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
91.2.0.1
\begin{equation} \label{defos-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 an ideal of square zero and is identified with $N$ such that $A' \to B'$ induces the given map $c$. We will say $B'$ is a solution to (91.2.0.1).
Lemma 91.2.1. 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 (91.2.0.1) we have
There exist a canonical element in $\mathop{\mathrm{Ext}}\nolimits ^1_{B_1}(\mathop{N\! L}\nolimits _{B_1/A_1}, N_2)$ whose vanishing is a necessary and sufficient condition for the existence of a ring map $B'_1 \to B'_2$ fitting into the diagram.
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 $\mathop{\mathrm{Hom}}\nolimits _{B_1}(\Omega _{B_1/A_1}, N_2)$.
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
\[ \mathop{N\! L}\nolimits _{B_1/A_1} = (J/J^2 \to \Omega _{A_1[E]/A_1} \otimes _{A_1[E]} B_1) \]
in Algebra, Section 10.134. Since $\Omega _{A_1[E]/A_1} \otimes B_1$ is a free $B_1$-module we have
\[ \mathop{\mathrm{Ext}}\nolimits ^1_{B_1}(\mathop{N\! L}\nolimits _{B_1/A_1}, N_2) = \frac{\mathop{\mathrm{Hom}}\nolimits _{B_1}(J/J^2, N_2)}{\mathop{\mathrm{Hom}}\nolimits _{B_1}(\Omega _{A_1[E]/A_1} \otimes B_1, N_2)} \]
We will construct an obstruction in the module on the right. Let $J' = \mathop{\mathrm{Ker}}(A'_1[E] \to B_1)$. 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.
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.
$\square$
Lemma 91.2.2. If there exists a solution to (91.2.0.1), then the set of isomorphism classes of solutions is principal homogeneous under $\mathop{\mathrm{Ext}}\nolimits ^1_ B(\mathop{N\! L}\nolimits _{B/A}, N)$.
Proof.
We observe right away that given two solutions $B'_1$ and $B'_2$ to (91.2.0.1) we obtain by Lemma 91.2.1 an obstruction element $o(B'_1, B'_2) \in \mathop{\mathrm{Ext}}\nolimits ^1_ B(\mathop{N\! L}\nolimits _{B/A}, N)$ 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 $\xi \in \mathop{\mathrm{Ext}}\nolimits ^1_ B(\mathop{N\! L}\nolimits _{B/A}, N)$ we can find a second solution $B'_\xi $ such that $o(B', B'_\xi ) = \xi $.
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
\[ \mathop{N\! L}\nolimits _{B/A} = (J/J^2 \to \Omega _{A[E]/A} \otimes _{A[E]} B) \]
in Algebra, Section 10.134. Since $\Omega _{A[E]/A} \otimes B$ is a free $B$-module we have
\[ \mathop{\mathrm{Ext}}\nolimits ^1_ B(\mathop{N\! L}\nolimits _{B/A}, N) = \frac{\mathop{\mathrm{Hom}}\nolimits _ B(J/J^2, N)}{\mathop{\mathrm{Hom}}\nolimits _ B(\Omega _{A[E]/A} \otimes B, N)} \]
Thus we may represent $\xi $ as the class of a morphism $\delta : J/J^2 \to N$.
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 $J' = \mathop{\mathrm{Ker}}(A'[E] \to B)$. 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.
$\square$
Lemma 91.2.3. Let $A$ be a ring. Let $B$ be an $A$-algebra. Let $N$ be a $B$-module. The set of isomorphism classes of extensions of $A$-algebras
\[ 0 \to N \to B' \to B \to 0 \]
where $N$ is an ideal of square zero is canonically bijective to $\mathop{\mathrm{Ext}}\nolimits ^1_ B(\mathop{N\! L}\nolimits _{B/A}, N)$.
Proof.
To prove this we apply the previous results to the case where (91.2.0.1) is given by the diagram
\[ \xymatrix{ 0 \ar[r] & N \ar[r] & {?} \ar[r] & B \ar[r] & 0 \\ 0 \ar[r] & 0 \ar[u] \ar[r] & A \ar[u] \ar[r]^{\text{id}} & A \ar[u] \ar[r] & 0 } \]
Thus our lemma follows from Lemma 91.2.2 and the fact that there exists a solution, namely $N \oplus B$. (See remark below for a direct construction of the bijection.)
$\square$
Lemma 91.2.5. Given ring maps $A \to B \to C$, a $B$-module $M$, a $C$-module $N$, a $B$-linear map $c : M \to N$, and extensions of $A$-algebras with square zero kernels
$0 \to M \to B' \to B \to 0$ corresponding to $\xi \in \mathop{\mathrm{Ext}}\nolimits ^1_ B(\mathop{N\! L}\nolimits _{B/A}, M)$, and
$0 \to N \to C' \to C \to 0$ corresponding to $\zeta \in \mathop{\mathrm{Ext}}\nolimits ^1_ C(\mathop{N\! L}\nolimits _{C/A}, N)$.
See Lemma 91.2.3. Then there is an $A$-algebra map $B' \to C'$ compatible with $B \to C$ and $c$ if and only if $\xi $ and $\zeta $ map to the same element of $\mathop{\mathrm{Ext}}\nolimits ^1_ B(\mathop{N\! L}\nolimits _{B/A}, N)$.
