness of $(f, f')$ and the assumption that
$\mathcal{F} = i^*\mathcal{F}'$ imply that
$$
\mathcal{F} = \mathcal{F}'/(f^{-1}\mathcal{J})\mathcal{F}'
$$
as sheaves on $\mathcal{C}$. Moreover, observe that
$f^*\mathcal{J} \otimes_\mathcal{O} \mathcal{F} =
f^{-1}\mathcal{J} \otimes_{f^{-1}\mathcal{O}_\mathcal{B}} \mathcal{F}$.
Hence the equivalence of (1) and (2) follows from
Modules on Sites, Lemma \ref{sites-modules-lemma-flat-over-thickening}.
\end{proof}

\begin{lemma}
\label{lemma-deform-fp-module-ringed-topoi}
Let $(f, f')$ be a morphism of first order thickenings of ringed topoi
as in Situation \ref{situation-morphism-thickenings-ringed-topoi}.
Let $\mathcal{F}'$ be an $\mathcal{O}'$-module
and set $\mathcal{F} = i^*\mathcal{F}'$.
Assume that $\mathcal{F}'$ is flat over $\mathcal{O}_{\mathcal{B}'}$
and that $(f, f')$ is a strict morphism of thickenings.
Then the following are equivalent
\begin{enumerate}
\item $\mathcal{F}'$ is an $\mathcal{O}'$-module of finite presentation, and
\item $\mathcal{F}$ is an $\mathcal{O}$-module of finite presentation.
\end{enumerate}
\end{lemma}

\begin{proof}
The implication (1) $\Rightarrow$ (2) follows from
Modules on Sites, Lemma \ref{sites-modules-lemma-local-pullback}.
For the converse, assume $\mathcal{F}$ of finite presentation.
We may and do assume that $\mathcal{C} = \mathcal{C}'$.
By Lemma \ref{lemma-deform-module-ringed-topoi} we have a short exact sequence
$$
0 \to \mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{F} \to
\mathcal{F}' \to \mathcal{F} \to 0
$$
Let $U$ be an object of $\mathcal{C}$ such that $\mathcal{F}|_U$ has a
presentation
$$
\mathcal{O}_U^{\oplus m} \to \mathcal{O}_U^{\oplus n} \to \mathcal{F}|_U \to 0
$$
After replacing $U$ by the members of a covering we may assume the
map $\mathcal{O}_U^{\oplus n} \to \mathcal{F}|_U$ lifts to a map
$(\mathcal{O}'_U)^{\oplus n} \to \mathcal{F}'|_U$. The induced map
$\mathcal{I}^{\oplus n} \to \mathcal{I} \otimes \mathcal{F}$ is
surjective by right exactness of $\otimes$. Thus after replacing $U$
by the members of a covering we can find a lift
$(\mathcal{O}'|_U)^{\oplus m} \to (\mathcal{O}'|_U)^{\oplus n}$
of the given map $\mathcal{O}_U^{\oplus m} \to \mathcal{O}_U^{\oplus n}$
such that
$$
(\mathcal{O}'_U)^{\oplus m} \to (\mathcal{O}'_U)^{\oplus n} \to
\mathcal{F}'|_U \to 0
$$
is a complex. Using right exactness of $\otimes$ once more it is seen
that this complex is exact.
\end{proof}

\begin{lemma}
\label{lemma-inf-map-rel-ringed-topoi}
Let $(f, f')$ be a morphism of first order thickenings as in
Situation \ref{situation-morphism-thickenings-ringed-topoi}.
Let $\mathcal{F}'$, $\mathcal{G}'$ be $\mathcal{O}'$-modules and set
$\mathcal{F} = i^*\mathcal{F}'$ and $\mathcal{G} = i^*\mathcal{G}'$.
Let $\varphi : \mathcal{F} \to \mathcal{G}$ be an $\mathcal{O}$-linear map.
Assume that $\mathcal{G}'$ is flat over $\mathcal{O}_{\mathcal{B}'}$ and that
$(f, f')$ is a strict morphism of thickenings.
The set of lifts of $\varphi$ to an $\mathcal{O}'$-linear map
$\varphi' : \mathcal{F}' \to \mathcal{G}'$ is, if nonempty, a principal
homogeneous space under
$$
\Hom_\mathcal{O}(\mathcal{F},
\mathcal{G} \otimes_\mathcal{O} f^*\mathcal{J})
$$
\end{lemma}

\begin{proof}
Combine Lemmas \ref{lemma-inf-map-special-ringed-topoi} and
\ref{lemma-deform-module-ringed-topoi}.
\end{proof}

\begin{lemma}
\label{lemma-inf-obs-map-special-ringed-topoi}
Let $i : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{D}), \mathcal{O}')$
be a first order thickening of ringed topoi.
Let $\mathcal{F}'$, $\mathcal{G}'$ be $\mathcal{O}'$-modules and set
$\mathcal{F} = i^*\mathcal{F}'$ and $\mathcal{G} = i^*\mathcal{G}'$.
Let $\varphi : \mathcal{F} \to \mathcal{G}$ be an $\mathcal{O}$-linear map.
There exists an element
$$
o(\varphi) \in
\text{Ext}^1_\mathcal{O}(Li^*\mathcal{F}', \mathcal{I}\mathcal{G}')
$$
whose vanishing is a necessary and sufficient condition for the
existence of a lift of $\varphi$ to an $\mathcal{O}'$-linear map
$\varphi' : \mathcal{F}' \to \mathcal{G}'$.
\end{lemma}

\begin{proof}
It is clear from the proof of Lemma \ref{lemma-inf-map-special-ringed-topoi}
that the vanishing of the boundary of $\varphi$ via the map
$$
\Hom_\mathcal{O}(\mathcal{F}, \mathcal{G}) =
\Hom_{\mathcal{O}'}(\mathcal{F}', \mathcal{G}) \longrightarrow
\text{Ext}^1_{\mathcal{O}'}(\mathcal{F}', \mathcal{I}\mathcal{G}')
$$
is a necessary and sufficient condition for the existence of a lift. We
conclude as
$$
\text{Ext}^1_{\mathcal{O}'}(\mathcal{F}', \mathcal{I}\mathcal{G}') =
\text{Ext}^1_\mathcal{O}(Li^*\mathcal{F}', \mathcal{I}\mathcal{G}')
$$
the adjointness of $i_* = Ri_*$ and $Li^*$ on the derived category
(Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-adjoint}).
\end{proof}

