In this post, we will discuss the definition and properties of quasi-finite morphisms, and its application to Zariski’s Main Theorem.

1. Finite Morphisms

Before diving into the theory of quasi-finite morphisms, we briefly recall the definition of finite morphisms in scheme theory.

Definition 1. A morphism \(f: X \to Y\) is a finite morphism of schemes if there exists a covering of \(Y\) by open affine subsets \(V_i = \Spec B_i\), such that for each \(i\), \(f^{-1}(V_i)\) is affine, equal to \(\Spec A_i\), where \(A_i\) is a \(B_i\)-algebra which is a finitely generated \(B_i\)-module.

Definition 2. A morphism of schemes \(f:X \to Y\) is affine if the inverse image of any open affine subset \(V\) of \(Y\) is an open affine subset of \(X\).

Lemma 1. A morphism \(f: X \to Y\) of schemes is finite if and only if \(f\) is affine and \(\Gamma(f^{-1}(V), \mathcal{O}_X)\) is a \(\Gamma(V,\mathcal{O}_Y)\)-algebra which is a finitely generated \(\Gamma(V,\mathcal{O}_Y)\)-module.

Proof. See Exercise 3.4(II) of (Hartshorne 1977).$\quad\square$

Finite morphisms have several nice properties.

Proposition 1.
1. A closed immersion is finite.
2. The composite of two finite morphisms is finite.
3. Any base change of a finite morphism is finite.

Proof. These reduce to statements about rings, which follow directly from standard results. For example, for (3), reducing to the local case, we need to show the following: if a ring homomorphism \(\varphi: B \to A\) is finite, then so is the induced map \(\varphi’: B’ \to A \otimes_B B’\) for any \(B\)-algebra \(B’\). $\quad\square$

Furthermore, by commutative algebra, it is easy to see that every finite morphism is integral; the precise definition follows.

Definition 3. Let \(f:X \to Y\) be a morphism of schemes. We say that \(f\) is integral if \(f\) is affine and if for every affine open \(\Spec B = V \subseteq Y \) with inverse image \(\Spec A = f^{-1}(V)\subseteq X\) the associated ring map \(B \to A\) is integral.

However, one of the most important properties of finite morphisms is that they are proper morphisms; that is, they behave similarly to continuous maps between compact Hausdorff spaces.

Proposition 2. Any finite morphism \(f : X \to Y\) is proper, that is, it is separated, of finite type, and universally closed.

Proof. It is known that every affine morphism is separated (Tag 01S7 (The Stacks project authors 2025)), so every finite morphism is automatically separated. By the “going up” theorem, one can easily show that every integral morphism is universally closed (Tag 01WM (The Stacks project authors 2025)).$\quad\square$

2. Quasi-Finite Morphisms

In general, we denote by \(X \times_SY\) the fiber product of two \(S\)-schemes \(X\) and \(Y\). When \(S = \Spec R\) is affine, we also write \(X \times_R Y\). When \(Y = \Spec B\), instead of writing \(X \times_R Y\) we may write \(X \times_S B\); and if, in addition, \(S = \Spec R\) is affine, we write \(X \otimes_R B\). We recall the definition of the fiber.

Definition 4. Let \(f: X \to Y\) be a morphism of schemes, and let \(y \in Y\) be a point. Let \(k(y)\) be the residue field of \(y\), and let \(\Spec k(y) \to Y\) be the natural morphism. Then we define the fiber of the morphism \(f\) over the point \(y\) to be the scheme \[ X_y = X \times_Y \Spec k(y). \]

The fiber \(X_y\) is a scheme over \(k(y)\), and one can show that its underlying topological space is homeomorphic to the subset \(f^{-1}(y)\) of \(X\) (Exer.3.10(II) of (Hartshorne 1977)).

In EGA II (Grothendieck 1961), Grothendieck defines quasi-finite morphisms.

Definition 5. Let \(f: X \to Y\) be a morphism of schemes. We say that \(f\) is quasi-finite if \(f\) is of finite type and \(X_y\) is discrete for all \(y \in Y\).

