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 $R' = B$. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)