Lemma 29.55.2. Let $f : X \to S$ be a morphism of schemes. The set of points of $X$ where $f$ is quasi-finite is an open $U \subset X$. The induced morphism $U \to S$ is locally quasi-finite.

** The locally quasi-finite locus of a morphism is open **

**Proof.**
Suppose $f$ is quasi-finite at $x$. Let $x \in U = \mathop{\mathrm{Spec}}(A) \subset X$, $V = \mathop{\mathrm{Spec}}(R) \subset S$ be affine opens as in Definition 29.20.1. By either Theorem 29.55.1 above or Algebra, Lemma 10.123.13, the set of primes $\mathfrak q$ at which $R \to A$ is quasi-finite is open in $\mathop{\mathrm{Spec}}(A)$. Since these all correspond to points of $X$ where $f$ is quasi-finite we get the first statement. The second statement is obvious.
$\square$

## 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Your email address will not be published.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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (4)

Comment #3963 by Manuel Hoff on

Comment #4099 by Johan on

Comment #5429 by slogan_bot on

Comment #5656 by Johan on

There are also: