The Stacks project

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:

  1. $f$ is finite,

  2. $f$ is proper and locally quasi-finite,

  3. $f$ is proper and $|X_ k|$ is a discrete space for every morphism $\mathop{\mathrm{Spec}}(k) \to Y$ where $k$ is a field,

  4. $f$ 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 ├ętale 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 ├ętale 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

  1. $f$ is proper, and

  2. $f$ 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

  1. $X \to B$ is a proper morphism,

  2. $Y \to B$ is separated and locally of finite type,

  3. 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. $\square$


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