Lemma 93.16.3 (0DZP)—The Stacks project

Table of contents
Part 6: Deformation Theory
Chapter 93: Deformation Problems
Section 93.16: Unobstructed deformation problems
Lemma 93.16.3 (cite)

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. $\square$

The Stacks project

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