The Stacks project

Lemma 27.13.4. Let $S$ be a scheme. The structure morphism $\mathbf{P}^ n_ S \to S$ is

  1. separated,

  2. quasi-compact,

  3. satisfies the existence and uniqueness parts of the valuative criterion, and

  4. universally closed.

Proof. All these properties are stable under base change (this is clear for the last two and for the other two see Schemes, Lemmas 26.21.12 and 26.19.3). Hence it suffices to prove them for the morphism $\mathbf{P}^ n_{\mathbf{Z}} \to \mathop{\mathrm{Spec}}(\mathbf{Z})$. Separatedness is Lemma 27.8.8. Quasi-compactness follows from Lemma 27.13.3. Existence and uniqueness of the valuative criterion follow from Lemma 27.8.11. Universally closed follows from the above and Schemes, Proposition 26.20.6. $\square$

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