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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