The Stacks project

66.9 Dimension at a point

We can use Descent, Lemma 35.21.2 to define the dimension of an algebraic space $X$ at a point $x$. This will give us a different notion than the topological one (i.e., the dimension of $|X|$ at $x$).

Definition 66.9.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \in |X|$ be a point of $X$. We define the dimension of $X$ at $x$ to be the element $\dim _ x(X) \in \{ 0, 1, 2, \ldots , \infty \} $ such that $\dim _ x(X) = \dim _ u(U)$ for any (equivalently some) pair $(a : U \to X, u)$ consisting of an étale morphism $a : U \to X$ from a scheme to $X$ and a point $u \in U$ with $a(u) = x$. See Definition 66.7.5, Lemma 66.7.4, and Descent, Lemma 35.21.2.

Warning: It is not the case that $\dim _ x(X) = \dim _ x(|X|)$ in general. A counter example is the algebraic space $X$ of Spaces, Example 65.14.9. Namely, let $x \in |X|$ be a point not equal to the generic point $x_0$ of $|X|$. Then we have $\dim _ x(X) = 0$ but $\dim _ x(|X|) = 1$. In particular, the dimension of $X$ (as defined below) is different from the dimension of $|X|$.

Definition 66.9.2. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The dimension $\dim (X)$ of $X$ is defined by the rule

\[ \dim (X) = \sup \nolimits _{x \in |X|} \dim _ x(X) \]

By Properties, Lemma 28.10.2 we see that this is the usual notion if $X$ is a scheme. There is another integer that measures the dimension of a scheme at a point, namely the dimension of the local ring. This invariant is compatible with étale morphisms also, see Section 66.10.


Comments (2)

Comment #6999 by Laurent Moret-Bailly on

In the warning after definition 04N5: if we take for the generic point, I think that is open , hence .


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Your email address will not be published.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 04N3. Beware of the difference between the letter 'O' and the digit '0'.