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:
\delta _{X/S}(\xi ),
\dim (Z), and
\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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