71.3 Generically finite morphisms

This section continues the discussion in Decent Spaces, Section 67.21 and the analogue for morphisms of algebraic spaces of Varieties, Section 33.17.

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. $\square$

Lemma 71.3.2. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is proper and $Y$ is locally Noetherian. Let $y \in Y$ be a point of codimension $\leq 1$ in $Y$. 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 there exists an open subspace $Y' \subset Y$ containing $y$ such that $Y' \times _ Y X \to Y'$ is finite.

Proof. By Lemma 71.3.1 the morphism $f$ is quasi-finite at every point lying over $y$. Let $\overline{y} : \mathop{\mathrm{Spec}}(k) \to Y$ be a geometric point lying over $y$. Then $|X_{\overline{y}}|$ is a discrete space (Decent Spaces, Lemma 67.18.10). Since $X_{\overline{y}}$ is quasi-compact as $f$ is proper we conclude that $|X_{\overline{y}}|$ is finite. Thus we can apply Cohomology of Spaces, Lemma 68.23.2 to conclude. $\square$

Lemma 71.3.3. Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. Let $f : Y \to X$ be a birational proper morphism of algebraic spaces with $Y$ reduced. Let $U \subset X$ be the maximal open over which $f$ is an isomorphism. Then $U$ contains

1. every point of codimension $0$ in $X$,

2. every $x \in |X|$ of codimension $1$ on $X$ such that the local ring of $X$ at $x$ is normal (Properties of Spaces, Remark 65.7.6), and

3. every $x \in |X|$ such that the fibre of $|Y| \to |X|$ over $x$ is finite and such that the local ring of $X$ at $x$ is normal.

Proof. Part (1) follows from Decent Spaces, Lemma 67.22.5 (and the fact that the Noetherian algebraic spaces $X$ and $Y$ are quasi-separated and hence decent). Part (2) follows from part (3) and Lemma 71.3.2 (and the fact that finite morphisms have finite fibres). Let $x \in |X|$ be as in (3). By Cohomology of Spaces, Lemma 68.23.2 (which applies by Decent Spaces, Lemma 67.18.10) we may assume $f$ is finite. Choose an affine scheme $X'$ and an étale morphism $X' \to X$ and a point $x' \in X$ mapping to $x$. It suffices to show there exists an open neighbourhood $U'$ of $x' \in X'$ such that $Y \times _ X X' \to X'$ is an isomorphism over $U'$ (namely, then $U$ contains the image of $U'$ in $X$, see Spaces, Lemma 64.5.6). Then $Y \times _ X X' \to X$ is a finite birational (Decent Spaces, Lemma 67.22.6) morphism. Since a finite morphism is affine we reduce to the case of a finite birational morphism of Noetherian affine schemes $Y \to X$ and $x \in X$ such that $\mathcal{O}_{X, x}$ is a normal domain. This is treated in Varieties, Lemma 33.17.3. $\square$

Comment #1776 by Weizhe Zheng on

I think in the first line it should be "Decent Spaces, Section 55.19". The two sections have the same \label{section-generically-finite}.

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