The Stacks project

Lemma 109.25.1. Let $g \geq 2$. The stack $\overline{\mathcal{M}}_ g$ is separated.

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

\[ \mathcal{M}_ g \to \overline{\mathcal{M}}_ g \quad \text{and}\quad \overline{\mathcal{M}}_ g \to \mathop{\mathrm{Spec}}(\mathbf{Z}) \]

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

\[ \xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r] \ar[d] & \mathcal{M}_ g \ar[r] & \overline{\mathcal{M}}_ g \ar[d] \\ \mathop{\mathrm{Spec}}(R) \ar[rr] \ar@{..>}[rru] & & \mathop{\mathrm{Spec}}(\mathbf{Z}) } \]

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. $\square$

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