Proof.
The stament makes sense as we have the maps
\[ \mathop{\mathrm{Ext}}\nolimits ^1_ B(\mathop{N\! L}\nolimits _{B/A}, M) \to \mathop{\mathrm{Ext}}\nolimits ^1_ B(\mathop{N\! L}\nolimits _{B/A}, N) \]
using the map $M \to N$ and
\[ \mathop{\mathrm{Ext}}\nolimits ^1_ C(\mathop{N\! L}\nolimits _{C/A}, N) \to \mathop{\mathrm{Ext}}\nolimits ^1_ B(\mathop{N\! L}\nolimits _{C/A}, N) \to \mathop{\mathrm{Ext}}\nolimits ^1_ B(\mathop{N\! L}\nolimits _{B/A}, N) \]
where the first arrows uses the restriction map $D(C) \to D(B)$ and the second arrow uses the canonical map of complexes $\mathop{N\! L}\nolimits _{B/A} \to \mathop{N\! L}\nolimits _{C/A}$. The statement of the lemma can be deduced from Lemma 91.2.1 applied to the diagram
\[ \xymatrix{ & 0 \ar[r] & N \ar[r] & C' \ar[r] & C \ar[r] & 0 \\ & 0 \ar[r]|\hole & 0 \ar[u] \ar[r] & A \ar[u] \ar[r]|\hole & A \ar[u] \ar[r] & 0 \\ 0 \ar[r] & M \ar[ruu] \ar[r] & B' \ar[r] & B \ar[ruu] \ar[r] & 0 \\ 0 \ar[r] & 0 \ar[ruu]|\hole \ar[u] \ar[r] & A \ar[ruu]|\hole \ar[u] \ar[r] & A \ar[ruu]|\hole \ar[u] \ar[r] & 0 } \]
and a compatibility between the constructions in the proofs of Lemmas 91.2.3 and 91.2.1 whose statement and proof we omit. (See remark below for a direct argument.)
$\square$
Lemma 91.2.7. Let $0 \to I \to A' \to A \to 0$, $A \to B$, and $c : I \to N$ be as in (91.2.0.1). Denote $\xi \in \mathop{\mathrm{Ext}}\nolimits ^1_ A(\mathop{N\! L}\nolimits _{A/A'}, I)$ the element corresponding to the extension $A'$ of $A$ by $I$ via Lemma 91.2.3. The set of isomorphism classes of solutions is canonically bijective to the fibre of
\[ \mathop{\mathrm{Ext}}\nolimits ^1_ B(\mathop{N\! L}\nolimits _{B/A'}, N) \to \mathop{\mathrm{Ext}}\nolimits ^1_ A(\mathop{N\! L}\nolimits _{A'/A}, N) \]
over the image of $\xi $.
Proof.
By Lemma 91.2.3 applied to $A' \to B$ and the $B$-module $N$ we see that elements $\zeta $ of $\mathop{\mathrm{Ext}}\nolimits ^1_ B(\mathop{N\! L}\nolimits _{B/A'}, N)$ parametrize extensions $0 \to N \to B' \to B \to 0$ of $A'$-algebras. By Lemma 91.2.5 applied to $A' \to A \to B$ and $c : I \to N$ we see that there is an $A'$-algebra map $A' \to B'$ compatible with $c$ and $A \to B$ if and only if $\zeta $ maps to $\xi $. Of course this is the same thing as saying $B'$ is a solution of (91.2.0.1).
$\square$
If our ring map $A \to B$ is a local complete intersection, then there is a solutuion. 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 16.3.2.
Lemma 91.2.9. If $A \to B$ is a local complete intersection ring map, then there exists a solution to (91.2.0.1).
First proof.
Write $B = A[x_1, \ldots , x_ n]/J$. By More on Algebra, Definition 15.33.2 the ideal $J$ is Koszul-regular. This implies $J$ is $H_1$-regular and quasi-regular, see More on Algebra, Section 15.32. 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 15.32.5 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 15.32.3) 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 (91.2.0.1) with $N = I \otimes _ A B$. The general case is obtained by doing a pushout along the given map $I \otimes _ A B \to N$.
$\square$
Second proof.
Please read Remark 91.2.8 before reading this proof. By More on Algebra, Lemma 15.33.6 the maps $\mathop{N\! L}\nolimits _{A'/A} \otimes _ A B \to \mathop{N\! L}\nolimits _{B/A'} \to \mathop{N\! L}\nolimits _{B/A}$ do form a distinguished triangle in $D(B)$. Hence it suffices to show that $\mathop{\mathrm{Ext}}\nolimits ^2_{B/A}(\mathop{N\! L}\nolimits _{B/A}, N)$ vanishes. By More on Algebra, Lemma 15.85.4 the complex $\mathop{N\! L}\nolimits _{B/A}$ is perfect of tor-amplitude in $[-1, 0]$. This implies our $\mathop{\mathrm{Ext}}\nolimits ^2$ vanishes for example by More on Algebra, Lemma 15.76.1 part (1).
$\square$
Comments (2)
Comment #8254 by s.goto on
Comment #8896 by Stacks project on