Lemma 89.20.1. In the situation above we have

1. 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 (89.20.0.1).

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

3. Given a solution $X'$, the set of automorphisms of $X'$ fitting into (89.20.0.1) 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 89.19.4) and $H^0(L_{X/S}) = \Omega _{X/S}$ (Lemma 89.19.2) we have seen parts (2) and (3) in Deformation Theory, Lemmas 88.7.1 and 88.7.3.

Proof of (1). We will use the results of Deformation Theory, Lemma 88.7.4 without further mention. Let $\alpha \in \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ S}(\mathop{N\! L}\nolimits _{S/\mathbf{Z}}, \mathcal{J})$ be the element corresponding to the isomorphism class of $S'$. The existence of $X'$ corresponds to an element $\beta \in \mathop{\mathrm{Ext}}\nolimits _{\mathcal{O}_ X}^1(\mathop{N\! L}\nolimits _{X/\mathbf{Z}}, \mathcal{G})$ which maps to the image of $\alpha$ in $\mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ X}(Lf^*\mathop{N\! L}\nolimits _{S/\mathbf{Z}}, \mathcal{G})$. Note that

$\mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ X}(Lf^*\mathop{N\! L}\nolimits _{S/\mathbf{Z}}, \mathcal{G}) = \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ X}(Lf^*L_{S/\mathbf{Z}}, \mathcal{G})$

and

$\mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ X}(\mathop{N\! L}\nolimits _{X/\mathbf{Z}}, \mathcal{G}) = \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ X}(L_{X/\mathbf{Z}}, \mathcal{G})$

by Lemma 89.19.4. The distinguished triangle of Lemma 89.19.3 for $X \to S \to (*, \mathbf{Z})$ gives rise to a long exact sequence

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

We obtain the result with $\xi$ the image of $\alpha$. $\square$

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