Lemma 10.122.8. Let $R \to S$ be a ring map of finite type. Let $R \to R'$ be any ring map. Set $S' = R' \otimes _ R S$.

The set $\{ \mathfrak q' \mid R' \to S' \text{ quasi-finite at }\mathfrak q'\} $ is the inverse image of the corresponding set of $\mathop{\mathrm{Spec}}(S)$ under the canonical map $\mathop{\mathrm{Spec}}(S') \to \mathop{\mathrm{Spec}}(S)$.

If $\mathop{\mathrm{Spec}}(R') \to \mathop{\mathrm{Spec}}(R)$ is surjective, then $R \to S$ is quasi-finite if and only if $R' \to S'$ is quasi-finite.

Any base change of a quasi-finite ring map is quasi-finite.

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