In fact, \(X_y\) is discrete if and only if it is finite. Since morphisms of finite type are stable under base change, the fiber \(X_y\) is a \(k(y)\)-scheme of finite type, and hence \(X_y\) is Noetherian (see Exer.3.13(II) of (Hartshorne 1977)). It is easy to see that every discrete Noetherian topological space must be finite. This fact explains why some books like (Hartshorne 1977) define quasi-finite morphisms by finite fibers instead of discrete fibers.

The notion of quasi-finite morphism is not purely topological due to the finite type condition. If \(K\) is a field extension of a field \(k\), the morphism \(\Spec K \to \Spec k\) is quasi-finite only when \(K\) is a finite extension of \(k\), even though both schemes involved are reduced and each consists of a single point.

Lemma 2. A finite morphism is quasi-finite.

Proof. We reduce to the following algebraic fact: if \(\varphi : B \to A\) be a finite ring homomorphism (i.e., \(A\) is a finitely generated \(B\)-module), then for a given prime ideal \(\qfr\) of \(B\), there are finitely many prime ideals \(\pfr\) of \(A\) such that \(\varphi^{-1}(\pfr) = \qfr\). Note that the induced homomorphism \(\varphi_\qfr : B_\qfr \to A_\qfr\) is also finite. Every prime ideal \(\pfr\) of \(A\) satisfying \(\varphi^{-1}(\pfr) = \qfr\) survives in \(A_\qfr\). Let \(\nfr\coloneqq \qfr B_\qfr\) be the unique maximal ideal of \(B_\qfr\). Since all the prime ideals of \(A_\qfr\), whose preimage is \(\nfr\), contains \(\nfr A_\qfr\), we can replace \(B_\qfr \to A_\qfr\) with \(B_\qfr / \nfr \to A_\qfr / \nfr A_\qfr\), which is still a finite ring homomorphism. Thus, \(A_\qfr / \nfr A_\qfr\) is a finite algebra over a field, so it is Artinian and has finitely many primes (Exercises 8.2 and 8.3 of (Atiyah and MacDonald 2018)).$\quad\square$

The proof of Lemma 2 can be rewritten in the following proposition.

Lemma 3. Let $f: X \to Y$ be a morphism of finite type. Then $f$ is quasi-finite if and only if for every $x \in X$, the scheme $X_{f(x)}$ is a finite $k(f(x))$-scheme.

The converse of Lemma 2 is false in general. For instance, consider the embedding \(\Abb_k^1 \setminus \{0\} \hookrightarrow \Abb_k^1\) of punctured affine line into the affine line. While it is clearly quasi-finite, it is not closed, so it is not finite by Proposition 2. Also, there are several non-trivial examples.

Example 1. A surjective and quasi-finite morphism need not be finite. Let \(X_1 = X_2 = \Abb_k^1\), let \(U_1 = U_2 = \Abb_k^1 \setminus \{P\}\), where \(P\) is the point corresponding to the maximal ideal \(x\), and let \(\varphi : U_1 \to U_2\) be the identity map. Let \(X\) be obtained by glueing \(X_1\) and \(X_2\) along \(U_1\) and \(U_2\) via \(\varphi\); i.e., \(X\) is the affine line with the point \(P\) doubled. Let \(Y = \Spec k[x] = \Abb_k^1\), and let \(f : X \to Y\) be the morphism obtained by the identity maps \(X_1 \to Y\) and \(X_2 \to Y\). Then this morphism is clearly surjective and has finite fibers. It is also of finite type, since \(f^{-1}(Y)\) has two affine coverings \(\Spec k[x]\), and clearly \(k[x]\) is itself a finite\(k[x]\)-algebra. However, \(X=f^{-1}(Y)\) is not affine, so \(f\) is not finite.

