The Stacks project

64.9 Dimension at a point

We can use Descent, Lemma 35.18.2 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 64.9.1. 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 ├ętale morphism $a : U \to X$ from a scheme to $X$ and a point $u \in U$ with $a(u) = x$. See Definition 64.7.5, Lemma 64.7.4, and Descent, Lemma 35.18.2.

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 63.14.9. Namely, in this example we have $\dim _ x(X) = 0$ and $\dim _ x(|X|) = 1$ (this holds for any $x \in |X|$). In particular, it also means that the dimension of $X$ (as defined below) is different from the dimension of $|X|$.

Definition 64.9.2. 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 ├ętale morphisms also, see Section 64.10.

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