\begin{lemma}
\label{lemma-inf-obs-map-rel-ringed-topoi}
Let $(f, f')$ be a morphism of first order thickenings as in
Situation \ref{situation-morphism-thickenings-ringed-topoi}.
Let $\mathcal{F}'$, $\mathcal{G}'$ be $\mathcal{O}'$-modules and set
$\mathcal{F} = i^*\mathcal{F}'$ and $\mathcal{G} = i^*\mathcal{G}'$.
Let $\varphi : \mathcal{F} \to \mathcal{G}$ be an $\mathcal{O}$-linear map.
Assume that $\mathcal{F}'$ and $\mathcal{G}'$ are flat over
$\mathcal{O}_{\mathcal{B}'}$ and
that $(f, f')$ is a strict morphism of thickenings. There exists an element
$$
o(\varphi) \in
\text{Ext}^1_\mathcal{O}(\mathcal{F},
\mathcal{G} \otimes_\mathcal{O} f^*\mathcal{J})
$$
whose vanishing is a necessary and sufficient condition for the
existence of a lift of $\varphi$ to an $\mathcal{O}'$-linear map
$\varphi' : \mathcal{F}' \to \mathcal{G}'$.
\end{lemma}

\begin{proof}[First proof]
This follows from Lemma \ref{lemma-inf-obs-map-special-ringed-topoi}
as we claim that under the assumptions of the lemma we have
$$
\text{Ext}^1_\mathcal{O}(Li^*\mathcal{F}', \mathcal{I}\mathcal{G}') =
\text{Ext}^1_\mathcal{O}(\mathcal{F},
\mathcal{G} \otimes_\mathcal{O} f^*\mathcal{J})
$$
Namely, we have
$\mathcal{I}\mathcal{G}' =
\mathcal{G} \otimes_\mathcal{O} f^*\mathcal{J}$
by Lemma \ref{lemma-deform-module-ringed-topoi}.
On the other hand, observe that
$$
H^{-1}(Li^*\mathcal{F}') =
\text{Tor}_1^{\mathcal{O}'}(\mathcal{F}', \mathcal{O})
$$
(local computation omitted). Using the short exact sequence
$$
0 \to \mathcal{I} \to \mathcal{O}' \to \mathcal{O} \to 0
$$
we see that this $\text{Tor}_1$ is computed by the kernel of the map
$\mathcal{I} \otimes_\mathcal{O} \mathcal{F} \to \mathcal{I}\mathcal{F}'$
which is zero by the final assertion of
Lemma \ref{lemma-deform-module-ringed-topoi}.
Thus $\tau_{\geq -1}Li^*\mathcal{F}' = \mathcal{F}$.
On the other hand, we have
$$
\text{Ext}^1_\mathcal{O}(Li^*\mathcal{F}',
\mathcal{I}\mathcal{G}') =
\text{Ext}^1_\mathcal{O}(\tau_{\geq -1}Li^*\mathcal{F}',
\mathcal{I}\mathcal{G}')
$$
by the dual of
Derived Categories, Lemma \ref{derived-lemma-negative-vanishing}.
\end{proof}

\begin{proof}[Second proof]
We can apply Lemma \ref{lemma-inf-obs-map-ringed-topoi} as follows. Note that
$\mathcal{K} = \mathcal{I} \otimes_\mathcal{O} \mathcal{F}$ and
$\mathcal{L} = \mathcal{I} \otimes_\mathcal{O} \mathcal{G}$
by Lemma \ref{lemma-deform-module-ringed-topoi}, that
$c_{\mathcal{F}'} = 1 \otimes 1$ and $c_{\mathcal{G}'} = 1 \otimes 1$
and taking $\psi = 1 \otimes \varphi$ the diagram of the lemma
commutes. Thus $o(\varphi) = o(\varphi, 1 \otimes \varphi)$
works.
\end{proof}

\begin{lemma}
\label{lemma-inf-ext-rel-ringed-topoi}
Let $(f, f')$ be a morphism of first order thickenings as in
Situation \ref{situation-morphism-thickenings-ringed-topoi}.
Let $\mathcal{F}$ be an $\mathcal{O}$-module.
Assume $(f, f')$ is a strict morphism of thickenings and
$\mathcal{F}$ flat over $\mathcal{O}_\mathcal{B}$. If there exists a pair
$(\mathcal{F}', \alpha)$ consisting of an
$\mathcal{O}'$-module $\mathcal{F}'$ flat over $\mathcal{O}_{\mathcal{B}'}$
and an isomorphism
$\alpha : i^*\mathcal{F}' \to \mathcal{F}$, then the set of
isomorphism classes of such pairs is principal homogeneous
under
$\text{Ext}^1_\mathcal{O}(
\mathcal{F}, \mathcal{I} \otimes_\mathcal{O} \mathcal{F})$.
\end{lemma}

\begin{proof}
If we assume there exists one such module, then the canonical map
$$
f^*\mathcal{J} \otimes_\mathcal{O} \mathcal{F} \to
\mathcal{I} \otimes_\mathcal{O} \mathcal{F}
$$
is an isomorphism by Lemma \ref{lemma-deform-module-ringed-topoi}. Apply
Lemma \ref{lemma-inf-ext-ringed-topoi} with $\mathcal{K} = 
\mathcal{I} \otimes_\mathcal{O} \mathcal{F}$
and $c = 1$. By Lemma \ref{lemma-deform-module-ringed-topoi}
the corresponding extensions
$\mathcal{F}'$ are all flat over $\mathcal{O}_{\mathcal{B}'}$.
\end{proof}

