The Stacks project

Proof. Let $f : X \to S$ be locally projective. In order to show that $f$ is proper we may work locally on the base, see Lemma 29.39.3. Hence, by Lemma 29.41.4 above we may assume there exists a closed immersion $X \to \mathbf{P}^ n_ S$. By Lemmas 29.39.4 and 29.39.6 it suffices to prove that $\mathbf{P}^ n_ S \to S$ is proper. Since $\mathbf{P}^ n_ S \to S$ is the base change of $\mathbf{P}^ n_{\mathbf{Z}} \to \mathop{\mathrm{Spec}}(\mathbf{Z})$ it suffices to show that $\mathbf{P}^ n_{\mathbf{Z}} \to \mathop{\mathrm{Spec}}(\mathbf{Z})$ is proper, see Lemma 29.39.5. By Constructions, Lemma 27.8.8 the scheme $\mathbf{P}^ n_{\mathbf{Z}}$ is separated. By Constructions, Lemma 27.8.9 the scheme $\mathbf{P}^ n_{\mathbf{Z}}$ is quasi-compact. It is clear that $\mathbf{P}^ n_{\mathbf{Z}} \to \mathop{\mathrm{Spec}}(\mathbf{Z})$ is locally of finite type since $\mathbf{P}^ n_{\mathbf{Z}}$ is covered by the affine opens $D_{+}(X_ i)$ each of which is the spectrum of the finite type $\mathbf{Z}$-algebra

\[ \mathbf{Z}[X_0/X_ i, \ldots , X_ n/X_ i]. \]

Finally, we have to show that $\mathbf{P}^ n_{\mathbf{Z}} \to \mathop{\mathrm{Spec}}(\mathbf{Z})$ is universally closed. This follows from Constructions, Lemma 27.8.11 and the valuative criterion (see Schemes, Proposition 26.20.6). $\square$


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