The Stacks project

Lemma 107.15.1. In the situation above the following are equivalent

  1. the classifying morphism $\mathop{\mathrm{Spec}}(k) \to \mathcal{C}\! \mathit{urves}$ factors through the open where $\mathcal{C}\! \mathit{urves}\to \mathop{\mathrm{Spec}}(\mathbf{Z})$ is smooth,

  2. the deformation category $\mathcal{D}\! \mathit{ef}_ X$ is unobstructed.

Proof. Since $\mathcal{C}\! \mathit{urves}\longrightarrow \mathop{\mathrm{Spec}}(\mathbf{Z})$ is locally of finite presentation (Lemma 107.5.3) formation of the open substack where $\mathcal{C}\! \mathit{urves}\longrightarrow \mathop{\mathrm{Spec}}(\mathbf{Z})$ is smooth commutes with flat base change (Morphisms of Stacks, Lemma 99.32.6). Since the Cohen ring $\Lambda $ is flat over $\mathbf{Z}$, we may work over $\Lambda $. In other words, we are trying to prove that

\[ \Lambda \text{-}\mathcal{C}\! \mathit{urves}\longrightarrow \mathop{\mathrm{Spec}}(\Lambda ) \]

is smooth in an open neighbourhood of the point $x_0 : \mathop{\mathrm{Spec}}(k) \to \Lambda \text{-}\mathcal{C}\! \mathit{urves}$ defined by $X/k$ if and only if $\mathcal{D}\! \mathit{ef}_ X$ is unobstructed.

The lemma now follows from Geometry of Stacks, Lemma 105.2.7 and the equality

\[ \mathcal{D}\! \mathit{ef}_ X = \mathcal{F}_{\Lambda \text{-}\mathcal{C}\! \mathit{urves}, k, x_0} \]

This equality is not completely trivial to esthablish. Namely, on the left hand side we have the deformation category classifying all flat deformations $Y \to \mathop{\mathrm{Spec}}(A)$ of $X$ as a scheme over $A \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}_\Lambda )$. On the right hand side we have the deformation category classifying all flat morphisms $Y \to \mathop{\mathrm{Spec}}(A)$ with special fibre $X$ where $Y$ is an algebraic space and $Y \to \mathop{\mathrm{Spec}}(A)$ is proper, of finite presentation, and of relative dimension $\leq 1$. Since $A$ is Artinian, we find that $Y$ is a scheme for example by Spaces over Fields, Lemma 70.9.3. Thus it remains to show: a flat deformation $Y \to \mathop{\mathrm{Spec}}(A)$ of $X$ as a scheme over an Artinian local ring $A$ with residue field $k$ is proper, of finite presentation, and of relative dimension $\leq 1$. Relative dimension is defined in terms of fibres and hence holds automatically for $Y/A$ since it holds for $X/k$. The morphism $Y \to \mathop{\mathrm{Spec}}(A)$ is proper and locally of finite presentation as this is true for $X \to \mathop{\mathrm{Spec}}(k)$, see More on Morphisms, Lemma 37.10.3. $\square$


Comments (2)

Comment #3174 by Anonymous on

The first sentence of the proof is missing the phrase "is smooth" before "commutes with flat base change".


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