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Universal property and uniqueness of a contraction

Statement

Assume the Axiom of Choice, inherited from the blowup suppliers. Let $X$ be a Noetherian scheme, $E \subseteq X$ an exceptional curve of the first kind and $b ⁣ : X \to X^{'}$ a contraction of $E$ (Exceptional curves of the first kind and their contractions). Write $x^{'} = b (E) \in X^{'}$ . Then:

$b$ is proper, surjective and closed; $b ⁣ : X \to X^{'}$ is a topological quotient map identifying $X^{'}$ with the quotient of $X$ obtained by collapsing $E$ to $x^{'}$ ; the canonical map $O_{X^{'}} \to b_{*} O_{X}$ is an isomorphism and $R^{1} b_{*} O_{X} = 0$ .
(Universal property) For every morphism $φ ⁣ : X \to Y$ of schemes with $φ (E)$ a single point there is a unique morphism $φ^{'} ⁣ : X^{'} \to Y$ with $φ = φ^{'} \circ b$ .
(Uniqueness) If $b_{i} ⁣ : X \to X_{i}^{'}$ , $i = 1, 2$ , are contractions of $E$ , there is a unique isomorphism $X_{1}^{'} \to X_{2}^{'}$ compatible with $b_{1}$ and $b_{2}$ .

Consequently a contraction of $E$ , when it exists, is unique and is characterized by the universal property.

Facts & Assumptions

Given: A Noetherian scheme $X$ , an exceptional curve of the first kind $E \subseteq X$ , a contraction $b ⁣ : X \to X^{'}$ of $E$ , and a morphism $φ ⁣ : X \to Y$ collapsing $E$ to a point.

[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-exceptional-curve-and-contraction. Assume the Axiom of Choice where it is inherited from the degree and intersection suppliers below (The Axiom of Choice). Let $X$ be a Noetherian scheme. (a) Exceptional curves of the first kind. A closed subscheme $E \subseteq X$ (def-closed-immersion-schemes) is an exceptional curve of the first kind if: 1. (Exceptional curves of the first kind and their contractions)

[F3]

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)

[F4]

thm-blowup-projective. Assume the Axiom of Choice, inherited from the relative Proj construction (The Axiom of Choice). Let $X$ be a scheme, let $I$ be a quasi-coherent ideal sheaf of finite type on $X$ (def-quasi-coherent-ideal-sheaf) and let $π ⁣ : {Bl}_{I} X \to X$ be the blowup of def-blowup-scheme-along-ideal. Then: 1. (Blowups of finite type ideals are locally H-projective, and proper)

[F5]

lem-blowup-isomorphism-off-center. Let $I$ be a quasi-coherent ideal sheaf of finite type with zero scheme $Z$ and let $π ⁣ : {Bl}_{I} X \to X$ be the blowup. (The blowup is an isomorphism off the center)

[F6]

lem-blowup-point-pushforward-vanishing. Assume the Axiom of Choice. Let $S$ be a regular surface over a field $k$ (more generally a locally Noetherian scheme of dimension two whose local rings at the center are regular of dimension two) and let $p$ be a closed point with residue field $κ (p)$ . Let $π ⁣ : S^{'} \to S$ be the blowup of $p$ with exceptional curve $E$ . (Pushforward and vanishing for point blowups on a surface)

[F7]

thm-proper-morphism-closed-image. Let $f : X \to S$ be a proper morphism of schemes. Then $f$ is a closed map of topological spaces: for every closed subset $Z \subseteq ∣ X ∣$ its image $f (Z)$ is closed in $∣ S ∣$ . In particular $f (X)$ is closed. (Proper morphisms are closed)

[F8]

cor-blowup-unique-up-to-unique-isomorphism. Assume the Axiom of Choice. Let $I$ be a quasi-coherent ideal sheaf of finite type with zero scheme $Z$ . If $π^{'} ⁣ : Y \to X$ is an $X$ -scheme such that $(π^{'})^{- 1} (Z)$ is an effective Cartier divisor and $Y$ carries the universal property of ${Bl}_{I} X$ (every $X$ -scheme in which the inverse image of $Z$ is an effective Cartier divisor maps un (Uniqueness of the blowup)

Proof

1.1F2F3F4F5F7given

By definition $b$ is the blowup of $X^{'}$ at the closed point $x^{'} = b (E)$ with regular two-dimensional local ring, so $b$ is proper and an isomorphism off the centre; its image contains that complement and the nonempty exceptional fibre over $x^{'}$ , so $b$ is surjective. Properness makes it closed.

2.1F6F7step 1.1

A continuous closed surjection is a quotient map, so $b$ identifies $X^{'}$ with the quotient of $X$ obtained by collapsing $E$ to $x^{'}$ set-theoretically and topologically; moreover for a point blowup the natural map $O_{X^{'}} \to b_{*} O_{X}$ is an isomorphism and $R^{1} b_{*} O_{X} = 0$ .

3.1F3F7step 2.1

Since $φ$ collapses $E$ to a point and $b$ is injective off $E$ , the map $φ$ is constant on the fibres of $b$ ; by the quotient property of step 2.1 it factors uniquely as a continuous map $φ^{'} ⁣ : X^{'} \to Y$ with $φ = φ^{'} \circ b$ .

4.1F5F6step 2.1step 3.1

For the morphism structure, the map of sheaves $φ^{♯} ⁣ : φ^{- 1} O_{Y} \to O_{X}$ is adjoint to a map $(φ^{'})^{- 1} O_{Y} \to b_{*} O_{X}$ along the quotient, and $b_{*} O_{X} = O_{X^{'}}$ by step 2.1, so it gives a map of sheaves of rings $(φ^{'})^{- 1} O_{Y} \to O_{X^{'}}$ ; locality is checked at $x^{'}$ : a germ vanishing at $φ (E) = φ^{'} (x^{'})$ pulls back under $φ$ to a function vanishing on $E$ , hence its image in $(b_{*} O_{X})_{x^{'}} = O_{X^{'}, x^{'}}$ lies in the maximal ideal. Thus $φ^{'}$ is a morphism of schemes with $φ = φ^{'} \circ b$ , unique because $b$ is a quotient map.

5.1F2F8step 4.1

For uniqueness of the contraction, apply the universal property to the two contractions $b_{1}, b_{2}$ of $E$ : each $b_{i}$ collapses $E$ , so $b_{2}$ factors uniquely through $b_{1}$ and vice versa, and the two factorizations are mutually inverse isomorphisms $X_{1}^{'} \to X_{2}^{'}$ compatible with the maps from $X$ .

6.1F1F8step 5.1∎

The Axiom of Choice is inherited from the blowup suppliers; the only uniqueness statement used is the universal property of the blowup up to unique isomorphism.

Remarks

The key sheaf input is $b_{*} O_{X} = O_{X^{'}}$ for a point blowup, which makes the adjunction computation of step 2.2 an honest map of structure sheaves.
Uniqueness of the contraction follows formally from the universal property and does not use any classification of exceptional curves.

Depends on

Used by

Dependency tree · two levels

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Sources

The Stacks Project, Resolution of Surfaces, Section 54.16 (Contracting exceptional curves) (standard reference, not scraped)
The Stacks Project, Resolution of Surfaces, Lemma 54.3.1 (Blowing up a regular surface at a point) (standard reference, not scraped)