Lemma 29.52.2. 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 $S$ is locally Noetherian and $f$ is locally of finite type, We have

29.52.2.1
\begin{equation} \label{morphisms-equation-dimension-formula-general} \dim (\mathcal{O}_{X, x}) \leq \dim (\mathcal{O}_{S, s}) + E - \text{trdeg}_{\kappa (s)} \kappa (x). \end{equation}

where $E$ is the maximum of $\text{trdeg}_{\kappa (f(\xi ))}(\kappa (\xi ))$ where $\xi$ runs over the generic points of irreducible components of $X$ containing $x$.

Proof. Let $X_1, \ldots , X_ n$ be the irreducible components of $X$ containing $x$ endowed with their reduced induced scheme structure. These correspond to the minimal primes $\mathfrak q_ i$ of $\mathcal{O}_{X, x}$ and hence there are finitely many of them (Schemes, Lemma 26.13.2 and Algebra, Lemma 10.31.6). Then $\dim (\mathcal{O}_{X, x}) = \max \dim (\mathcal{O}_{X, x}/\mathfrak q_ i) = \max \dim (\mathcal{O}_{X_ i, x})$. The $\xi$'s occurring in the definition of $E$ are exactly the generic points $\xi _ i \in X_ i$. Let $Z_ i = \overline{\{ f(\xi _ i)\} } \subset S$ endowed with the reduced induced scheme structure. The composition $X_ i \to X \to S$ factors through $Z_ i$ (Schemes, Lemma 26.12.7). Thus we may apply the dimension formula (Lemma 29.52.1) to see that $\dim (\mathcal{O}_{X_ i, x}) \leq \dim (\mathcal{O}_{Z_ i, x}) + \text{trdeg}_{\kappa (f(\xi ))}(\kappa (\xi )) - \text{trdeg}_{\kappa (s)} \kappa (x)$. Putting everything together we obtain the lemma. $\square$

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