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The Stacks project

Lemma 29.28.2. Let f : X \to Y and g : Y \to S be morphisms of schemes. Let x \in X and set y = f(x), s = g(y). Assume f and g locally of finite type. Then

\dim _ x(X_ s) \leq \dim _ x(X_ y) + \dim _ y(Y_ s).

Moreover, equality holds if \mathcal{O}_{X_ s, x} is flat over \mathcal{O}_{Y_ s, y}, which holds for example if \mathcal{O}_{X, x} is flat over \mathcal{O}_{Y, y}.

Proof. Note that \text{trdeg}_{\kappa (s)}(\kappa (x)) = \text{trdeg}_{\kappa (y)}(\kappa (x)) + \text{trdeg}_{\kappa (s)}(\kappa (y)). Thus by Lemma 29.28.1 the statement is equivalent to

\dim (\mathcal{O}_{X_ s, x}) \leq \dim (\mathcal{O}_{X_ y, x}) + \dim (\mathcal{O}_{Y_ s, y}).

For this see Algebra, Lemma 10.112.6. For the flat case see Algebra, Lemma 10.112.7. \square


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