Lemma 93.16.1. In Example 93.8.1 let P be a local complete intersection over k (Algebra, Definition 10.135.1). Then \mathcal{D}\! \mathit{ef}_ P is unobstructed.
93.16 Unobstructed deformation problems
Let p : \mathcal{F} \to \mathcal{C}_\Lambda be a category cofibred in groupoids. Recall that we say \mathcal{F} is smooth or unobstructed if p is smooth. This means that given a surjection \varphi : A' \to A in \mathcal{C}_\Lambda and x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{F}(A)) there exists a morphism f : x' \to x in \mathcal{F} with p(f) = \varphi . See Formal Deformation Theory, Section 90.9. In this section we give some geometrically meaningful examples.
Proof. Let (A, Q) \to (k, P) be an object of \mathcal{D}\! \mathit{ef}_ P. Then we see that A \to Q is a syntomic ring map by Algebra, Definition 10.136.1. Hence for any surjection A' \to A in \mathcal{C}_\Lambda we see that there is a morphism (A', Q') \to (A, Q) lifting A' \to A by Smoothing Ring Maps, Proposition 16.3.2. This proves the lemma. \square
Lemma 93.16.2. In Situation 93.9.9 if U_{12} \to \mathop{\mathrm{Spec}}(k) is smooth, then the morphism
is smooth. If in addition U_1 is a local complete intersection over k, then
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
see Formal Deformation Theory, Remarks 90.5.2 (14). Using Lemma 93.9.10 the first statement is that the functor
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
Lemma 93.16.3. In Example 93.9.1 let X be a scheme over k. Assume
X is separated, finite type over k and \dim (X) \leq 1,
X \to \mathop{\mathrm{Spec}}(k) is smooth except at the closed points p_1, \ldots , p_ n \in X.
Let \mathcal{O}_{X, p_1}, \mathcal{O}_{X, p_1}^ h, \mathcal{O}_{X, p_1}^\wedge be the local ring, henselization, completion. Consider the maps of deformation categories
The first arrow is smooth and the second and third arrows are smooth and induce isomorphisms on tangent spaces.
Proof. Choose an affine open U_2 \subset X containing p_1, \ldots , p_ n and the generic point of every irreducible component of X. This is possible by Varieties, Lemma 33.43.3 and Properties, Lemma 28.29.5. Then X \setminus U_2 is finite and we can choose an affine open U_1 \subset X \setminus \{ p_1, \ldots , p_ n\} such that X = U_1 \cup U_2. Set U_{12} = U_1 \cap U_2. Then U_1 and U_{12} are smooth affine schemes over k. We conclude that
is smooth by Lemma 93.16.2. Applying Lemmas 93.9.7 and 93.15.1 we win. \square
Lemma 93.16.4. In Example 93.9.1 let X be a scheme over k. Assume
X is separated, finite type over k and \dim (X) \leq 1,
X is a local complete intersection over k, and
X \to \mathop{\mathrm{Spec}}(k) is smooth except at finitely many points.
Then \mathcal{D}\! \mathit{ef}_ X is unobstructed.
Proof. Let p_1, \ldots , p_ n \in X be the points where X \to \mathop{\mathrm{Spec}}(k) isn't smooth. Choose an affine open U_2 \subset X containing p_1, \ldots , p_ n and the generic point of every irreducible component of X. This is possible by Varieties, Lemma 33.43.3 and Properties, Lemma 28.29.5. Then X \setminus U_2 is finite and we can choose an affine open U_1 \subset X \setminus \{ p_1, \ldots , p_ n\} such that X = U_1 \cup U_2. Set U_{12} = U_1 \cap U_2. Then U_1 and U_{12} are smooth affine schemes over k. We conclude that
is smooth by Lemma 93.16.2. Applying Lemmas 93.9.7 and 93.16.1 we win. \square
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