\begin{lemma}
\label{lemma-inf-obs-ext-rel-ringed-topoi}
Let $(f, f')$ be a morphism of first order thickenings as in
Situation \ref{situation-morphism-thickenings-ringed-topoi}.
Let $\mathcal{F}$ be an $\mathcal{O}$-module. Assume
$(f, f')$ is a strict morphism of thickenings
and $\mathcal{F}$ flat over $\mathcal{O}_\mathcal{B}$. There exists an
$\mathcal{O}'$-module $\mathcal{F}'$ flat over $\mathcal{O}_{\mathcal{B}'}$
with $i^*\mathcal{F}' \cong \mathcal{F}$, if and only if
\begin{enumerate}
\item the canonical map
$f^*\mathcal{J} \otimes_\mathcal{O} \mathcal{F} \to
\mathcal{I} \otimes_\mathcal{O} \mathcal{F}$
is an isomorphism, and
\item the class
$o(\mathcal{F}, \mathcal{I} \otimes_\mathcal{O} \mathcal{F}, 1)
\in \text{Ext}^2_\mathcal{O}(
\mathcal{F}, \mathcal{I} \otimes_\mathcal{O} \mathcal{F})$
of Lemma \ref{lemma-inf-obs-ext-ringed-topoi} is zero.
\end{enumerate}
\end{lemma}

\begin{proof}
This follows immediately from the characterization of
$\mathcal{O}'$-modules flat over $\mathcal{O}_{\mathcal{B}'}$ of 
Lemma \ref{lemma-deform-module-ringed-topoi} and
Lemma \ref{lemma-inf-obs-ext-ringed-topoi}.
\end{proof}






\section{Application to flat modules on flat thickenings of ringed topoi}
\label{section-flat-ringed-topoi}

\noindent
Consider a commutative diagram
$$
\xymatrix{
(\Sh(\mathcal{C}), \mathcal{O}) \ar[r]_i \ar[d]_f &
(\Sh(\mathcal{D}), \mathcal{O}') \ar[d]^{f'} \\
(\Sh(\mathcal{B}), \mathcal{O}_\mathcal{B}) \ar[r]^t &
(\Sh(\mathcal{B}'), \mathcal{O}_{\mathcal{B}'})
}
$$
of ringed topoi whose horizontal arrows are first order thickenings
as in Situation \ref{situation-morphism-thickenings-ringed-topoi}. Set
$\mathcal{I} = \text{Ker}(i^\sharp) \subset \mathcal{O}'$ and
$\mathcal{J} = \text{Ker}(t^\sharp) \subset \mathcal{O}_{\mathcal{B}'}$.
Let $\mathcal{F}$ be an $\mathcal{O}$-module. Assume that
\begin{enumerate}
\item $(f, f')$ is a strict morphism of thickenings,
\item $f'$ is flat, and
\item $\mathcal{F}$ is flat over $\mathcal{O}_\mathcal{B}$.
\end{enumerate}
Note that (1) $+$ (2) imply that $\mathcal{I} = f^*\mathcal{J}$
(apply Lemma \ref{lemma-deform-module-ringed-topoi} to $\mathcal{O}'$).
The theory of the preceding section is especially nice
under these assumptions. We summarize the results already obtained
in the following lemma.

\begin{lemma}
\label{lemma-flat-ringed-topoi}
In the situation above.
\begin{enumerate}
\item There exists an $\mathcal{O}'$-module $\mathcal{F}'$ flat over
$\mathcal{O}_{\mathcal{B}'}$ with $i^*\mathcal{F}' \cong \mathcal{F}$,
if and only if
the class $o(\mathcal{F}, f^*\mathcal{J} \otimes_\mathcal{O} \mathcal{F}, 1)
\in \text{Ext}^2_\mathcal{O}(
\mathcal{F}, f^*\mathcal{J} \otimes_\mathcal{O} \mathcal{F})$
of Lemma \ref{lemma-inf-obs-ext-ringed-topoi} is zero.
\item If such a module exists, then the set of isomorphism classes
of lifts is principal homogeneous under
$\text{Ext}^1_\mathcal{O}(
\mathcal{F}, f^*\mathcal{J} \otimes_\mathcal{O} \mathcal{F})$.
\item Given a lift $\mathcal{F}'$, the set of automorphisms of
$\mathcal{F}'$ which pull back to $\text{id}_\mathcal{F}$ is canonically
isomorphic to $\text{Ext}^0_\mathcal{O}(
\mathcal{F}, f^*\mathcal{J} \otimes_\mathcal{O} \mathcal{F})$.
\end{enumerate}
\end{lemma}

\begin{proof}
Part (1) follows from Lemma \ref{lemma-inf-obs-ext-rel-ringed-topoi}
as we have seen above that $\mathcal{I} = f^*\mathcal{J}$.
Part (2) follows from Lemma \ref{lemma-inf-ext-rel-ringed-topoi}.
Part (3) follows from Lemma \ref{lemma-inf-map-rel-ringed-topoi}.
\end{proof}

\begin{situation}
\label{situation-ses-flat-thickenings-ringed-topoi}
Let $f : (\Sh(\mathcal{C}), \mathcal{O}) \to
(\Sh(\mathcal{B}), \mathcal{O}_\mathcal{B})$ be a morphism of
ringed topoi. Consider a commutative diagram
$$
\xymatrix{
(\Sh(\mathcal{C}'_1), \mathcal{O}'_1) \ar[r]_h \ar[d]_{f'_1} &
(\Sh(\mathcal{C}'_2), \mathcal{O}'_2) \ar[r] \ar[d]_{f'_2} &
(\Sh(\mathcal{C}'_3), \mathcal{O}'_3) \ar[d]_{f'_3} \\
(\Sh(\mathcal{B}'_1), \mathcal{O}_{\mathcal{B}'_1}) \ar[r] &
(\Sh(\mathcal{B}'_2), \mathcal{O}_{\mathcal{B}'_2}) \ar[r] &
(\Sh(\mathcal{B}'_3), \mathcal{O}_{\mathcal{B}'_3})
}
$$
where (a) the top row is a short exact sequence of first order thickenings
of $(\Sh(\mathcal{C}), \mathcal{O})$, (b) the lower row is a short exact
sequence of first order thickenings of
$(\Sh(\mathcal{B}), \mathcal{O}_\mathcal{B})$, (c) each $f'_i$ restricts
to $f$, (d) each pair $(f, f_i')$ is a strict morphism of thickenings, and
(e) each $f'_i$ is flat. Finally, let $\mathcal{F}'_2$ be an
$\mathcal{O}'_2$-module flat over $\mathcal{O}_{\mathcal{B}'_2}$ and
set $\mathcal{F} = \mathcal{F}'_2 \otimes \mathcal{O}$. Let
$\pi : (\Sh(\mathcal{C}'_1), \mathcal{O}'_1) \to
(\Sh(\mathcal{C}), \mathcal{O})$ be the canonical splitting
(Remark \ref{remark-short-exact-sequence-thickenings-ringed-topoi}).
\end{situation}

