The Stacks project

Proof. Since $\mathcal{C}\! \mathit{urves}\longrightarrow \mathop{\mathrm{Spec}}(\mathbf{Z})$ is locally of finite presentation (Lemma 109.5.3) 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 101.33.6). Since the Cohen ring $\Lambda $ is flat over $\mathbf{Z}$, we may work over $\Lambda $. In other words, we are trying to prove that

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 107.2.7 and the equality

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 72.9.3. 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. $\square$