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.
\[ \mathcal{D}\! \mathit{ef}_ X \longrightarrow \mathcal{D}\! \mathit{ef}_{U_1} \times \mathcal{D}\! \mathit{ef}_{U_2} = \mathcal{D}\! \mathit{ef}_{P_1} \times \mathcal{D}\! \mathit{ef}_{P_2} \]
\[ \mathcal{D}\! \mathit{ef}_ X \longrightarrow \mathcal{D}\! \mathit{ef}_{U_2} = \mathcal{D}\! \mathit{ef}_{P_2} \]
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. $\square$
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