The Stacks project

Lemma 66.20.2. Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Let $T \subset |X|$ be an irreducible closed subset. Let $\xi \in T$ be the generic point (Proposition 66.12.4). Then $\text{codim}(T, |X|)$ (Topology, Definition 5.11.1) is the dimension of the local ring of $X$ at $\xi $ (Properties of Spaces, Definition 64.10.2).

Proof. Choose a scheme $U$, a point $u \in U$, and an ├ętale morphism $U \to X$ sending $u$ to $\xi $. Then any sequence of nontrivial specializations $\xi _ e \leadsto \ldots \leadsto \xi _0 = \xi $ can be lifted to a sequence $u_ e \leadsto \ldots \leadsto u_0 = u$ in $U$ by Lemma 66.12.2. Conversely, any sequence of nontrivial specializations $u_ e \leadsto \ldots \leadsto u_0 = u$ in $U$ maps to a sequence of nontrivial specializations $\xi _ e \leadsto \ldots \leadsto \xi _0 = \xi $ by Lemma 66.12.1. Because $|X|$ and $U$ are sober topological spaces we conclude that the codimension of $T$ in $|X|$ and of $\overline{\{ u\} }$ in $U$ are the same. In this way the lemma reduces to the schemes case which is Properties, Lemma 28.10.3. $\square$


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