The Stacks project

Example 42.7.4. Here $S$ is a Cohen-Macaulay scheme. Then $S$ is universally catenary by Morphisms, Lemma 29.17.5. We set $\delta (s) = -\dim (\mathcal{O}_{S, s})$. If $s' \leadsto s$ is a nontrivial specialization of points of $S$, then $\mathcal{O}_{S, s'}$ is the localization of $\mathcal{O}_{S, s}$ at a nonmaximal prime ideal $\mathfrak p \subset \mathcal{O}_{S, s}$, see Schemes, Lemma 26.13.2. Thus $\dim (\mathcal{O}_{S, s}) = \dim (\mathcal{O}_{S, s'}) + \dim (\mathcal{O}_{S, s}/\mathfrak p) > \dim (\mathcal{O}_{S, s'})$ by Algebra, Lemma 10.104.4. Hence $\delta (s') > \delta (s)$. If $s' \leadsto s$ is an immediate specialization, then there is no prime ideal strictly between $\mathfrak p$ and $\mathfrak m_ s$ and we find $\delta (s') = \delta (s) + 1$. Thus $\delta $ is a dimension function. In other words, the pair $(S, \delta )$ is an example of Situation 42.7.1.

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