Lemma 29.20.17 (03WR)—The Stacks project

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Part 2: Schemes
Chapter 29: Morphisms of Schemes
Section 29.20: Quasi-finite morphisms
Lemma 29.20.17 (cite)

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Lemma 29.20.17. Let $X \to Y$ be a morphism of schemes over a base scheme $S$ . Let $x \in X$ . If $X \to S$ is quasi-finite at $x$ , then $X \to Y$ is quasi-finite at $x$ . If $X$ is locally quasi-finite over $S$ , then $X \to Y$ is locally quasi-finite.

Proof. Via Lemma 29.20.11 this translates into the following algebra fact: Given ring maps $A \to B \to C$ such that $A \to C$ is quasi-finite, then $B \to C$ is quasi-finite. This follows from Algebra, Lemma 10.122.6 with $R = A$ , $S = S' = C$ and $R' = B$ . $\square$

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