Lemma 93.16.2. In Situation 93.9.9 if $U_{12} \to \mathop{\mathrm{Spec}}(k)$ is smooth, then the morphism
\[ \mathcal{D}\! \mathit{ef}_ X \longrightarrow \mathcal{D}\! \mathit{ef}_{U_1} \times \mathcal{D}\! \mathit{ef}_{U_2} = \mathcal{D}\! \mathit{ef}_{P_1} \times \mathcal{D}\! \mathit{ef}_{P_2} \]
is smooth. If in addition $U_1$ is a local complete intersection over $k$, then
\[ \mathcal{D}\! \mathit{ef}_ X \longrightarrow \mathcal{D}\! \mathit{ef}_{U_2} = \mathcal{D}\! \mathit{ef}_{P_2} \]
is smooth.
Proof.
The equality signs hold by Lemma 93.9.7. Let us think of $\mathcal{C}_\Lambda $ as a deformation category over $\mathcal{C}_\Lambda $ as in Formal Deformation Theory, Section 90.9. Then
\[ \mathcal{D}\! \mathit{ef}_{P_1} \times \mathcal{D}\! \mathit{ef}_{P_2} = \mathcal{D}\! \mathit{ef}_{P_1} \times _{\mathcal{C}_\Lambda } \mathcal{D}\! \mathit{ef}_{P_2}, \]
see Formal Deformation Theory, Remarks 90.5.2 (14). Using Lemma 93.9.10 the first statement is that the functor
\[ \mathcal{D}\! \mathit{ef}_{P_1} \times _{\mathcal{D}\! \mathit{ef}_{P_{12}}} \mathcal{D}\! \mathit{ef}_{P_2} \longrightarrow \mathcal{D}\! \mathit{ef}_{P_1} \times _{\mathcal{C}_\Lambda } \mathcal{D}\! \mathit{ef}_{P_2} \]
is smooth. This follows from Formal Deformation Theory, Lemma 90.20.2 as long as we can show that $T\mathcal{D}\! \mathit{ef}_{P_{12}} = (0)$. This vanishing follows from Lemma 93.8.4 as $P_{12}$ is smooth over $k$. For the second statement it suffices to show that $\mathcal{D}\! \mathit{ef}_{P_1} \to \mathcal{C}_\Lambda $ is smooth, see Formal Deformation Theory, Lemma 90.8.7. In other words, we have to show $\mathcal{D}\! \mathit{ef}_{P_1}$ is unobstructed, which is Lemma 93.16.1.
$\square$
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