In this post, we continue our discussion of the application of Zariski’s Main Theorem, following ‘On Quasi-finite Morphisms (Part 1)’.

Let’s recall Grothendieck’s 2nd version of Zariski’s Main Theorem.

Theorem 1 (Zariski’s Main Theorem–Grothendieck’s 2nd Version). Let $Y$ be a quasi-compact and quasi-separated scheme. If $f : X \to Y$ is quasi-finite, separated, and of finite presentation, then $f$ factors as $X \xrightarrow{f’} Y’ \xrightarrow{u} Y $, where $f’$ is an open immersion and $u$ is a finite morphism.

Proof. The proof requires a deep understanding of the underlying mathematics. See 8.12 of (Grothendieck 1966) or Tag 02LQ(The Stacks project authors 2025).$\quad\square$

The goal of this post is to show the following: if a separated morphism of finite type has a finite generic fiber, then it is ‘generically finite’.

Definition 1. A generic point of the topological space $X$ is a point $x \in X$ such that $X = \overline{\{x\}}$.

Recall that a topological space $X$ is reducible if it can be written as a union $X = X_1 \cup X_2$ of two closed proper subsets $X_1$ and $X_2$ of $X$. A topological space is irreducible if it is not reducible. An irreducible component of a topological space is a maximal irreducible subset. If a subset is irreducible, its closure is also irreducible, so irreducible components are closed.

Proposition 1. Let $X$ be a scheme. Any irreducible closed subset of $X$ has a unique generic point.

Proof. See Exercise 2.9(II) of (Hartshorne 1977) or Tag 01IS(The Stacks project authors 2025). $\quad\square$

To deal with questions about generic points, let’s recall the concepts of specialization and generalization

Definition 2. Let $X$ be a topological space. If $x,x’ \in X$, then we say that $x$ is a specialization of $x’$ (or equivalently, $x’$ is a generalization of $x$) if $x \in \overline{\{x’\}}$, and we denote this by $x’\leadsto x$.

Recall that for any continuous map $f : X \to Y$ between topological spaces, for $A \subseteq X$ we have $f\left(\overline{A}\right) \subseteq \overline{f(A)}$. Thus, continuous maps preserve the relations of specialization and generalization.

Recall that a morphism of scheme $f: X \to Y$ is dominant if $\overline{f(X)} = Y$. We also define scheme-theoretically dominant morphisms.

Definition 3. Let $f : Z \to X$ be a morphism of schemes. When the homomorphism $\Ocal_X \to f_\ast\Ocal_Z$ is injective, we say that $f$ is scheme-theoretically dominant. When $f$ is the canonical injection of a subscheme $Z \subseteq X$, we also say that $Z$ is scheme-theoretically dense in $X$.

Lemma 1. A scheme-theoretically dominant morphism is dominant.

Proof. For any nonempty open subset $U \subseteq X$, the homomorphism $\Gamma(U,\Ocal_X) \to \Gamma(f^{-1}(U), \Ocal_Z)$ is injective. Hence, $f^{-1}(U) \neq \varnothing$, implying that $U \cap f(Z) \neq \varnothing$. This shows that $\overline{f(Z)}=X$. $\quad\square$

Now we study the definition and properties of scheme-theoretic images and closures.

Definition 4. Let $f : X \to Y$ be a morphism of schemes. If there exists the smallest closed subscheme $Y’ \subseteq Y$ such that $f$ factors through the canonical injection $j : Y’ \to Y$, then $Y’$ is called the scheme-theoretic image of $X$ in $Y$ under $f$.

Definition 5. Let $U$ be a subscheme of of a scheme $X$. If there exists a smallest closed subscheme $V \subseteq X$ that contains $V$, then $V$ is called the scheme-theoretic closure of $U$ in $X$.

