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Blowing up an effective Cartier divisor does nothing

Statement

Assume the Axiom of Choice, inherited from the blowup construction (The Axiom of Choice). Let $I$ be a quasi-coherent ideal sheaf of finite type on a scheme $X$ that is invertible as an $O_{X}$ -module (Invertible sheaves) — equivalently, $Z = V (I)$ is an effective Cartier divisor on $X$ (Effective cartier divisor, Cartier divisor). Then the blowup $π ⁣ : {Bl}_{I} X \to X$ of Blowup of a scheme along an ideal sheaf is an isomorphism. If $I$ is generated by a single nonzerodivisor $f$ on an affine open $U = Spec A$ , then the blowup is $Spec A [I / f] = Spec A$ over $U$ .

Facts & Assumptions

Given: A scheme $X$ , a quasi-coherent ideal sheaf $I$ of finite type with zero scheme $Z$ , and the blowup $π ⁣ : {Bl}_{I} X \to X$ .

[F1]

Effective cartier divisor and Cartier divisor: An effective Cartier divisor on $X$ is given by an open cover ${U_{i}}$ with regular sections $f_{i} \in Γ (U_{i}, O_{X})$ , regular meaning that multiplication by the germ $(f_{i})_{x}$ is injective for every $x \in U_{i}$ , and the local principal ideals $f_{i} O_{U_{i}}$ glue to an ideal sheaf $I_{D}$ ; conversely an invertible ideal sheaf is locally generated by one element and that generator is a nonzerodivisor, because the map $O_{U_{i}} \to I ∣_{U_{i}}$ , $1 \mapsto f_{i}$ , is an isomorphism.

[F2]

The sheaf of a Cartier divisor is invertible: For a Cartier divisor $D$ the sheaf $O_{X} (D)$ is invertible and locally freely generated by $f_{i}^{- 1}$ ; in particular the ideal sheaf of an effective Cartier divisor is an invertible $O_{X}$ -module.

[F3]

Affine blowup standard charts and overlaps and Affine blowup algebras: normal form, nonzerodivisors, reducedness, domains: For $I = (f_{0}, \dots, f_{r})$ on an affine $U = Spec A$ the standard charts are $Spec A [I / f_{i}]$ and they cover ${Bl}_{I} U$ ; for a principal ideal $I = (f)$ with $f$ a nonzerodivisor the affine blowup algebra is $A [I / f] = A$ .

[F4]

Universal property of the blowup: For every $X$ -scheme $Y \to X$ whose inverse image of $Z$ is an effective Cartier divisor there is a unique $X$ -morphism $Y \to {Bl}_{I} X$ ; equivalently ${Bl}_{I} X$ is final among such $X$ -schemes, and Uniqueness of the blowup makes the resulting identifications unique.

Proof

1.1F1F2

If $I$ is invertible then it is locally generated by a single element $f_{i}$ on an open cover ${U_{i}}$ and the generator is a nonzerodivisor by [F1], so the zero scheme $Z$ is an effective Cartier divisor with local equations $f_{i}$ ; conversely if $Z = V (I)$ is an effective Cartier divisor with local equations $f_{i}$ , then $I ∣_{U_{i}} = (f_{i})$ and [F2] exhibits $I$ as invertible.

2.1F3step 1.1

Suppose $I$ is invertible. On each affine open $U = Spec A$ with $I ∣_{U} = (f)$ and $f$ a nonzerodivisor, the single standard chart of the blowup is $Spec A [I / f] = Spec A = U$ by [F3], and it covers ${Bl}_{I} U = π^{- 1} (U)$ ; hence $π^{- 1} (U) \to U$ is an isomorphism.

3.1F4step 2.1

The restriction $π^{- 1} (U) \to U$ is an isomorphism for the members of an affine open cover of $X$ by step 2.1, and being an isomorphism is local on the target, so $π ⁣ : {Bl}_{I} X \to X$ is an isomorphism; its inverse is characterized by the universal property [F4] applied to the identity of $X$ , whose inverse image of $Z$ is the effective Cartier divisor $Z$ itself, so the inverse is the unique $X$ -morphism $X \to {Bl}_{I} X$ supplied there, and it is unique by Uniqueness of the blowup.

3.2F3step 2.1

In the affine case $X = Spec A$ and $I = \tilde{I}$ with $I = (f)$ for a nonzerodivisor $f$ , step 2.1 with $U = X$ gives ${Bl}_{I} Spec A = Spec A [I / f] = Spec A$ , as claimed.

4.1step 1.1step 3.1step 3.2∎

Steps 1.1, 3.1 and 3.2 prove the statement: invertible centers are exactly effective Cartier divisors, and the blowup along such an ideal sheaf is an isomorphism, computed on an affine chart as $Spec A [I / f] = Spec A$ when $I$ is generated by one nonzerodivisor.

Remarks

This is the case in which a blowup changes nothing at all: it is an isomorphism precisely when the centre is already Cartier, so nontrivial blowups require a centre that fails to be Cartier in a neighbourhood of itself.
Combining the statement with Invariance of the blowup under invertible (fractional) rescaling of the ideal shows that ${Bl}_{I} X$ depends only on the class of $I$ modulo invertible rescaling, a class that is trivial exactly in the Cartier case.

Depends on

Used by

Dependency tree · two levels

33 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, Divisors, Sections 31.33-31.36 (Blowing up; Strict transform; Admissible blowups; Blowing up and flatness) (standard reference, not scraped)
Ravi Vakil, Foundations of Algebraic Geometry, June 27, 2011 draft (author-hosted 'Early (out-of-date) version of The Rising Sea') (standard reference, not scraped)