Lemma 29.27.2 (02JS)—The Stacks project

Table of contents
Part 2: Schemes
Chapter 29: Morphisms of Schemes
Section 29.27: Morphisms and dimensions of fibres
Lemma 29.27.2 (cite)

Lemma 29.27.2. 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 29.27.1 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 10.111.6. For the flat case see Algebra, Lemma 10.111.7. $\square$

The Stacks project

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