Proposition 2 ((6.10.5)(Grothendieck and Dieudonné 1971)). Let $f: X \to Y$ be a morphism of schemes. If the direct-image sheaf $f_\ast\mathcal{O}_X$ is a quasi-coherent $\mathcal{O}_Y$-module (this is the case when $f$ is quasi-compact and quasi-separated), then the scheme-theoretic image of $Y’ \subseteq Y$ of $X$ under $f$ exists. The scheme-theoretic image $Y’$ contains $\overline{f(X)}$ as an underlying topological space, and in the factorization \[ X \xrightarrow{g} Y’ \xrightarrow{j} Z, \] where $j$ is the canonical immersion, $g$ is scheme-theoretically dominant.

Proof. Note that $f$ is quasi-compact and quasi-separated, then $f_\ast\mathcal{O}_X$ is a quasi-coherent $\mathcal{O}_Y$-module by Tag 01LC (The Stacks project authors 2025). By the assumption, the sheaf of ideal \[ \Jcal = \ker(\Ocal_Y \to f_\ast \Ocal_X) \] is quasi-coherent (Prop.5.7(II)(Hartshorne 1977)). Let $Y’$ be the closed subscheme of $Y$ determined by $\Jcal$. It is easy to check that $Y’$ is the smallest by the construction. Because the morphism $\Ocal_Y / \Jcal \to f_\ast \Ocal_X$ is injective, the induced ring morphism $g : X \to Y’$ is scheme-theoretically dominant.$\quad\square$

Corollary. Let $X$ be a subscheme of a scheme $Y$, and let $j: X \to Y$ be the canonical injection. If $j_\ast(\Ocal_X)$ is a quasi-coherent $\Ocal_Y$-module (which is the case when $j$ is quasi-compact), then the scheme-theoretic closure $Y’$ of $X$ in $Y$ exists. The scheme-theoretic closure $Y’$ contains $\overline{X}$ as an underlying topological space, and $X$ is scheme-theoretically dense in $Y’$.

As a last preparation step, we state some technical lemmas about generic points.

Lemma 2. Let $f:X \to Y$ be a morphism of schemes, and let $\eta$ be the generic point of an irreducible component of $Y$. Let $S$ be the set of all generic points of irreducible components of $X$. If $f^{-1}(\eta)$ is discrete, then $f^{-1}(\eta) \subseteq S$.

Proof. Let $\eta$ be the generic point of $Y$. Suppose that $x \in f^{-1}(\eta)$. Then there exists some generic point $\xi \in S$ such that $\xi \leadsto x$. Thus, we get $f(\xi) \leadsto \eta$, implying that $f(\xi) = \eta$. However, since $f^{-1}(\eta)$ is discrete, there is no non-trivial specialization in $f^{-1}(\eta)$, so we get $x = \xi$. This shows that $f^{-1}(\eta) \subseteq S$. $\quad\square$

Lemma 3 (Tag 02NE(The Stacks project authors 2025)). Let $f:X \to Y$ be a quasi-compact morphism of schemes. Let $\eta \in Y$ be a generic point of an irreducible component of $Y$. If $\eta \notin f(X)$ then there exists an open neighborhood $V \subseteq Y$ of $\eta$ such that $f^{−1}(V)=\varnothing$.

Now we state the main proposition of this post.

Proposition 3. Let \(f: X \to Y\) be separated and of finite type with \(Y\) irreducible and locally Noetherian. If the fiber over the generic point \(\eta \in Y\) is finite, then there is an open neighborhood \(V\) of \(\eta\) in \(Y\) such that \(f^{-1}(V) \to V\) is finite.

Proof. Consider the set \[ U \coloneqq \{ x \in X \mid f^{-1}(f(x)) \text{ is finite} \} \] (in fact, $U$ is open by (13.1.4)(Grothendieck 1966)). We claim that $\eta \notin \overline{f(X \setminus U)}$. If $\eta \in \overline{f(X \setminus U)}$, then $\overline{f(X \setminus U)} = Y$, so $f(X \setminus U)$ is dense in $Y$. However, a quasi-compact morphism is dominant if and only if all generic points of irreducible components are in the image of $f$ (Tag 01RL(The Stacks project authors 2025)). Thus, we obtain $\eta \in f(X \setminus U)$, which is a contradiction.