Example 2. Let \(A\) be a Dedekind domain with field of fractions \(K\). Then \(\Spec K \to \Spec A\) is never finite, it is quasi-finite if it is of finite type, and it is of finite type if and only if \(A\) has only finitely many prime ideals.
Suppose for contradiction that \(a_1/b_1, \ldots, a_n/b_n \in K\) generate \(K\) as an \(A\)-module, where \(a_i, b_i \in A\). Since \(A\) is not a field, there exists a non-zero prime \(\pfr\), and \(A_\pfr \subseteq K\) is a discrete valuation ring with a discrete valuation \(v\). Because \(K\) is also a field of fractions of \(A_\pfr\), and \(v\) gives the valuation on \(K\). Thus, for any \(x \in K\), we have \(v(x) \ge - \max_i\{v(b_i)\}\). However, clearly the valuation of \(K\) is not bounded below (e.g., for sufficiently large \(N\), the valuation of \(\pi^{-N}\) is \(-N\) where \(\pi\) is a uniformiser of \(\pfr\) in \(A_\pfr\)), a contradiction. This shows that \(\Spec K \to \Spec A\) is never finite. Meanwhile, it is quasi-finite by the definition if it is of finite type. Now we show that it is of finite type if and only if \(A\) has only finitely many prime ideals. Suppose that \(A\) has only finitely many prime ideals. Let the non-zero prime ideals of \(A\) be \(\pfr_1, \ldots, \pfr_r\), and fix a uniformiser \(\pi_i \in \pfr_i \setminus \pfr_i^2\) for each \(i\). Then it is easy to see that every element \(x\) in \(K\) can be expressed as \[x = u\pi_1^{n_1} \cdots \pi_r^{n_r} \] where \(u \in A^\times\) and \(n_i \in \mathbb{Z}\). This shows that \(K = A[\pi_1^{-1}, \ldots, \pi_r^{-1}]\). For the forward direction, assume \(K = A[x_1, \ldots, x_n]\) for some \(x_1, \ldots, x_n \in K\). Write each generators in reduced from: \(x_i = a_i/b_i\) where \(a_i,b_i \in A\) and \(b_i \neq 0\). Let \(S\) be the set of prime ideals of \(A\) dividing some \((b_i)\). Since \(A\) is Dedekind, for each \(b_i\), only finitely many prime ideals can contain \(b_i\), so \(S\) is finite. We claim that there is no nonzero prime of \(A\) not in \(S\). Suppose there exists a prime \(\pfr\) not in \(S\), and let \(\pi_\pfr\) be a uniformiser of \(\pfr\). Since \(K = A[x_1, \ldots, x_n]\), there exists some polynomial \(P(T_1, \ldots, F_n) \in A[T_1, \ldots, T_n]\) such that \(\pi_\pfr = P(x_1, \ldots, x_n)\). Thus, we have an expression \[ \pi_{\pfr}^{-1} = \frac{c}{b_1^{e_1} \cdots b_n^{e_n}} \quad (c \in A, \ e_i \ge 0) \] in \(K\), so we get \(c\pi_{\pfr} = b_1^{e_1} \cdots b_n^{e_n}\), a contradiction.

However, if we force a quasi-finite morphism to be proper, then it must be finite.

Proposition 3. Let $f: X \to Y$ be a morphism of schemes. Then $f$ is finite if and only if $f$ is proper and quasi-finite.

Proof. See (8.11.1)(Grothendieck 1966) under the assumption that the morphism is locally of finite presentation, or Tag 02LS (The Stacks project authors 2025) for any morphism (in fact, we can prove it using Zariski’s Main Theorem). $\quad\square$

Proposition 3 is extremely useful in practice. In general, showing directly that a morphism is finite is not easy. However, for properness, we have the valuative criterion, and quasi-finiteness is not difficult to check.

Quasi-finite morphisms also have several good properties.

Proposition 4. 1. Any immersion is quasi-finite.
2. The composite of two quasi-finite morphisms is quasi-finite.
3. Any base change of a quasi-finite morphism is quasi-finite.

