Lemma 10.121.7. A composition of quasi-finite ring maps is quasi-finite.
Proof. Suppose $A \to B$ and $B \to C$ are quasi-finite ring maps. By Lemma 10.6.2 we see that $A \to C$ is of finite type. Let $\mathfrak r \subset C$ be a prime of $C$ lying over $\mathfrak q \subset B$ and $\mathfrak p \subset A$. Since $A \to B$ and $B \to C$ are quasi-finite at $\mathfrak q$ and $\mathfrak r$ respectively, then there exist $b \in B$ and $c \in C$ such that $\mathfrak q$ is the only prime of $D(b)$ which maps to $\mathfrak p$ and similarly $\mathfrak r$ is the only prime of $D(c)$ which maps to $\mathfrak q$. If $c' \in C$ is the image of $b \in B$, then $\mathfrak r$ is the only prime of $D(cc')$ which maps to $\mathfrak p$. Therefore $A \to C$ is quasi-finite at $\mathfrak r$. $\square$
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