The Stacks project

Lemma 33.15.1. Let $k$ be a field. Let $X$ be a scheme over $k$. If there exists an ample invertible sheaf on $X_ K$ for some field extension $K/k$, then $X$ has an ample invertible sheaf.

Proof. Let $K/k$ be a field extension such that $X_ K$ has an ample invertible sheaf $\mathcal{L}$. 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 quasi-separated (by Properties, Lemma 28.26.7) 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 $X_ i = X \times _{\mathop{\mathrm{Spec}}(k)} \mathop{\mathrm{Spec}}(A_ i)$. Since $X_ K = \mathop{\mathrm{lim}}\nolimits X_ i$ we find an $i$ and an invertible sheaf' $\mathcal{L}_ i$ on $X_ i$ whose pullback to $X_ K$ is $\mathcal{L}$ (Limits, Lemma 32.10.3; here and below we use that $X$ is quasi-compact and quasi-separated as just shown). By Limits, Lemma 32.4.15 we may assume $\mathcal{L}_ i$ is ample after possibly increasing $i$. Fix such an $i$ and let $\mathfrak m \subset A_ i$ be a maximal ideal. By the Hilbert Nullstellensatz (Algebra, Theorem 10.34.1) the residue field $k' = A_ i/\mathfrak m$ is a finite extension of $k$. Hence $X_{k'} \subset X_ i$ is a closed subscheme hence has an ample invertible sheaf (Properties, Lemma 28.26.3). Since $X_{k'} \to X$ is finite locally free we conclude that $X$ has an ample invertible sheaf by Divisors, Proposition 31.17.9. $\square$

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Your email address will not be published.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 0BDC. Beware of the difference between the letter 'O' and the digit '0'.