\begin{lemma}
\label{lemma-verify-iv-ringed-topoi}
In Situation \ref{situation-ses-flat-thickenings-ringed-topoi} the modules
$\pi^*\mathcal{F}$ and $h^*\mathcal{F}'_2$ are $\mathcal{O}'_1$-modules
flat over $\mathcal{O}_{\mathcal{B}'_1}$ restricting to $\mathcal{F}$ on
$(\Sh(\mathcal{C}), \mathcal{O})$.
Their difference (Lemma \ref{lemma-flat-ringed-topoi}) is an element
$\theta$ of
$\text{Ext}^1_\mathcal{O}(\mathcal{F},
f^*\mathcal{J}_1 \otimes_\mathcal{O} \mathcal{F})$
whose boundary in
$\text{Ext}^2_\mathcal{O}(\mathcal{F},
f^*\mathcal{J}_3 \otimes_\mathcal{O} \mathcal{F})$
equals the obstruction (Lemma \ref{lemma-flat-ringed-topoi})
to lifting $\mathcal{F}$ to an $\mathcal{O}'_3$-module flat over
$\mathcal{O}_{\mathcal{B}'_3}$.
\end{lemma}

\begin{proof}
Note that both $\pi^*\mathcal{F}$ and $h^*\mathcal{F}'_2$
restrict to $\mathcal{F}$ on $(\Sh(\mathcal{C}), \mathcal{O})$
and that the kernels of
$\pi^*\mathcal{F} \to \mathcal{F}$ and $h^*\mathcal{F}'_2 \to \mathcal{F}$
are given by $f^*\mathcal{J}_1 \otimes_\mathcal{O} \mathcal{F}$.
Hence flatness by Lemma \ref{lemma-deform-module-ringed-topoi}.
Taking the boundary makes sense as the sequence of modules
$$
0 \to f^*\mathcal{J}_3 \otimes_\mathcal{O} \mathcal{F} \to
f^*\mathcal{J}_2 \otimes_\mathcal{O} \mathcal{F} \to
f^*\mathcal{J}_1 \otimes_\mathcal{O} \mathcal{F} \to 0
$$
is short exact due to the assumptions in
Situation \ref{situation-ses-flat-thickenings-ringed-topoi}
and the fact that $\mathcal{F}$ is flat over $\mathcal{O}_\mathcal{B}$.
The statement on the obstruction class is a direct translation
of the result of
Remark \ref{remark-complex-thickenings-and-ses-modules-ringed-topoi}
to this particular situation.
\end{proof}







\section{Deformations of ringed topoi and the naive cotangent complex}
\label{section-deformations-ringed-topoi}

\noindent
In this section we use the naive cotangent complex to do a little bit
of deformation theory. We start with a first order thickening
$t : (\Sh(\mathcal{B}), \mathcal{O}_\mathcal{B}) \to
(\Sh(\mathcal{B}'), \mathcal{O}_{\mathcal{B}'})$ of ringed topoi.
We denote $\mathcal{J} = \text{Ker}(t^\sharp)$ and we
identify the underlying topoi of $\mathcal{B}$ and $\mathcal{B}'$.
Moreover we assume given a morphism of ringed topoi
$f : (\Sh(\mathcal{C}), \mathcal{O}) \to
(\Sh(\mathcal{B}), \mathcal{O}_\mathcal{B})$, an $\mathcal{O}$-module
$\mathcal{G}$, and a map $f^{-1}\mathcal{J} \to \mathcal{G}$
of sheaves of $f^{-1}\mathcal{O}_\mathcal{B}$-modules.
In this section we ask ourselves whether we can find
the question mark fitting into the following diagram
\begin{equation}
\label{equation-to-solve-ringed-topoi}
\vcenter{
\xymatrix{
0 \ar[r] & \mathcal{G} \ar[r] & {?} \ar[r] & \mathcal{O} \ar[r] & 0 \\
0 \ar[r] & f^{-1}\mathcal{J} \ar[u]^c \ar[r] &
f^{-1}\mathcal{O}_{\mathcal{B}'} \ar[u] \ar[r] &
f^{-1}\mathcal{O}_\mathcal{B} \ar[u] \ar[r] & 0
}
}
\end{equation}
and moreover how unique the solution is (if it exists). More precisely,
we look for a first order thickening
$i : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{C}'), \mathcal{O}')$
and a morphism of thickenings $(f, f')$ as in
(\ref{equation-morphism-thickenings-ringed-topoi})
where $\text{Ker}(i^\sharp)$ is identified with $\mathcal{G}$
such that $(f')^\sharp$ induces the given map $c$.
We will say $(\Sh(\mathcal{C}'), \mathcal{O}')$ is a {\it solution} to
(\ref{equation-to-solve-ringed-topoi}).


