36.39 Applications of Zariski's Main Theorem, I

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

Lemma 36.39.1. Let $f : X \to S$ be a morphism of schemes. The following are equivalent:

1. $f$ is finite,

2. $f$ is proper with finite fibres,

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

4. $f$ is universally closed, separated, locally of finite type and has finite fibres.

Proof. We have (1) implies (2) by Morphisms, Lemmas 28.42.11, 28.19.10, and 28.42.10. We have (2) implies (3) by Morphisms, Lemma 28.19.7. We have (3) implies (4) by the definition of proper morphisms and Morphisms, Lemmas 28.19.9 and 28.19.10.

Assume (3). Pick $s \in S$. By Morphisms, Lemma 28.19.7 we see that all the finitely many points of $X_ s$ are isolated in $X_ s$. Choose an elementary étale neighbourhood $(U, u) \to (S, s)$ and decomposition $X_ U = V \amalg W$ as in Lemma 36.36.6. Note that $W_ u = \emptyset$ because all points of $X_ s$ are isolated. 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 $s \in S$ was arbitrary this means there exists a family $\{ U_ i \to S\}$ of étale morphisms whose images cover $S$ such that the base changes $X_{U_ i} \to U_ i$ are finite. Note that $\{ U_ i \to S\}$ is an étale covering, see Topologies, Definition 33.4.1. Hence it is an fpqc covering, see Topologies, Lemma 33.9.6. Hence we conclude $f$ is finite by Descent, Lemma 34.20.23. $\square$

As a consequence we have the following useful results.

Lemma 36.39.2. Let $f : X \to S$ be a morphism of schemes. Let $s \in S$. Assume that $f$ is proper and $f^{-1}(\{ s\} )$ is a finite set. Then there exists an open neighbourhood $V \subset S$ of $s$ such that $f|_{f^{-1}(V)} : f^{-1}(V) \to V$ is finite.

Proof. The morphism $f$ is quasi-finite at all the points of $f^{-1}(\{ s\} )$ by Morphisms, Lemma 28.19.7. By Morphisms, Lemma 28.53.2 the set of points at which $f$ is quasi-finite is an open $U \subset X$. Let $Z = X \setminus U$. Then $s \not\in f(Z)$. Since $f$ is proper the set $f(Z) \subset S$ is closed. Choose any open neighbourhood $V \subset S$ of $s$ with $f(Z) \cap V = \emptyset$. Then $f^{-1}(V) \to V$ is locally quasi-finite and proper. Hence it is quasi-finite (Morphisms, Lemma 28.19.9), hence has finite fibres (Morphisms, Lemma 28.19.10), hence is finite by Lemma 36.39.1. $\square$

Lemma 36.39.3. Consider a commutative diagram of schemes

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

Let $s \in S$. Assume

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

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

3. the image of $X_ s \to Y_ s$ is finite.

Then there is an open subspace $U \subset S$ containing $s$ such that $X_ U \to Y_ U$ factors through a closed subscheme $Z \subset Y_ U$ finite over $U$.

Proof. Let $Z \subset Y$ be the scheme theoretic image of $h$, see Morphisms, Section 28.6. By Morphisms, Lemma 28.39.10 the morphism $X \to Z$ is surjective and $Z \to S$ is proper. Thus $X_ s \to Z_ s$ is surjective. We see that either (3) implies $Z_ s$ is finite. Hence $Z \to S$ is finite in an open neighbourhood of $s$ by Lemma 36.39.2. $\square$

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