The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

5.11 Codimension and catenary spaces

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

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

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

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

\[ T = T_0 \subset T_1 \subset \ldots \subset T_ e = T' \]

has the same length (equal to the codimension).

Lemma 5.11.5. Let $X$ be a topological space. The following are equivalent:

  1. $X$ is catenary,

  2. $X$ 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:

  1. $X$ is catenary, and

  2. 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. Omitted. $\square$


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