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
the image of |X_ k| \to |Y_ k| is finite,
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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