Proof. We only show (3). Let \(f: X \to S\) be quasi-finite and \(S’ \to S\) arbitrary, and assume \(s’ \mapsto s\). Then \( \Spec k(s’) \to S’ \to S \) factors through the point \(s\), so we get \[ \begin{aligned} \left(X \times_S S’\right)_{s’} &= \left(X \times_{S} S’\right) \times_{S’} k(s’) \\ &\cong X \times_S \left( S’ \times_{S’} k(s’) \right) \\ &\cong X \times_S k(s’) \\ &\cong X \times_S \left(\Spec k(s) \otimes_{k(s)} k(s’) \right) \\ &\cong \left(X \times_S \Spec k(s) \right) \otimes_{k(s)} k(s’) \\ &= X_s \otimes_{k(s)} k(s’). \end{aligned} \] Since \(X_s\) is finite over \(k(s)\) by Lemma 3, and finite morphism is stable under base change, we conclude that \(X_s \otimes_{k(s)} k(s’)\) is finite over \(k(s’)\). By Lemma 3, this shows that \( \left(X \times_S S’\right)_{s’} \) is discrete. $\quad\square$

In most cases, the previous definition of quasi-finite morphisms is sufficient. Nevertheless, Grothendieck added definitions of quasi-finiteness in local sense in the 2nd edition of EGA I (Grothendieck and Dieudonné 1971)

Definition 6. Let \(f: X \to Y\) be a morphism of schemes.
1. We say that \(f\) is quasi-finite at a point \(x \in X\) if there are affine open neighborhoods \(V\) of \(y = f(x)\) and \(U\) of \(x\) such that \(f(U)\subseteq V\) and the restricted morphism \(U \to V\) is quasi-finite.
2. We say that \(f\) is locally quasi-finite if \(f\) is quasi-finite at every point \(x\) of \(X\).

Lemma 4. Let \(f: X \to Y\) be a morphism of schemes. The set \(U\) of points of \(X\) where \(f\) is quasi-finite is open in \(X\). The induced morphism \(U\) is locally quasi-finite.

Proof. See Tag 01TI (The Stacks project authors 2025).$\quad\square$

I think he might realize that ‘quasi-finite at a point’ is a useful notion in practice when applying theorems about quasi-finite morphisms.

3. Zariski’s Main Theorem

The classical statement of Zariski’s Main Theorem in (Zariski 1943) is as follows.

Theorem 1 (Zariski’s Main Theorem–Original Version). If \(W\) is an irreducible fundamental variety on \(V\) of a birational correspondence $T$ between $V$ and $V’$, and if $T$ has no fundamental elements on $V’$, then–under the assumption that $V$ is locally normal at $W$–each irreducible component of the transform $T[W]$ is of higher dimension than $W$.

This theorem contains several terminologies not used in now days. Very roughly, the theorem says that if the target has no ‘bad points(fundamental elements)’, then a birational maps can not contract any ‘bad parts(fundamental variety)’ in the source: every such ‘bad part’ is sent to a subvariety of strictly higher dimension.

Historically, there have been numerous variants of Zariski’s Main Theorem, and it can be difficult to verify how they are related to one another. One such version appears in (Mumford 1999), where the author provides an intuitive explanation of Zariski’s Main Theorem for varieties over $\mathbb{C}$ and explains how several versions of Zariski’s Matin Theorem are realted.

Theorem 2 (Zariski’s Main Theorem–The Red Book Version). Let $X$ be a normal variety over an algebraically closed field $k$ and let $f : X’ \to X$ be a birational morphism with finite fibres from a variety $X’$ to $X$. Then $f$ is an isomorphism of $X’$ with an open subset $U \subseteq X$.

In EGA III, Grothendieck states The Red Book Version in the language of scheme theory.

Theorem 3 (Zariski’s Main Theorem–Grothendieck’s 1st Version). Let $Y$ be a Noetherian scheme and let $f : X \to Y$ be a quasi-projective morphism. Denote by $X’$ the set of points $x \in X$ that are isolated in their fiber, i.e., such that $x$ is an isolated point of the fiber $f^{-1}(f(x))$. Then $X’$ is an open subset of $X$, and the induced subscheme $X’$ is isomorphic to the subscheme induced on an open subset of a $Y$-scheme $Y’$ that is finite over $Y$.

Proof. (4.4.3)(Grothendieck 1961)$\quad\square$

Grothendieck eventually realized that the first version suggests a deeper structural property of quasi-finite morphisms, and subsequently stated a more general version in EGA IV.

