Lemma 29.41.5. A locally projective morphism is proper.

**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

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). $\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)