We only define the codimension of irreducible closed subsets.
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 of irreducible closed subsets in $X$ starting with $Y$. We will denote this $\text{codim}(Y, X)$.
\[ Y = Y_0 \subset Y_1 \subset \ldots \subset Y_ e \subset X \]
The codimension is an element of $\{ 0, 1, 2, \ldots \} \cup \{ \infty \} $. If $\text{codim}(Y, X) < \infty $, then every chain can be extended to a maximal chain (but these do not all have to have the same length).
Lemma 5.11.2. Let $X$ be a topological space. Let $Y \subset X$ be an irreducible closed subset. Let $U \subset X$ be an open subset such that $Y \cap U$ is nonempty. Then
\[ \text{codim}(Y, X) = \text{codim}(Y \cap U, U) \]
Proof. The rule $T \mapsto \overline{T}$ defines a bijective inclusion preserving map between the closed irreducible subsets of $U$ and the closed irreducible subsets of $X$ which meet $U$. Using this the lemma easily follows. Details omitted. $\square$
Example 5.11.3. Let $X = [0, 1]$ be the unit interval with the following topology: The sets $[0, 1]$, $(1 - 1/n, 1]$ for $n \in \mathbf{N}$, and $\emptyset $ are open. So the closed sets are $\emptyset $, $\{ 0\} $, $[0, 1 - 1/n]$ for $n > 1$ and $[0, 1]$. This is clearly a Noetherian topological space. But the irreducible closed subset $Y = \{ 0\} $ has infinite codimension $\text{codim}(Y, X) = \infty $. To see this we just remark that all the closed sets $[0, 1 - 1/n]$ are irreducible.
Definition 5.11.4. Let $X$ be a topological space. We say $X$ is catenary if for every pair of irreducible closed subsets $T \subset T'$ we have $\text{codim}(T, T') < \infty $ and every maximal chain of irreducible closed subsets has the same length (equal to the codimension).
\[ T = T_0 \subset T_1 \subset \ldots \subset T_ e = T' \]
Lemma 5.11.5. Let $X$ be a topological space. The following are equivalent:
$X$ is catenary,
$X$ has an open covering by catenary spaces.
Moreover, in this case any locally closed subspace of $X$ is catenary.
Proof. Suppose that $X$ is catenary and that $U \subset X$ is an open subset. The rule $T \mapsto \overline{T}$ defines a bijective inclusion preserving map between the closed irreducible subsets of $U$ and the closed irreducible subsets of $X$ which meet $U$. Using this the lemma easily follows. Details omitted. $\square$
Lemma 5.11.6. Let $X$ be a topological space. The following are equivalent:
$X$ is catenary, and
for every pair of irreducible closed subsets $Y \subset Y'$ we have $\text{codim}(Y, Y') < \infty $ and for every triple $Y \subset Y' \subset Y''$ of irreducible closed subsets we have
\[ \text{codim}(Y, Y'') = \text{codim}(Y, Y') + \text{codim}(Y', Y''). \]
Proof. Let suppose that $X$ is catenary. According to Definition 5.11.4, for every pair of irreducible closed subsets $Y \subset Y'$ we have $\text{codim}(Y,Y') < \infty $. Let $Y \subset Y' \subset Y''$ be a triple of irreducible closed subsets of $X$. Let
be a maximal chain of irreducible closed subsets between $Y$ and $Y'$ where $e_1 = \text{codim}(Y,Y')$. Let also
be a maximal chain of irreducible closed subsets between $Y'$ and $Y''$ where $e_2 = \text{codim}(Y',Y'')$. As the two chains are maximal, the concatenation
is maximal too (between $Y$ and $Y''$) and its length equals to $e_1 + e_2$. As $X$ is catenary, each maximal chain has the same length equals to the codimension. Thus the point (2) that $\text{codim}(Y,Y'') = e_1 + e_2 = \text{codim}(Y,Y') + \text{codim}(Y',Y'')$ is verified.
For the reciprocal, we show by induction that : if $Y = Y_1 \subset ... \subset Y_ n = Y'$, then $ \text{codim}(Y,Y') = \text{codim}(Y_1,Y_2) + ... + \text{codim}(Y_{n-1},Y_ n)$. Therefore, it forces maximal chains to have the same length. $\square$
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