This section is the continuation of Deformation Theory, Section 90.7 which we urge the reader to read first. We briefly recall the setup. We have a first order thickening $t : (S, \mathcal{O}_ S) \to (S', \mathcal{O}_{S'})$ of ringed spaces with $\mathcal{J} = \mathop{\mathrm{Ker}}(t^\sharp )$, 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. We ask whether we can find the question mark fitting into the following diagram

91.21.0.1
\begin{equation} \label{cotangent-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}

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 Deformation Theory, Equation (90.3.1.1) where $\mathop{\mathrm{Ker}}(i^\sharp )$ is identified with $\mathcal{G}$ such that $(f')^\sharp $ induces the given map $c$. We will say $X'$ is a *solution* to (91.21.0.1).

Lemma 91.21.1. In the situation above we have

There is a canonical element $\xi \in \mathop{\mathrm{Ext}}\nolimits ^2_{\mathcal{O}_ X}(L_{X/S}, \mathcal{G})$ whose vanishing is a sufficient and necessary condition for the existence of a solution to (91.21.0.1).

If there exists a solution, then the set of isomorphism classes of solutions is principal homogeneous under $\mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ X}(L_{X/S}, \mathcal{G})$.

Given a solution $X'$, the set of automorphisms of $X'$ fitting into (91.21.0.1) is canonically isomorphic to $\mathop{\mathrm{Ext}}\nolimits ^0_{\mathcal{O}_ X}(L_{X/S}, \mathcal{G})$.

**Proof.**
Via the identifications $\mathop{N\! L}\nolimits _{X/S} = \tau _{\geq -1}L_{X/S}$ (Lemma 91.20.4) and $H^0(L_{X/S}) = \Omega _{X/S}$ (Lemma 91.20.2) we have seen parts (2) and (3) in Deformation Theory, Lemmas 90.7.1 and 90.7.3.

Proof of (1). Roughly speaking, this follows from the discussion in Deformation Theory, Remark 90.7.9 by replacing the naive cotangent complex by the full cotangent complex. Here is a more detailed explanation. By Deformation Theory, Lemma 90.7.8 there exists an element

\[ \xi ' \in \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ X}(Lf^*\mathop{N\! L}\nolimits _{S/S'}, \mathcal{G}) = \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ X}(Lf^*L_{S/S'}, \mathcal{G}) \]

such that a solution exists if and only if this element is in the image of the map

\[ \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ X}(NL_{X/S'}, \mathcal{G}) = \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ X}(L_{X/S'}, \mathcal{G}) \longrightarrow \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ X}(Lf^*L_{S/S'}, \mathcal{G}) \]

The distinguished triangle of Lemma 91.20.3 for $X \to S \to S'$ gives rise to a long exact sequence

\[ \ldots \to \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ X}(L_{X/S'}, \mathcal{G}) \to \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ X}(Lf^*L_{S/S'}, \mathcal{G}) \to \mathop{\mathrm{Ext}}\nolimits ^2_{\mathcal{O}_ X}(L_{X/S}, \mathcal{G}) \to \ldots \]

Hence taking $\xi $ the image of $\xi '$ works.
$\square$

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