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A proper birational map to a normal target is an isomorphism near a quasi-finite point

Statement

Assume AC. Let $f : X \to S$ be a proper birational morphism of integral Noetherian schemes with normal $S$ . If $f$ is quasi-finite at $x \in X$ , then $f$ is an isomorphism over an open neighbourhood of $f (x)$ ; in particular that fibre consists of $x$ .

Facts & Assumptions

Given: A proper birational morphism $f ⁣ : X \to S$ of integral Noetherian schemes with normal $S$ , and a point $x \in X$ at which $f$ is quasi-finite.

[F1]

def-axiom-of-choice. The Axiom of Choice (AC) is the following statement. > Every family of nonempty sets has a choice function > (def-choice-function). Written out: for every set $F$ all of whose members are nonempty, there exists a function $g$ with domain $F$ satisfying $g (S) \in S$ for all $S \in F$ . (The Axiom of Choice)

[F2]

def-normal-surface-modification-and-normalized-point-blowup. Normal schemes. A locally Noetherian scheme is normal if every local ring $O_{X, x}$ is an integrally closed domain (def-normal-noetherian-ring). This is a local condition on the local rings and is checked on an affine open cover; it does not require the global section ring to be a domain. The empty scheme is normal vacuously. (Normal scheme modifications and normalized point blowups)

[F3]

cor-quasi-finite-locus-open-finite-type-algebra. Assume the Axiom of Choice (The Axiom of Choice). Let $R \to S$ be a ring map of finite type (def-finite-type-and-module-finite-algebras). (The quasi-finite locus of a finite-type algebra is open)

[F4]

cor-quasi-finite-algebra-is-source-locally-a-localization-of-a-finite-algebra. Assume the Axiom of Choice (The Axiom of Choice). Let $R \to S$ be a ring map of finite type (def-finite-type-and-module-finite-algebras) that is quasi-finite at every prime of $S$ (def-quasi-finite-at-a-prime-for-finite-type-algebras), and let $S^{'} \subseteq S$ be the integral closure of the image of $R$ in $S$ (def-integral-subalgebra-of-an-arbitrary-ring-map). (Quasi-finite algebras are source locally localizations of finite algebras)

[F5]

def-proper-morphism. A morphism of schemes $f : X \to S$ is proper if and only if it is separated, of finite type, and universally closed. Here separatedness has the meaning of def-separated-morphism-schemes, finite type has the meaning of def-locally-finite-type-and-finite-type-morphism, and universally closed has the meaning of def-universally-closed-morphism. (Proper morphisms)

Proof

1.1F3F5given

Choose affine open neighbourhoods $Spec A$ of $f (x)$ and $Spec C \subseteq f^{- 1} (Spec A)$ of $x$ ; the quasi-finite locus of $f$ is open, so after shrinking $C$ we may assume $f$ is quasi-finite at every point of $Spec C$ , and $C$ is a domain contained in the common function field $K$ of $X$ and $S$ .

2.1F2step 1.1

The relative integral closure of $A$ in $C$ equals $A$ : every element of $C$ integral over $A$ lies in $K = Frac C$ , and $A$ is normal, hence integrally closed in $K$ .

3.1F4step 2.1

By the source-local structure theorem for a quasi-finite algebra over a normal domain, the finite intermediate algebra between $A$ and $C$ is contained in the relative integral closure, hence equals $A$ ; therefore there is $g \in A$ with $x \in D (g) \subseteq Spec C$ and $C_{g} = A_{g}$ as subalgebras of $K$ , so $f$ restricted to the principal open $D (g)$ is an open immersion onto $D (g) \subseteq Spec A$ .

4.1F5step 3.1

The inverse of this open immersion gives a section $s ⁣ : D (g) \to X$ of $f$ over $D (g)$ ; a section of a separated morphism is a closed immersion, obtained by base changing the diagonal, and $f$ is separated because it is proper. The image of $s$ contains the generic point of the integral scheme $f^{- 1} (D (g))$ , over which $f$ is an isomorphism by birationality, and being closed it is the whole underlying space; the defining ideal has radical zero and the source is reduced, so the section is an isomorphism onto $f^{- 1} (D (g))$ .

5.1F1step 4.1∎

Hence $f$ is an isomorphism over the open neighbourhood $D (g)$ of $f (x)$ , and in particular the fibre of $f$ over $f (x)$ consists of the single point $x$ ; the Axiom of Choice is inherited from the cited suppliers.

Remarks

The normality of the target is used exactly in step 1.2 to conclude that the relative integral closure is trivial.
The argument is the Zariski main theorem in this restricted setting and avoids any connected-fibre assumption.

Depends on

Used by

A birational morphism of regular surfaces factors through the blowup of a point where its inverse is undefined Lemma

Dependency tree · two levels

29 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, Resolution of Surfaces, proof of Lemma 54.17.1; local Zariski Main argument (standard reference, not scraped)