The Stacks project

Typos in the proof. It sould read "...of finite type and of dimension $0$. Hence it is finite over $\kappa(\mathfrak{p})$ for example by Lemma 10.114.4."

THanks, fixed here.

Perhaps Lemma 00PM can be re-stated as the following equivalence: Let $R \to S$ be a finite type ring map and $\mathfrak p$ be a prime ideal of $R$. Then the following are equivalent: 1. $R \to S$ is quasi-finite at all primes of $S$ lying over $\mathfrak p$. 2. $S \otimes_R \kappa(\mathfrak p)$ is a finite $\mathfrak p$-algebra. 3. $\text{Spec}(S \otimes_R \kappa(\mathfrak p))$ is a finite set.

All three equivalences are essentially noether normalization over a field (since a polynomial ring with at least one variable over a field has infinitely many primes).

Then the connection between quasi-finiteness and Lemma 02ML is clear and one can omit the following sentence before Lemma 02ML: "The following lemma is not quite about quasi-finite ring maps, but it does not seem to fit anywhere else so well."

Sorry, part 2. in my comment above should say $S \otimes_R \kappa(\mathfrak p)$ is a finite $\kappa(\mathfrak p)$-algebra.

OK, thanks. I added this here.