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A finite birational morphism onto a normal variety is an isomorphism

Statement

Assume the Axiom of Choice. Let $f ⁣ : Y \to X$ be a finite birational morphism of irreducible classical varieties over an algebraically closed field. If $X$ is normal, then $f$ is an isomorphism. The normality of the target is essential: the normalization of the cusp is finite and birational but not an isomorphism.

Facts & Assumptions

Given: AC, the algebraically closed field $k$ , the irreducible varieties $Y, X$ , the finite birational morphism $f ⁣ : Y \to X$ , and the assumption that $X$ is normal. Also a finite affine cover of $X$ by affine opens $U$ with $f^{- 1} (U)$ affine and $k [f^{- 1} (U)]$ a finite $k [U]$ -module.

[F1]

By the definition of finiteness and its affine-locality, such a cover exists, and over every affine open $U \subseteq X$ the preimage $f^{- 1} (U)$ is affine with $k [f^{- 1} (U)]$ a finite $k [U]$ -module (Finite morphisms of classical varieties, Principal opens form a basis for the Zariski topology on an affine variety).

[F2]

On an affine variety, the coordinate ring is a domain, and the local ring at a point is the localisation of the coordinate ring at its maximal ideal; normality of $X$ thus makes each coordinate ring an integrally closed domain, by the localisation criterion for integrally closed domains (A classical affine variety has a domain coordinate ring, and conversely, The local ring at a point of an affine variety is the localization at its maximal ideal, Normal points and normal varieties, A domain is integrally closed if and only if its prime localisations are, equivalently if and only if its maximal localisations are).

[F3]

Birationality gives an isomorphism of function fields: passing to a nonempty affine open $U \subseteq X$ , the pullback identifies $Frac (k [U]) = k (U)$ with $Frac (k [f^{- 1} (U)])$ (Irreducible affine varieties are birational exactly when their function fields are isomorphic, The Axiom of Choice).

[F4]

A finite algebra is integral over its base, integrally closed domains contain the integral elements of their fraction fields, and pullback identifies morphisms of affine varieties with $k$ -algebra homomorphisms (Integrality and finite-module characterizations for one element, Integral closure in an extension ring and integrally closed domains, Affine morphisms are contravariantly equivalent to coordinate-ring homomorphisms).

[F5]

$X$ has a finite affine cover and every open subvariety of $X$ has one (Classical varieties have finite irreducible decompositions).

Proof

1.1F1F2F3F4given

The affine case. Let $U \subseteq X$ be a nonempty affine open with $A = k [U]$ , and put $V = f^{- 1} (U)$ , $B = k [V]$ , so that $B$ is a finite $A$ -module and the pullback $A \to B$ exhibits $f ∣_{V}$ [F1]. By [F2], $A$ is an integrally closed domain, and $B$ is a domain. By [F3], $Frac (B) = Frac (A)$ under the pullback, so every $b \in B$ lies in the fraction field of $A$ and is integral over $A$ because $B$ is a finite, hence integral, $A$ -module [F4]. Since $A$ is integrally closed, $b \in A$ ; hence $B \subseteq A$ , and with $A \subseteq B$ we get $A = B$ . By the anti-equivalence [F4] the map $f ∣_{V}$ is an isomorphism.

2.1F1F5step 1.1∎

The general case. Cover $X$ by finitely many nonempty affine opens $U_{1}, \dots, U_{n}$ as in [F1] and [F5]; by step 1.1 each restriction $f_{i} ⁣ : f^{- 1} (U_{i}) \to U_{i}$ is an isomorphism, with inverse $g_{i} ⁣ : U_{i} \to f^{- 1} (U_{i})$ . On an overlap $U_{i} \cap U_{j}$ , the maps $g_{i}$ and $g_{j}$ both invert the same map $f$ on that overlap, so they agree there. Since being a morphism is a local condition on the source and the $U_{i}$ cover $X$ , the $g_{i}$ glue to a morphism $g ⁣ : X \to Y$ ; the identities $f \circ g = {i d}_{X}$ and $g \circ f = {i d}_{Y}$ hold because they hold locally on the cover. Thus $f$ is an isomorphism.

Depends on

Used by

Birational quasi-finite maps to normal targets are open immersions Corollary
The cusp normalization is bijective but not an isomorphism Counterexample

Dependency tree · two levels

51 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

The Stacks Project, Lemma 29.55.8: a finite birational morphism onto a normal scheme is an isomorphism (standard reference, not scraped)
J. S. Milne, Algebraic Geometry (2025 version), Ch. 8 §a and §c: normal points and finite maps (Definition 8.1, Definition 8.17, Lemma 8.19, Summary 8.22) (standard reference, not scraped)