Remove the closed set $f\overline{(X \setminus U)}$ from $Y$ and shrink the remainder to an affine neighborhood of $\eta$. After this replacement we may assume that \(f: X \to Y\) is separated, quasi-finite, and of finite type, where \(Y\) is an irreducible Noetherian affine scheme with the generic point $\eta$. This also forces $X$ to be Noetherian (Exercise 3.13(II) of (Hartshorne 1977)). Thus, $X$ has finitely many irreducible components.

Let $S = \{\xi_1, \ldots, \xi_r, \xi_{r+1}, \ldots, \xi_{r+s}\}$ be the set of generic points of all irreducible components of $X$, where $f^{-1}(\eta) = \{\xi_1, \ldots, \xi_r\}$. Let $C \coloneqq \overline{\{\xi_{r+1}, \ldots, \xi_{r+s}\}}$. Note that $\eta \notin f(C)$, since there must not be any non trivial specialization in $S$. Hence, there is an open neighborhood of $\eta$ such that the its preimage does not intersect with $C$. Shrinking $Y$ to such an affine neighborhood of $\eta$, we may say that $f^{-1}(\eta) = \{\xi_1, \ldots, \xi_r\}$ is exactly the set of generic points of all irreducible components of $X$ (because irreducibility is preserved under restriction to non-empty open subsets, no new irreducible components appear.).

Applying Zariski’s Main Theorem, we obtain the factorization $X \xrightarrow{j} Z \xrightarrow{g} Y$ of $f$, where $j$ is an open immersion and $g$ is finite. Since $f$ and $g$ are quasi-compact and $j$ is separated, the open immersion $j$ is also quasi-compact (see Tag 01L7 and Tag 03GI of (The Stacks project authors 2025)). Let $Z’$ be the scheme-theoretic image of $X$ under $j$. Then we get the following factorization \[ X \xrightarrow{j} Z’ \xrightarrow{g’} Y, \] where $j$ is a scheme-theoretically dominant open immersion, and $g’$ is finite. Let $(g’)^{-1}(\eta) = \{\zeta_1, \ldots, \zeta_r, \zeta_{r+1}, \ldots, \zeta_{r+t}\}$, where $\zeta_i = j(\xi_i)$ for $1 \le i \le r$. However, since $\overline{j(X)} = Z’$, we have $\overline{\{\zeta_1, \ldots, \zeta_r\}} = Z’$, so $(g’)^{-1}(\eta) = \{\zeta_1, \ldots, \zeta_r\}$ by Lemma 2. Because $ Z’\setminus j(X)$ is closed in $Z’$, its image $g’(Z\setminus j(X) )$ is also closed in $Y$. Note that $(g’)^{-1}(\eta)$ and $f^{-1}(\eta)$ are isomorphic; therefore, $\eta$ is not in $g’(Z\setminus j(X) )$. Take $V \coloneqq Y \setminus g’(Z\setminus j(X) )$. Then the induced map $(g’)^{-1}(V) \to V$ is still finite, and $f^{-1}(V) \to (g’)^{-1}(V)$ is an isomorphism. This ends the proof. $\quad\square$


Acknowledgment I would like to thank my friend Jeongwoo Park for her helpful discussions and suggestions while writing this post.


References

  1. Grothendieck, Alexander. 1966. “Éléments De Géométrie Algébrique : IV. Étude Locale Des Schémas Et Des Morphismes De Schémas, Troisième Partie.” Publications Mathématiques De l’IHÉS 28. Institut des Hautes Études Scientifiques: 5–255. https://www.numdam.org/item/PMIHES_1966__28__5_0/.
  2. Grothendieck, Alexandre, and Jean A. Dieudonné. 1971. Éléments De Géométrie Algébrique. I: Le Langage Des Schémas. 2nd ed. Vol. 166. Grundlehren Der Mathematischen Wissenschaften. Berlin and New York: Springer-Verlag.
  3. Hartshorne, Robin. 1977. Algebraic Geometry. 1st ed. Vol. 52. Graduate Texts in Mathematics. New York, NY: Springer New York. https://doi.org/10.1007/978-1-4757-3849-0.
  4. The Stacks project authors. 2025. “The Stacks Project.” https://stacks.math.columbia.edu.