\begin{lemma}
\label{lemma-huge-diagram-ringed-topoi}
Assume given a commutative diagram of morphisms ringed topoi
\begin{equation}
\label{equation-huge-1-ringed-topoi}
\vcenter{
\xymatrix{
& (\Sh(\mathcal{C}_2), \mathcal{O}_2) \ar[r]_{i_2} \ar[d]_{f_2} \ar[ddl]_g &
(\Sh(\mathcal{C}'_2), \mathcal{O}'_2) \ar[d]^{f'_2} \\
&
(\Sh(\mathcal{B}_2), \mathcal{O}_{\mathcal{B}_2}) \ar[r]^{t_2} \ar[ddl]|\hole &
(\Sh(\mathcal{B}'_2), \mathcal{O}_{\mathcal{B}'_2}) \ar[ddl] \\
(\Sh(\mathcal{C}_1), \mathcal{O}_1) \ar[r]_{i_1} \ar[d]_{f_1} &
(\Sh(\mathcal{C}'_1), \mathcal{O}'_1) \ar[d]^{f'_1} \\
(\Sh(\mathcal{B}_1), \mathcal{O}_{\mathcal{B}_1}) \ar[r]^{t_1} &
(\Sh(\mathcal{B}'_1), \mathcal{O}_{\mathcal{B}'_1})
}
}
\end{equation}
whose horizontal arrows are first order thickenings. Set
$\mathcal{G}_j = \text{Ker}(i_j^\sharp)$ and assume given a
map of $g^{-1}\mathcal{O}_1$-modules
$\nu : g^{-1}\mathcal{G}_1 \to \mathcal{G}_2$
giving rise to the commutative diagram
\begin{equation}
\label{equation-huge-2-ringed-topoi}
\vcenter{
\xymatrix{
& 0 \ar[r] & \mathcal{G}_2 \ar[r] &
\mathcal{O}'_2 \ar[r] &
\mathcal{O}_2 \ar[r] & 0 \\
& 0 \ar[r]|\hole &
f_2^{-1}\mathcal{J}_2 \ar[u]_{c_2} \ar[r] &
f_2^{-1}\mathcal{O}_{\mathcal{B}'_2} \ar[u] \ar[r]|\hole &
f_2^{-1}\mathcal{O}_{\mathcal{B}_2} \ar[u] \ar[r] & 0 \\
0 \ar[r] &
\mathcal{G}_1 \ar[ruu] \ar[r] &
\mathcal{O}'_1 \ar[r] &
\mathcal{O}_1 \ar[ruu] \ar[r] & 0 \\
0 \ar[r] &
f_1^{-1}\mathcal{J}_1 \ar[ruu]|\hole \ar[u]^{c_1} \ar[r] &
f_1^{-1}\mathcal{O}_{\mathcal{B}'_1} \ar[ruu]|\hole \ar[u] \ar[r] &
f_1^{-1}\mathcal{O}_{\mathcal{B}_1} \ar[ruu]|\hole \ar[u] \ar[r] & 0
}
}
\end{equation}
with front and back solutions to (\ref{equation-to-solve-ringed-topoi}).
(The north-north-west arrows are maps on $\mathcal{C}_2$ after applying
$g^{-1}$ to the source.)
\begin{enumerate}
\item There exist a canonical element in
$\text{Ext}^1_{\mathcal{O}_2}(
Lg^*\NL_{\mathcal{O}_1/\mathcal{O}_{\mathcal{B}_1}}, \mathcal{G}_2)$
whose vanishing is a necessary and sufficient condition for the existence
of a morphism of ringed topoi
$(\Sh(\mathcal{C}'_2), \mathcal{O}'_2) \to
(\Sh(\mathcal{C}'_1), \mathcal{O}'_1)$ fitting into
(\ref{equation-huge-1-ringed-topoi}) compatibly with $\nu$.
\item If there exists a morphism
$(\Sh(\mathcal{C}'_2), \mathcal{O}'_2) \to
(\Sh(\mathcal{C}'_1), \mathcal{O}'_1)$
fitting into
(\ref{equation-huge-1-ringed-topoi}) compatibly with $\nu$ the set
of all such morphisms is a principal homogeneous space under
$$
\Hom_{\mathcal{O}_1}(
\Omega_{\mathcal{O}_1/\mathcal{O}_{\mathcal{B}_1}}, g_*\mathcal{G}_2) =
\Hom_{\mathcal{O}_2}(
g^*\Omega_{\mathcal{O}_1/\mathcal{O}_{\mathcal{B}_1}}, \mathcal{G}_2) =
\text{Ext}^0_{\mathcal{O}_2}(
Lg^*\NL_{\mathcal{O}_1/\mathcal{O}_{\mathcal{B}_1}}, \mathcal{G}_2).
$$
\end{enumerate}
\end{lemma}

\begin{proof}
The proof of this lemma is identical to the proof of
Lemma \ref{lemma-huge-diagram-ringed-spaces}.
We urge the reader to read that proof instead of this one.
We will identify the underlying topoi for every
thickening in sight (we have already used this convention
in the statement). The equalities in the last statement of the
lemma are immediate from the definitions. Thus we will work with the groups
$\text{Ext}^k_{\mathcal{O}_2}(
Lg^*\NL_{\mathcal{O}_1/\mathcal{O}_{\mathcal{B}_1}}, \mathcal{G}_2)$,
$k = 0, 1$ in the rest of the proof. We first argue that we can reduce
to the case where the underlying topos of all ringed topoi in the lemma
is the same.

\medskip\noindent
To do this, observe that
$g^{-1}\NL_{\mathcal{O}_1/\mathcal{O}_{\mathcal{B}_1}}$ is equal to the naive
cotangent complex of the homomorphism of sheaves of rings
$g^{-1}f_1^{-1}\mathcal{O}_{\mathcal{B}_1} \to g^{-1}\mathcal{O}_1$, see
Modules on Sites, Lemma \ref{sites-modules-lemma-pullback-differentials}.
Moreover, the degree $0$ term of
$\NL_{\mathcal{O}_1/\mathcal{O}_{\mathcal{B}_1}}$ is a flat
$\mathcal{O}_1$-module, hence the canonical map
$$
Lg^*\NL_{\mathcal{O}_1/\mathcal{O}_{\mathcal{B}_1}}
\longrightarrow
g^{-1}\NL_{\mathcal{O}_1/\mathcal{O}_{\mathcal{B}_1}}
\otimes_{g^{-1}\mathcal{O}_1} \mathcal{O}_2
$$
induces an isomorphism on cohomology sheaves in degrees $0$ and $-1$.
Thus we may replace the Ext groups of the lemma with
$$
\text{Ext}^k_{g^{-1}\mathcal{O}_1}(
g^{-1}\NL_{\mathcal{O}_1/\mathcal{O}_{\mathcal{B}_1}}, \mathcal{G}_2) =
\text{Ext}^k_{g^{-1}\mathcal{O}_1}(
\NL_{g^{-1}\mathcal{O}_1/g^{-1}f_1^{-1}\mathcal{O}_{\mathcal{B}_1}},
\mathcal{G}_2)
$$
The set of morphism of ringed topoi
$(\Sh(\mathcal{C}'_2), \mathcal{O}'_2) \to
(\Sh(\mathcal{C}'_1), \mathcal{O}'_1)$ fitting into
(\ref{equation-huge-1-ringed-topoi}) compatibly with $\nu$ is in
one-to-one bijection with the set of homomorphisms of
$g^{-1}f_1^{-1}\mathcal{O}_{\mathcal{B}'_1}$-algebras
$g^{-1}\mathcal{O}'_1 \to \mathcal{O}'_2$ which are compatible with
$f^\sharp$ and $\nu$. In this way we see that we may assume we have a
diagram (\ref{equation-huge-2-ringed-topoi}) of sheaves on a site
$\mathcal{C}$ (with $f_1 = f_2 = \text{id}$ on underlying topoi)
and we are looking to find a homomorphism of sheaves of rings
$\mathcal{O}'_1 \to \mathcal{O}'_2$ fitting into it.

