Lemma 70.10.2. Let $k$ be a field with algebraic closure $\overline{k}$. Let $X$ be a quasi-separated algebraic space over $k$.

1. If there exists a field extension $k \subset K$ such that $X_ K$ is a scheme, then $X_{\overline{k}}$ is a scheme.

2. If $X$ is quasi-compact and there exists a field extension $k \subset K$ such that $X_ K$ is a scheme, then $X_{k'}$ is a scheme for some finite separable extension $k'$ of $k$.

Proof. Since every algebraic space is the union of its quasi-compact open subspaces, we see that the first part of the lemma follows from the second part (some details omitted). Thus we assume $X$ is quasi-compact and we assume given an extension $k \subset K$ with $K_ K$ representable. Write $K = \bigcup A$ as the colimit of finitely generated $k$-subalgebras $A$. By Limits of Spaces, Lemma 68.5.11 we see that $X_ A$ is a scheme for some $A$. Choose a maximal ideal $\mathfrak m \subset A$. By the Hilbert Nullstellensatz (Algebra, Theorem 10.33.1) the residue field $k' = A/\mathfrak m$ is a finite extension of $k$. Thus we see that $X_{k'}$ is a scheme. If $k' \supset k$ is not separable, let $k' \supset k'' \supset k$ be the subextension found in Fields, Lemma 9.14.6. Since $k'/k''$ is purely inseparable, by Lemma 70.10.1 the algebraic space $X_{k''}$ is a scheme. Since $k''|k$ is separable the proof is complete. $\square$

## 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Your email address will not be published.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.

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 0B84. Beware of the difference between the letter 'O' and the digit '0'.