The Stacks project

Remark 105.5.1. In general, the dimension of the algebraic space $X$ at a point $x$ may not coincide with the dimension of the underlying topological space $|X|$ at $x$. E.g. if $k$ is a field of characteristic zero and $X = \mathbf{A}^1_ k / \mathbf{Z}$, then $X$ has dimension $1$ (the dimension of $\mathbf{A}^1_ k$) at each of its points, while $|X|$ has the indiscrete topology, and hence is of Krull dimension zero. On the other hand, in Algebraic Spaces, Example 63.14.9 there is given an example of an algebraic space which is of dimension $0$ at each of its points, while $|X|$ is irreducible of Krull dimension $1$, and admits a generic point (so that the dimension of $|X|$ at any of its points is $1$); see also the discussion of this example in Properties of Spaces, Section 64.9.

On the other hand, if $X$ is a decent algebraic space, in the sense of Decent Spaces, Definition 66.6.1 (in particular, if $X$ is quasi-separated; see Decent Spaces, Section 66.6) then in fact the dimension of $X$ at $x$ does coincide with the dimension of $|X|$ at $x$; see Decent Spaces, Lemma 66.12.5.


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Your email address will not be published.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