\medskip\noindent
In the rest of the proof of the lemma we assume
all underlying topological spaces are the
same, i.e., we have a diagram (\ref{equation-huge-2-ringed-topoi})
of sheaves on a site $\mathcal{C}$ (with $f_1 = f_2 = \text{id}$
on underlying topoi) and we are looking for
homomorphisms of sheaves of rings
$\mathcal{O}'_1 \to \mathcal{O}'_2$ fitting into it.
As ext groups we will use
$\text{Ext}^k_{\mathcal{O}_1}(
\NL_{\mathcal{O}_1/\mathcal{O}_{\mathcal{B}_1}}, \mathcal{G}_2)$, $k = 0, 1$.

\medskip\noindent
Step 1. Construction of the obstruction class. Consider the sheaf
of sets
$$
\mathcal{E} = \mathcal{O}'_1 \times_{\mathcal{O}_2} \mathcal{O}'_2
$$
This comes with a surjective map $\alpha : \mathcal{E} \to \mathcal{O}_1$
and hence we can use $\NL(\alpha)$ instead of
$\NL_{\mathcal{O}_1/\mathcal{O}_{\mathcal{B}_1}}$, see
Modules on Sites, Lemma \ref{sites-modules-lemma-NL-up-to-qis}.
Set
$$
\mathcal{I}' =
\text{Ker}(\mathcal{O}_{\mathcal{B}'_1}[\mathcal{E}] \to \mathcal{O}_1)
\quad\text{and}\quad
\mathcal{I} =
\text{Ker}(\mathcal{O}_{\mathcal{B}_1}[\mathcal{E}] \to \mathcal{O}_1)
$$
There is a surjection $\mathcal{I}' \to \mathcal{I}$ whose kernel
is $\mathcal{J}_1\mathcal{O}_{\mathcal{B}'_1}[\mathcal{E}]$.
We obtain two homomorphisms of $\mathcal{O}_{\mathcal{B}'_2}$-algebras
$$
a : \mathcal{O}_{\mathcal{B}'_1}[\mathcal{E}] \to \mathcal{O}'_1
\quad\text{and}\quad
b : \mathcal{O}_{\mathcal{B}'_1}[\mathcal{E}] \to \mathcal{O}'_2
$$
which induce maps $a|_{\mathcal{I}'} : \mathcal{I}' \to \mathcal{G}_1$ and
$b|_{\mathcal{I}'} : \mathcal{I}' \to \mathcal{G}_2$. Both $a$ and $b$
annihilate $(\mathcal{I}')^2$. Moreover $a$ and $b$ agree on
$\mathcal{J}_1\mathcal{O}_{\mathcal{B}'_1}[\mathcal{E}]$
as maps into $\mathcal{G}_2$
because the left hand square of (\ref{equation-huge-2-ringed-topoi})
is commutative. Thus the difference
$b|_{\mathcal{I}'} - \nu \circ a|_{\mathcal{I}'}$
induces a well defined $\mathcal{O}_1$-linear map
$$
\xi : \mathcal{I}/\mathcal{I}^2 \longrightarrow \mathcal{G}_2
$$
which sends the class of a local section $f$ of $\mathcal{I}$ to
$a(f') - \nu(b(f'))$ where $f'$ is a lift of $f$ to a local
section of $\mathcal{I}'$. We let
$[\xi] \in \text{Ext}^1_{\mathcal{O}_1}(\NL(\alpha), \mathcal{G}_2)$
be the image (see below).

\medskip\noindent
Step 2. Vanishing of $[\xi]$ is necessary. Let us write $\Omega =
\Omega_{\mathcal{O}_{\mathcal{B}_1}[\mathcal{E}]/\mathcal{O}_{\mathcal{B}_1}}
\otimes_{\mathcal{O}_{\mathcal{B}_1}[\mathcal{E}]} \mathcal{O}_1$.
Observe that $\NL(\alpha) = (\mathcal{I}/\mathcal{I}^2 \to \Omega)$
fits into a distinguished triangle
$$
\Omega[0] \to
\NL(\alpha) \to
\mathcal{I}/\mathcal{I}^2[1] \to
\Omega[1]
$$
Thus we see that $[\xi]$ is zero if and only if $\xi$
is a composition $\mathcal{I}/\mathcal{I}^2 \to \Omega \to \mathcal{G}_2$
for some map $\Omega \to \mathcal{G}_2$. Suppose there exists a
homomorphisms of sheaves of rings
$\varphi : \mathcal{O}'_1 \to \mathcal{O}'_2$ fitting into
(\ref{equation-huge-2-ringed-topoi}). In this case consider the map
$\mathcal{O}'_1[\mathcal{E}] \to \mathcal{G}_2$,
$f' \mapsto b(f') - \varphi(a(f'))$. A calculation
shows this annihilates $\mathcal{J}_1\mathcal{O}_{\mathcal{B}'_1}[\mathcal{E}]$
and induces a derivation
$\mathcal{O}_{\mathcal{B}_1}[\mathcal{E}] \to \mathcal{G}_2$.
The resulting linear map $\Omega \to \mathcal{G}_2$ witnesses the
fact that $[\xi] = 0$ in this case.

