The Stacks project

Lemma 72.10.8. Let $k$ be a field. Let $X$ be an algebraic space over $k$. The following are equivalent

  1. $X$ is locally quasi-finite over $k$,

  2. $X$ is locally of finite type over $k$ and has dimension $0$,

  3. $X$ is a scheme and is locally quasi-finite over $k$,

  4. $X$ is a scheme and is locally of finite type over $k$ and has dimension $0$, and

  5. $X$ is a disjoint union of spectra of Artinian local $k$-algebras $A$ over $k$ with $\dim _ k(A) < \infty $.

Proof. Because we are over a field relative dimension of $X/k$ is the same as the dimension of $X$. Hence by Morphisms of Spaces, Lemma 67.34.6 we see that (1) and (2) are equivalent. Hence it follows from Lemma 72.9.1 (and trivial implications) that (1) – (4) are equivalent. Finally, Varieties, Lemma 33.20.2 shows that (1) – (4) are equivalent with (5). $\square$

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Your email address will not be published.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 06S0. Beware of the difference between the letter 'O' and the digit '0'.