The Stacks project

Lemma 109.5.3. The morphism $\mathcal{C}\! \mathit{urves}\to \mathop{\mathrm{Spec}}(\mathbf{Z})$ is quasi-separated and locally of finite presentation.

Proof. To check $\mathcal{C}\! \mathit{urves}\to \mathop{\mathrm{Spec}}(\mathbf{Z})$ is quasi-separated we have to show that its diagonal is quasi-compact and quasi-separated. This is immediate from Lemma 109.5.1. To prove that $\mathcal{C}\! \mathit{urves}\to \mathop{\mathrm{Spec}}(\mathbf{Z})$ is locally of finite presentation, it suffices to show that $\mathcal{C}\! \mathit{urves}$ is limit preserving, see Limits of Stacks, Proposition 102.3.8. This is Quot, Lemma 99.15.6. $\square$

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