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
\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
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
\mathcal{D}\! \mathit{ef}_ X \longrightarrow \prod \mathcal{D}\! \mathit{ef}_{\mathcal{O}_{X, p_ i}} \longrightarrow \prod \mathcal{D}\! \mathit{ef}_{\mathcal{O}_{X, p_ i}^ h} \longrightarrow \prod \mathcal{D}\! \mathit{ef}_{\mathcal{O}_{X, p_ i}^\wedge }
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
\mathcal{D}\! \mathit{ef}_ X \longrightarrow \mathcal{D}\! \mathit{ef}_{U_2}
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
\mathcal{D}\! \mathit{ef}_ X \longrightarrow \mathcal{D}\! \mathit{ef}_{U_2}
is smooth by Lemma 93.16.2. Applying Lemmas 93.9.7 and 93.16.1 we win. \square
Comments (0)