The Stacks project

Lemma 10.122.5. Let $R \to S$ be a finite type ring map. Let $\mathfrak q \subset S$ be a prime lying over $\mathfrak p \subset R$. Let $f \in R$, $f \not\in \mathfrak p$ and $g \in S$, $g \not\in \mathfrak q$. Then $R \to S$ is quasi-finite at $\mathfrak q$ if and only if $R_ f \to S_{fg}$ is quasi-finite at $\mathfrak qS_{fg}$.

Proof. The fibre of $\mathop{\mathrm{Spec}}(S_{fg}) \to \mathop{\mathrm{Spec}}(R_ f)$ is homeomorphic to an open subset of the fibre of $\mathop{\mathrm{Spec}}(S) \to \mathop{\mathrm{Spec}}(R)$. Hence the lemma follows from part (1) of the equivalent conditions of Lemma 10.122.2. $\square$

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