The Stacks project

Lemma 42.7.5. Let $(S, \delta )$ be as in Situation 42.7.1. Assume in addition $S$ is a Jacobson scheme, and $\delta (s) = 0$ for every closed point $s$ of $S$. Let $X$ be locally of finite type over $S$. Let $Z \subset X$ be an integral closed subscheme and let $\xi \in Z$ be its generic point. The following integers are the same:

  1. $\delta _{X/S}(\xi )$,

  2. $\dim (Z)$, and

  3. $\dim (\mathcal{O}_{Z, z})$ where $z$ is a closed point of $Z$.

Proof. Let $X \to S$, $\xi \in Z \subset X$ be as in the lemma. Since $X$ is locally of finite type over $S$ we see that $X$ is Jacobson, see Morphisms, Lemma 29.16.9. Hence closed points of $X$ are dense in every closed subset of $Z$ and map to closed points of $S$. Hence given any chain of irreducible closed subsets of $Z$ we can end it with a closed point of $Z$. It follows that $\dim (Z) = \sup _ z(\dim (\mathcal{O}_{Z, z})$ (see Properties, Lemma 28.10.3) where $z \in Z$ runs over the closed points of $Z$. Note that $\dim (\mathcal{O}_{Z, z}) = \delta (\xi ) - \delta (z)$ by the properties of a dimension function. For each closed $z \in Z$ the field extension $\kappa (z)/\kappa (f(z))$ is finite, see Morphisms, Lemma 29.16.8. Hence $\delta _{X/S}(z) = \delta (f(z)) = 0$ for $z \in Z$ closed. It follows that all three integers are equal. $\square$

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