Lemma 33.19.3. Let f : X \to Y be locally of finite type. Let x \in X be a point with image y \in Y such that \mathcal{O}_{Y, y} is Noetherian. Let d \geq 0 be an integer such that for every generic point \eta of an irreducible component of X which contains x, we have f(\eta ) \not= y and \dim _\eta (X_{f(\eta )}) = d. Then \dim _ x(X_ y) \leq d + \dim (\mathcal{O}_{Y, y}) - 1.
Proof. Exactly as in the proof of Lemma 33.19.1 we reduce to the case X = \mathop{\mathrm{Spec}}(A) with A a domain and Y = \mathop{\mathrm{Spec}}(B) where B is a Noetherian local ring whose maximal ideal corresponds to y. After replacing B by B/\mathop{\mathrm{Ker}}(B \to A) we may assume that B is a domain and that B \subset A. Then we use the dimension formula (Morphisms, Lemma 29.52.1) to get
\dim (\mathcal{O}_{X, x}) + \text{trdeg}_{\kappa (y)} \kappa (x) \leq \dim (B) + \text{trdeg}_ B(A)
We have \text{trdeg}_ B(A) = d by our assumption that \dim _\eta (X_\xi ) = d, see Morphisms, Lemma 29.28.1. Since \mathcal{O}_{X, x} \to \mathcal{O}_{X_ y, x} has a kernel (as f(\eta ) \not= y) and since \mathcal{O}_{X, x} is a Noetherian domain we see that \dim (\mathcal{O}_{X, x}) > \dim (\mathcal{O}_{X_ y, x}). We conclude that
\dim _ x(X_ y) = \dim (\mathcal{O}_{X_ y, x}) + \text{trdeg}_{\kappa (y)} \kappa (x) < \dim (B) + d
(equality by Morphisms, Lemma 29.28.1) which proves what we want. \square
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