The Stacks project

Lemma 29.20.4. Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $g : S' \to S$ be any morphism. Denote $f' : X' \to S'$ the base change. If $x' \in X'$ maps to a point $x \in X$ which is closed in $X_{f(x)}$ then $x'$ is closed in $X'_{f'(x')}$.

Proof. The residue field $\kappa (x')$ is a quotient of $\kappa (f'(x')) \otimes _{\kappa (f(x))} \kappa (x)$, see Schemes, Lemma 26.17.5. Hence it is a finite extension of $\kappa (f'(x'))$ as $\kappa (x)$ is a finite extension of $\kappa (f(x))$ by Lemma 29.20.3. Thus we see that $x'$ is closed in its fibre by applying that lemma one more time. $\square$

Comments (1)

Comment #777 by Keenan Kidwell on

In the last sentence of the statement of the lemma, should be .

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