\medskip\noindent
Step 3. Vanishing of $[\xi]$ is sufficient. Let
$\theta : \Omega \to \mathcal{G}_2$ be a $\mathcal{O}_1$-linear map
such that $\xi$ is equal to
$\theta \circ (\mathcal{I}/\mathcal{I}^2 \to \Omega)$.
Then a calculation shows that
$$
b + \theta \circ d :
\mathcal{O}_{\mathcal{B}'_1}[\mathcal{E}]
\longrightarrow
\mathcal{O}'_2
$$
annihilates $\mathcal{I}'$ and hence defines a map
$\mathcal{O}'_1 \to \mathcal{O}'_2$ fitting into
(\ref{equation-huge-2-ringed-topoi}).

\medskip\noindent
Proof of (2) in the special case above. Omitted. Hint:
This is exactly the same as the proof of (2) of
Lemma \ref{lemma-huge-diagram}.
\end{proof}

\begin{lemma}
\label{lemma-NL-represent-ext-class-ringed-topoi}
Let $\mathcal{C}$ be a site. Let $\mathcal{A} \to \mathcal{B}$ be a
homomorphism of sheaves of rings on $\mathcal{C}$.
Let $\mathcal{G}$ be a $\mathcal{B}$-module.
Let
$\xi \in \text{Ext}^1_\mathcal{B}(\NL_{\mathcal{B}/\mathcal{A}}, \mathcal{G})$. 
There exists a map of sheaves of sets $\alpha : \mathcal{E} \to \mathcal{B}$
such that $\xi \in \text{Ext}^1_\mathcal{B}(\NL(\alpha), \mathcal{G})$
is the class of a map $\mathcal{I}/\mathcal{I}^2 \to \mathcal{G}$
(see proof for notation).
\end{lemma}

\begin{proof}
Recall that given $\alpha : \mathcal{E} \to \mathcal{B}$
such that $\mathcal{A}[\mathcal{E}] \to \mathcal{B}$ is surjective
with kernel $\mathcal{I}$ the complex
$\NL(\alpha) = (\mathcal{I}/\mathcal{I}^2 \to 
\Omega_{\mathcal{A}[\mathcal{E}]/\mathcal{A}}
\otimes_{\mathcal{A}[\mathcal{E}]} \mathcal{B})$ is canonically
isomorphic to $\NL_{\mathcal{B}/\mathcal{A}}$, see
Modules on Sites, Lemma \ref{sites-modules-lemma-NL-up-to-qis}.
Observe moreover, that
$\Omega = \Omega_{\mathcal{A}[\mathcal{E}]/\mathcal{A}}
\otimes_{\mathcal{A}[\mathcal{E}]} \mathcal{B}$ is the sheaf
associated to the presheaf
$U \mapsto \bigoplus_{e \in \mathcal{E}(U)} \mathcal{B}(U)$.
In other words, $\Omega$ is the free $\mathcal{B}$-module on the
sheaf of sets $\mathcal{E}$ and in particular there is a canonical
map $\mathcal{E} \to \Omega$.

\medskip\noindent
Having said this, pick some $\mathcal{E}$ (for example
$\mathcal{E} = \mathcal{B}$ as in the definition of the naive cotangent
complex). The obstruction to writing $\xi$ as the class of a map
$\mathcal{I}/\mathcal{I}^2 \to \mathcal{G}$ is an element in
$\text{Ext}^1_\mathcal{B}(\Omega, \mathcal{G})$. Say this is represented
by the extension $0 \to \mathcal{G} \to \mathcal{H} \to \Omega \to 0$
of $\mathcal{B}$-modules. Consider the sheaf of sets
$\mathcal{E}' = \mathcal{E} \times_\Omega \mathcal{H}$
which comes with an induced map $\alpha' : \mathcal{E}' \to \mathcal{B}$.
Let $\mathcal{I}' = \text{Ker}(\mathcal{A}[\mathcal{E}'] \to \mathcal{B})$
and $\Omega' = \Omega_{\mathcal{A}[\mathcal{E}']/\mathcal{A}}
\otimes_{\mathcal{A}[\mathcal{E}']} \mathcal{B}$.
The pullback of $\xi$ under the quasi-isomorphism
$\NL(\alpha') \to \NL(\alpha)$ maps to zero in
$\text{Ext}^1_\mathcal{B}(\Omega', \mathcal{G})$
because the pullback of the extension $\mathcal{H}$
by the map $\Omega' \to \Omega$ is split as $\Omega'$ is the free
$\mathcal{B}$-module on the sheaf of sets $\mathcal{E}'$ and since
by construction there is a commutative diagram
$$
\xymatrix{
\mathcal{E}' \ar[r] \ar[d] & \mathcal{E} \ar[d] \\
\mathcal{H} \ar[r] & \Omega
}
$$
This finishes the proof.
\end{proof}

\begin{lemma}
\label{lemma-choices-ringed-topoi}
If there exists a solution to (\ref{equation-to-solve-ringed-topoi}),
then the set of isomorphism classes of solutions is principal homogeneous
under $\text{Ext}^1_\mathcal{O}(
\NL_{\mathcal{O}/\mathcal{O}_\mathcal{B}}, \mathcal{G})$.
\end{lemma}

\begin{proof}
We observe right away that given two solutions $\mathcal{O}'_1$ and
$\mathcal{O}'_2$ to (\ref{equation-to-solve-ringed-topoi}) we obtain by
Lemma \ref{lemma-huge-diagram-ringed-topoi} an obstruction element
$o(\mathcal{O}'_1, \mathcal{O}'_2) \in \text{Ext}^1_\mathcal{O}(
\NL_{\mathcal{O}/\mathcal{O}_\mathcal{B}}, \mathcal{G})$
to the existence of a map $\mathcal{O}'_1 \to \mathcal{O}'_2$.
Clearly, this element
is the obstruction to the existence of an isomorphism, hence separates
the isomorphism classes. To finish the proof it therefore suffices to
show that given a solution $\mathcal{O}'$ and an element
$\xi \in \text{Ext}^1_\mathcal{O}(
\NL_{\mathcal{O}/\mathcal{O}_\mathcal{B}}, \mathcal{G})$
we can find a second solution $\mathcal{O}'_\xi$ such that
$o(\mathcal{O}', \mathcal{O}'_\xi) = \xi$.

