The Stacks project

Proof. The statement means that the morphism $\overline{\mathcal{M}}_ g \to \mathop{\mathrm{Spec}}(\mathbf{Z})$ is separated. We will prove this using the refined Noetherian valuative criterion as stated in More on Morphisms of Stacks, Lemma 106.11.4

Since $\overline{\mathcal{M}}_ g$ is an open substack of $\mathcal{C}\! \mathit{urves}$, we see $\overline{\mathcal{M}}_ g \to \mathop{\mathrm{Spec}}(\mathbf{Z})$ is quasi-separated and locally of finite presentation by Lemma 109.5.3. In particular the stack $\overline{\mathcal{M}}_ g$ is locally Noetherian (Morphisms of Stacks, Lemma 101.17.5). By Lemma 109.22.8 the open immersion $\mathcal{M}_ g \to \overline{\mathcal{M}}_ g$ has dense image. Also, $\mathcal{M}_ g \to \overline{\mathcal{M}}_ g$ is quasi-compact (Morphisms of Stacks, Lemma 101.8.2), hence of finite type. Thus all the preliminary assumptions of More on Morphisms of Stacks, Lemma 106.11.4 are satisfied for the morphisms

and it suffices to check the following: given any $2$-commutative diagram

where $R$ is a discrete valuation ring with field of fractions $K$ the category of dotted arrows is either empty or a setoid with exactly one isomorphism class. (Observe that we don't need to worry about $2$-arrows too much, see Morphisms of Stacks, Lemma 101.39.3). Unwinding what this means using that $\mathcal{M}_ g$, resp. $\overline{\mathcal{M}}_ g$ are the algebraic stacks parametrizing smooth, resp. stable families of genus $g$ curves, we find that what we have to prove is exactly the uniqueness result stated and proved in Lemma 109.24.2. $\square$