The Stacks project

70.13 Geometrically irreducible algebraic spaces

Spaces, Example 63.14.9 shows that it is best not to think about irreducible algebraic spaces in complete generality1. For decent (for example quasi-separated) algebraic spaces this kind of disaster doesn't happen. Thus we make the following definition only under the assumption that our algebraic space is decent.

Definition 70.13.1. Let $k$ be a field. Let $X$ be a decent algebraic space over $k$. We say $X$ is geometrically irreducible if the topological space $|X_{k'}|$ is irreducible2 for any field extension $k'$ of $k$.

Observe that $X_{k'}$ is a decent algebraic space (Decent Spaces, Lemma 66.6.5). Hence the topological space $|X_{k'}|$ is sober. Decent Spaces, Proposition 66.12.4.

[1] To be sure, if we say “the algebraic space $X$ is irreducible”, we probably mean to say “the topological space $|X|$ is irreducible”.
[2] An irreducible space is nonempty.

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Your email address will not be published.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