\medskip\noindent
Pick $\alpha : \mathcal{E} \to \mathcal{O}$ as in
Lemma \ref{lemma-NL-represent-ext-class-ringed-topoi}
for the class $\xi$. Consider the surjection
$f^{-1}\mathcal{O}_\mathcal{B}[\mathcal{E}] \to \mathcal{O}$
with kernel $\mathcal{I}$ and corresponding naive cotangent complex
$\NL(\alpha) = (\mathcal{I}/\mathcal{I}^2 \to
\Omega_{f^{-1}\mathcal{O}_\mathcal{B}[\mathcal{E}]/
f^{-1}\mathcal{O}_\mathcal{B}}
\otimes_{f^{-1}\mathcal{O}_\mathcal{B}[\mathcal{E}]} \mathcal{O})$.
By the lemma $\xi$ is the class of a morphism
$\delta : \mathcal{I}/\mathcal{I}^2 \to \mathcal{G}$.
After replacing $\mathcal{E}$ by
$\mathcal{E} \times_\mathcal{O} \mathcal{O}'$ we may also assume
that $\alpha$ factors through a map
$\alpha' : \mathcal{E} \to \mathcal{O}'$.

\medskip\noindent
These choices determine an $f^{-1}\mathcal{O}_{\mathcal{B}'}$-algebra map
$\varphi : \mathcal{O}_{\mathcal{B}'}[\mathcal{E}] \to \mathcal{O}'$.
Let $\mathcal{I}' = \text{Ker}(\varphi)$.
Observe that $\varphi$ induces a map
$\varphi|_{\mathcal{I}'} : \mathcal{I}' \to \mathcal{G}$
and that $\mathcal{O}'$ is the pushout, as in the following
diagram
$$
\xymatrix{
0 \ar[r] & \mathcal{G} \ar[r] & \mathcal{O}' \ar[r] &
\mathcal{O} \ar[r] & 0 \\
0 \ar[r] & \mathcal{I}' \ar[u]^{\varphi|_{\mathcal{I}'}} \ar[r] &
f^{-1}\mathcal{O}_{\mathcal{B}'}[\mathcal{E}] \ar[u] \ar[r] &
\mathcal{O} \ar[u]_{=} \ar[r] & 0
}
$$
Let $\psi : \mathcal{I}' \to \mathcal{G}$ be the sum of the map
$\varphi|_{\mathcal{I}'}$ and the composition
$$
\mathcal{I}' \to \mathcal{I}'/(\mathcal{I}')^2 \to
\mathcal{I}/\mathcal{I}^2 \xrightarrow{\delta} \mathcal{G}.
$$
Then the pushout along $\psi$ is an other ring extension
$\mathcal{O}'_\xi$ fitting into a diagram as above.
A calculation (omitted) shows that $o(\mathcal{O}', \mathcal{O}'_\xi) = \xi$
as desired.
\end{proof}

\begin{lemma}
\label{lemma-extensions-of-ringed-topoi}
Let $(\Sh(\mathcal{B}), \mathcal{O}_\mathcal{B})$ be a ringed topos
and let $\mathcal{J}$ be an $\mathcal{O}_\mathcal{B}$-module.
\begin{enumerate}
\item The set of extensions of sheaves of rings
$0 \to \mathcal{J} \to \mathcal{O}_{\mathcal{B}'} \to
\mathcal{O}_\mathcal{B} \to 0$
where $\mathcal{J}$ is an ideal of square zero is canonically bijective to
$\text{Ext}^1_{\mathcal{O}_\mathcal{B}}(
\NL_{\mathcal{O}_\mathcal{B}/\mathbf{Z}}, \mathcal{J})$.
\item Given a morphism of ringed topoi
$f : (\Sh(\mathcal{C}), \mathcal{O}) \to
(\Sh(\mathcal{B}), \mathcal{O}_\mathcal{B})$, an $\mathcal{O}$-module
$\mathcal{G}$, an $f^{-1}\mathcal{O}_\mathcal{B}$-module map
$c : f^{-1}\mathcal{J} \to \mathcal{G}$, and
given extensions of sheaves of rings with square zero kernels:
\begin{enumerate}
\item[(a)] $0 \to \mathcal{J} \to \mathcal{O}_{\mathcal{B}'} \to
\mathcal{O}_\mathcal{B} \to 0$ corresponding to
$\alpha \in \text{Ext}^1_{\mathcal{O}_\mathcal{B}}(
\NL_{\mathcal{O}_\mathcal{B}/\mathbf{Z}}, \mathcal{J})$,
\item[(b)] $0 \to \mathcal{G} \to \mathcal{O}' \to \mathcal{O} \to 0$
corresponding to
$\beta \in \text{Ext}^1_\mathcal{O}(\NL_{\mathcal{O}/\mathbf{Z}}, \mathcal{G})$
\end{enumerate}
then there is a morphism $(\Sh(\mathcal{C}), \mathcal{O}') \to
(\Sh(\mathcal{B}, \mathcal{O}_{\mathcal{B}'})$ fitting into a diagram
(\ref{equation-to-solve-ringed-topoi}) if and only if $\beta$ and $\alpha$
map to the same element of
$\text{Ext}^1_\mathcal{O}(
Lf^*\NL_{\mathcal{O}_\mathcal{B}/\mathbf{Z}}, \mathcal{G})$.
\end{enumerate}
\end{lemma}

\begin{proof}
To prove this we apply the previous results where we work over
the base ringed topos $(\Sh(*), \mathbf{Z})$ with trivial thickening.
Part (1) follows from Lemma \ref{lemma-choices-ringed-topoi}
and the fact that there exists a solution, namely
$\mathcal{J} \oplus \mathcal{O}_\mathcal{B}$.
Part (2) follows from Lemma \ref{lemma-huge-diagram-ringed-topoi}
and a compatibility between the constructions in the proofs
of Lemmas \ref{lemma-choices-ringed-topoi} and
\ref{lemma-huge-diagram-ringed-topoi}
whose statement and proof we omit.
\end{proof}






\input{chapters}

\bibliography{my}
\bibliographystyle{amsalpha}

\end{document}