Lemma 29.50.1. Let $S$ be a scheme. Let $f : X \to S$ be a morphism of schemes. Let $x \in X$, and set $s = f(x)$. Assume

1. $S$ is locally Noetherian,

2. $f$ is locally of finite type,

3. $X$ and $S$ integral, and

4. $f$ dominant.

We have

29.50.1.1
$$\label{morphisms-equation-dimension-formula} \dim (\mathcal{O}_{X, x}) \leq \dim (\mathcal{O}_{S, s}) + \text{trdeg}_{R(S)}R(X) - \text{trdeg}_{\kappa (s)} \kappa (x).$$

Moreover, equality holds if $S$ is universally catenary.

Proof. The corresponding algebra statement is Algebra, Lemma 10.112.1. $\square$

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