Lemma 71.3.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is locally of finite type and $Y$ is locally Noetherian. Let $y \in |Y|$ be a point of codimension $\leq 1$ on $Y$. Let $X^0 \subset |X|$ be the set of points of codimension $0$ on $X$. Assume in addition one of the following conditions is satisfied

1. for every $x \in X^0$ the transcendence degree of $x/f(x)$ is $0$,

2. for every $x \in X^0$ with $f(x) \leadsto y$ the transcendence degree of $x/f(x)$ is $0$,

3. $f$ is quasi-finite at every $x \in X^0$,

4. $f$ is quasi-finite at a dense set of points of $|X|$,

Then $f$ is quasi-finite at every point of $X$ lying over $y$.

Proof. We want to reduce the proof to the case of schemes. To do this we choose a commutative diagram

$\xymatrix{ U \ar[r] \ar[d]_ g & X \ar[d]^ f \\ V \ar[r] & Y }$

where $U$, $V$ are schemes and where the horizontal arrows are étale and surjective. Pick $v \in V$ mapping to $y$. Observe that $V$ is locally Noetherian and that $\dim (\mathcal{O}_{V, v}) \leq 1$ (see Properties of Spaces, Definitions 65.10.2 and Remark 65.7.3). The fibre $U_ v$ of $U \to V$ over $v$ surjects onto $f^{-1}(\{ y\} ) \subset |X|$. The inverse image of $X^0$ in $U$ is exactly the set of generic points of irreducible components of $U$ (Properties of Spaces, Lemma 65.11.1). If $\eta \in U$ is such a point with image $x \in X^0$, then the transcendence degree of $x / f(x)$ is the transcendence degree of $\kappa (\eta )$ over $\kappa (g(\eta ))$ (Morphisms of Spaces, Definition 66.33.1). Observe that $U \to V$ is quasi-finite at $u \in U$ if and only if $f$ is quasi-finite at the image of $u$ in $X$.

Case (1). Here case (1) of Varieties, Lemma 33.17.1 applies and we conclude that $U \to V$ is quasi-finite at all points of $U_ v$. Hence $f$ is quasi-finite at every point lying over $y$.

Case (2). Let $u \in U$ be a generic point of an irreducible component whose image in $V$ specializes to $v$. Then the image $x \in X^0$ of $u$ has the property that $f(x) \leadsto y$. Hence we see that case (2) of Varieties, Lemma 33.17.1 applies and we conclude as before.

Case (3) follows from case (3) of Varieties, Lemma 33.17.1.

In case (4), since $|U| \to |X|$ is open, we see that the set of points where $U \to V$ is quasi-finite is dense as well. Hence case (4) of Varieties, Lemma 33.17.1 applies. $\square$

There are also:

• 1 comment(s) on Section 71.3: Generically finite morphisms

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.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0AD1. Beware of the difference between the letter 'O' and the digit '0'.