Lemma 5.20.4. Let $X$ be locally Noetherian, sober and catenary. Then any point has an open neighbourhood $U \subset X$ which has a dimension function.

Proof. We will use repeatedly that an open subspace of a catenary space is catenary, see Lemma 5.11.5 and that a Noetherian topological space has finitely many irreducible components, see Lemma 5.9.2. In the proof of Lemma 5.20.3 we saw how to construct such a function. Namely, we first replace $X$ by a Noetherian open neighbourhood of $x$. Next, we let $Z_1, \ldots , Z_ r \subset X$ be the irreducible components of $X$. Let

$Z_ i \cap Z_ j = \bigcup Z_{ijk}$

be the decomposition into irreducible components. We replace $X$ by

$X \setminus \left( \bigcup \nolimits _{x \not\in Z_ i} Z_ i \cup \bigcup \nolimits _{x \not\in Z_{ijk}} Z_{ijk} \right)$

so that we may assume $x \in Z_ i$ for all $i$ and $x \in Z_{ijk}$ for all $i, j, k$. For $y \in X$ choose any $i$ such that $y \in Z_ i$ and set

$\delta (y) = - \text{codim}(\overline{\{ x\} }, Z_ i) + \text{codim}(\overline{\{ y\} }, Z_ i).$

We claim this is a dimension function. First we show that it is well defined, i.e., independent of the choice of $i$. Namely, suppose that $y \in Z_{ijk}$ for some $i, j, k$. Then we have (using Lemma 5.11.6)

\begin{align*} \delta (y) & = - \text{codim}(\overline{\{ x\} }, Z_ i) + \text{codim}(\overline{\{ y\} }, Z_ i) \\ & = - \text{codim}(\overline{\{ x\} }, Z_{ijk}) - \text{codim}(Z_{ijk}, Z_ i) + \text{codim}(\overline{\{ y\} }, Z_{ijk}) + \text{codim}(Z_{ijk}, Z_ i) \\ & = - \text{codim}(\overline{\{ x\} }, Z_{ijk}) + \text{codim}(\overline{\{ y\} }, Z_{ijk}) \end{align*}

which is symmetric in $i$ and $j$. We omit the proof that it is a dimension function. $\square$

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