The Stacks project

Lemma 33.15.4. Let $k$ be a field. Let $X$ be a scheme over $k$. If $X_ K$ is proper over $K$ for some field extension $K/k$, then $X$ is proper over $k$.

Proof. Let $K/k$ be a field extension such that $X_ K$ is proper over $K$. Recall that this implies $X_ K$ is separated and quasi-compact (Morphisms, Definition 29.41.1). The morphism $X_ K \to X$ is surjective. Hence $X$ is quasi-compact as the image of a quasi-compact scheme (Properties, Definition 28.26.1). Since $X_ K$ is separated we see that $X$ is quasi-separated: If $U, V \subset X$ are affine open, then $(U \cap V)_ K = U_ K \cap V_ K$ is quasi-compact and $(U \cap V)_ K \to U \cap V$ is surjective. Thus Schemes, Lemma 26.21.6 applies.

Write $K = \mathop{\mathrm{colim}}\nolimits A_ i$ as the colimit of the subalgebras of $K$ which are of finite type over $k$. Denote $X_ i = X \times _{\mathop{\mathrm{Spec}}(k)} \mathop{\mathrm{Spec}}(A_ i)$. By Limits, Lemma 32.13.1 there exists an $i$ such that $X_ i \to \mathop{\mathrm{Spec}}(A_ i)$ is proper. Here we use that $X$ is quasi-compact and quasi-separated as just shown. Choose a maximal ideal $\mathfrak m \subset A_ i$. By the Hilbert Nullstellensatz (Algebra, Theorem 10.34.1) the residue field $k' = A_ i/\mathfrak m$ is a finite extension of $k$. The base change $X_{k'} \to \mathop{\mathrm{Spec}}(k')$ is proper (Morphisms, Lemma 29.41.5). Since $k'/k$ is finite both $X_{k'} \to X$ and the composition $X_{k'} \to \mathop{\mathrm{Spec}}(k)$ are proper as well (Morphisms, Lemmas 29.44.11, 29.41.5, and 29.41.4). The first implies that $X$ is separated over $k$ as $X_{k'}$ is separated (Morphisms, Lemma 29.41.11). The second implies that $X \to \mathop{\mathrm{Spec}}(k)$ is proper by Morphisms, Lemma 29.41.9. $\square$


Comments (0)


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Your email address will not be published.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.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0BDF. Beware of the difference between the letter 'O' and the digit '0'.