Theorem 4 (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 demands a deep understanding of the underlying mathematics. See 8.12 of (Grothendieck 1966) or Tag 02LQ(The Stacks project authors 2025).$\quad\square$

The theorem says that quasi-finite morphisms are essentially “almost finite” morphisms. Since quasi-finite morphisms arise naturally in several circumstances, Zariski’s Main Theorem enables us to apply powerful theories (e.g., étale cohomology theory). We close this post with two toy applications of Zariski’s Main Theorem. The first example is a proof of Proposition 3.

Proof of Proposition 3. Note that finiteness is a local property, so we may assume \(X\) is affine, which is quasi-compact. Let \(f = u \circ f’\) be the factorization as in Zariski’s Main Theorem. Since the composite of two finite morphisms is finite, it suffices to show that \(f’\) is finite. Consider the graph morphism \(\Gamma_{f’}: X \to X \times_X Y’\) defined by \(x \mapsto (x,f’(x))\). It is known that any graph of a separated morphism is a closed immersion (Tag 024T(The Stacks project authors 2025) or (5.2.4)(Grothendieck and Dieudonné 1971), so \(\Gamma_{f’}\) is a closed immersion (hence finite and proper). Thus, \(f’\) is proper, since \(f’ = p_{Y’} \circ \Gamma_f\) with the projection \( p_{Y’} : X \times_X Y’ \to Y’ \). This shows that \(f’\) is an immersion with a closed image, that is, a closed immersion (Tag 01IQ(The Stacks project authors 2025)). This ends the proof.$\quad\square$

Definition 7. A scheme $X$ is called quasi-affine if it is quasi-compact and isomorphic to an open subscheme of an affine scheme.

Definition 8. A morphism of schemes $f:X \to Y$ is called quasi-affine if the inverse image of every affine open of $Y$ is a quasi-affine scheme.

Corollary 1. A separated quasi-finite morphism is quasi-affine.

Proof. Let $f : X \to Y$ be a separated quasi-finite morphism. Then $f$ factors as $X \xrightarrow{f’} Y’ \xrightarrow{u} Y $ where $f’$ is an open immersion and $u$ is a finite morphism. Let $V$ be an affine open of $Y$. Then $u^{-1}(V)$ is an affine open of $Y’$, and $f^{-1}(V)$ is isomorphic to $f(V) \cap u^{-1}(V)$, which is an open subscheme of $u^{-1}(V)$. $\quad\square$


References

  1. Atiyah, Michael F., and I. G. MacDonald. 2018. Introduction To Commutative Algebra. Boca Raton: CRC Press. https://doi.org/10.1201/9780429493638.
  2. Grothendieck, Alexander. 1961. “Éléments De Géométrie Algébrique : II. Étude Globale Élémentaire De Quelques Classes De Morphismes.” Publications Mathématiques De l’IHÉS 8. Institut des Hautes Études Scientifiques: 5–222. https://www.numdam.org/item/PMIHES_1961__8__5_0/.
  3. ———. 1961. “Éléments De Géométrie Algébrique : III. Étude Cohomologique Des Faisceaux Cohérents, Première Partie.” Publications Mathématiques De l’IHÉS 11. Institut des Hautes Études Scientifiques: 5–167. https://www.numdam.org/item/PMIHES_1961__11__5_0/.
  4. ———. 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/.
  5. 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.
  6. 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.
  7. Mumford, David. 1999. The Red Book of Varieties and Schemes: Includes the Michigan Lectures (1974) on Curves and Their Jacobians. 2nd ed. Vol. 1358. Lecture Notes in Mathematics. Berlin / Heidelberg: Springer–Verlag. https://doi.org/10.1007/b62130.
  8. Zariski, Oscar. 1943. “Foundations of a General Theory of Birational Correspondences.” Transactions of the American Mathematical Society 53 (3): 490–542. https://doi.org/10.1090/S0002-9947-1943-0008468-9.
  9. The Stacks project authors. 2025. “The Stacks Project.” https://stacks.math.columbia.edu.