Section 76.35 (0F43): Applications of Zariski's Main Theorem, I

76.35 Applications of Zariski's Main Theorem, I

A first application is the characterization of finite morphisms as proper morphisms with finite fibres.

Lemma 76.35.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ . The following are equivalent:

$f$ is finite,
$f$ is proper and locally quasi-finite,
$f$ is proper and $|X_ k|$ is a discrete space for every morphism $\mathop{\mathrm{Spec}}(k) \to Y$ where $k$ is a field,
$f$ is universally closed, separated, locally of finite type and $|X_ k|$ is a discrete space for every morphism $\mathop{\mathrm{Spec}}(k) \to Y$ where $k$ is a field.

Proof. We have (1) $\Rightarrow$ (2) by Morphisms of Spaces, Lemmas 67.45.9, 67.45.8. We have (2) $\Rightarrow$ (3) by Morphisms of Spaces, Lemma 67.27.5. By definition (3) implies (4).

Assume (4). Since $f$ is universally closed it is quasi-compact (Morphisms of Spaces, Lemma 67.9.7). Pick a point $y$ of $|Y|$ . We represent $y$ by a morphism $\mathop{\mathrm{Spec}}(k) \to Y$ . Note that $|X_ k|$ is finite discrete as a quasi-compact discrete space. The map $|X_ k| \to |X|$ surjects onto the fibre of $|X| \to |Y|$ over $y$ (Properties of Spaces, Lemma 66.4.3). By Morphisms of Spaces, Lemma 67.34.8 we see that $X \to Y$ is quasi-finite at all the points of the fibre of $|X| \to |Y|$ over $y$ . Choose an elementary étale neighbourhood $(U, u) \to (Y, y)$ and decomposition $X_ U = V \amalg W$ as in Lemma 76.33.1 adapted to all the points of $|X|$ lying over $y$ . Note that $W_ u = \emptyset$ because we used all the points in the fibre of $|X| \to |Y|$ over $y$ . Since $f$ is universally closed we see that the image of $|W|$ in $|U|$ is a closed set not containing $u$ . After shrinking $U$ we may assume that $W = \emptyset$ . In other words we see that $X_ U = V$ is finite over $U$ . Since $y \in |Y|$ was arbitrary this means there exists a family $\{ U_ i \to Y\}$ of étale morphisms whose images cover $Y$ such that the base changes $X_{U_ i} \to U_ i$ are finite. We conclude that $f$ is finite by Morphisms of Spaces, Lemma 67.45.3. $\square$

As a consequence we have the following useful result.

Lemma 76.35.2. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ . Let $y \in |Y|$ . Assume

$f$ is proper, and
$f$ is quasi-finite at all $x \in |X|$ lying over $y$ (Decent Spaces, Lemma 68.18.10).

Then there exists an open neighbourhood $V \subset Y$ of $y$ such that $f|_{f^{-1}(V)} : f^{-1}(V) \to V$ is finite.

Proof. By Morphisms of Spaces, Lemma 67.34.7 the set of points at which $f$ is quasi-finite is an open $U \subset X$ . Let $Z = X \setminus U$ . Then $y \not\in f(Z)$ . Since $f$ is proper the set $f(Z) \subset Y$ is closed. Choose any open neighbourhood $V \subset Y$ of $y$ with $Z \cap V = \emptyset$ . Then $f^{-1}(V) \to V$ is locally quasi-finite and proper. Hence $f^{-1}(V) \to V$ is finite by Lemma 76.35.1. $\square$

slogan

Lemma 76.35.3. Let $S$ be a scheme. Let

$\xymatrix{ X \ar[rr]_ h \ar[rd]_ f & & Y \ar[ld]^ g \\ & B }$

be a commutative diagram of morphism of algebraic spaces over $S$ . Let $b \in B$ and let $\mathop{\mathrm{Spec}}(k) \to B$ be a morphism in the equivalence class of $b$ . Assume

$X \to B$ is a proper morphism,
$Y \to B$ is separated and locally of finite type,
one of the following is true
1. the image of $|X_ k| \to |Y_ k|$ is finite,
2. the image of $|f|^{-1}(\{ b\} )$ in $|Y|$ is finite and $B$ is decent.

Then there is an open subspace $B' \subset B$ containing $b$ such that $X_{B'} \to Y_{B'}$ factors through a closed subspace $Z \subset Y_{B'}$ finite over $B'$ .

Proof. Let $Z \subset Y$ be the scheme theoretic image of $h$ , see Morphisms of Spaces, Section 67.16. By Morphisms of Spaces, Lemma 67.40.8 the morphism $X \to Z$ is surjective and $Z \to B$ is proper. Thus

$\{ x \in |X|\text{ lying over }b\} \to \{ z \in |Z|\text{ lying over }b\}$

and $|X_ k| \to |Z_ k|$ are surjective. We see that either (3)(a) or (3)(b) imply that $Z \to B$ is quasi-finite all points of $|Z|$ lying over $b$ by Decent Spaces, Lemma 68.18.10. Hence $Z \to B$ is finite in an open neighbourhood of $b$ by Lemma 76.35.2. $\square$

The Stacks project

76.35 Applications of Zariski's Main Theorem, I

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