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
Comments (1)
Comment #10058 by Ulrich Görtz on