The Stacks project

Lemma 72.3.1. 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 $|X|$,

  5. add more here.

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 ├ętale 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 66.10.2 and Remark 66.7.3). 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 66.11.1). 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 67.33.1). 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. $\square$

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