The Stacks project

Definition 5.11.1. Let $X$ be a topological space. Let $Y \subset X$ be an irreducible closed subset. The codimension of $Y$ in $X$ is the supremum of the lengths $e$ of chains

\[ Y = Y_0 \subset Y_1 \subset \ldots \subset Y_ e \subset X \]

of irreducible closed subsets in $X$ starting with $Y$. We will denote this $\text{codim}(Y, X)$.

Comments (2)

Comment #10116 by Jacob Scharmberg on

Should the last inclusion be a non-strict inclusion, ie. ? If I understand correctly, if X is irreducible itself, we would want that to be part of the chain. That's also how it seems to be used in Definition 5.11.4.

Comment #10592 by on

The convention in the stacks project is that means that is a subset of . In particular, can be equal to .

There are also:

  • 2 comment(s) on Section 5.11: Codimension and catenary spaces

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View Definition 5.11.1 as pdf