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The pulled-back center ideal is the relative twist; the exceptional divisor is Cartier

Statement

Assume the Axiom of Choice, inherited from the relative Proj construction (The Axiom of Choice). Let $I$ be a quasi-coherent ideal sheaf of finite type on a scheme $X$ (Quasi-coherent ideal sheaves), let $π ⁣ : {Bl}_{I} X \to X$ be its blowup and let $E = π^{- 1} (Z)$ be the exceptional subscheme, with the convention that $O (1)$ is the positive relative twist of the Rees algebra $R (I)$ and $O (- 1) : = O (1)^{\lor}$ . Then:

$O (1)$ is invertible;
the natural degree-one map $π^{*} I \to O (1)$ is surjective, its image is the inverse image ideal $I O_{Bl}$ , and the induced map $I O_{Bl} \to O (1)$ of invertible sheaves is an isomorphism;
$I O_{Bl}$ is invertible and $E = V (I O_{Bl})$ is an effective Cartier divisor on ${Bl}_{I} X$ , with $O_{Bl} (- E) = I O_{Bl} = O (1)$ and $O_{Bl} (E) = O (- 1)$ .

Facts & Assumptions

Given: A scheme $X$ , a quasi-coherent ideal sheaf $I$ of finite type, the Rees algebra sheaf $R (I) = ⨁_{n \geq 0} I^{n}$ (Rees algebra sheaf of a finite type ideal), the blowup $π ⁣ : {Bl}_{I} X = {Proj}_{X} R (I) \to X$ with relative twists $O (n)$ (Blowup of a scheme along an ideal sheaf, Relative Proj of a graded quasi-coherent algebra), and the exceptional subscheme $E = π^{- 1} (Z)$ with ideal sheaf the inverse image ideal $I O_{Bl}$ (Exceptional subscheme of a blowup).

[F1]

Affine blowup standard charts and overlaps and Affine blowup algebras: normal form, nonzerodivisors, reducedness, domains: For $a \in I = Γ (U, I)$ on an affine open $U = Spec A$ , the chart $Spec A [I / a]$ of ${Bl}_{I} U$ has $I A [I / a] = a A [I / a]$ with $a$ a nonzerodivisor of $A [I / a]$ , and the charts over a finite generating family cover the blowup.

[F2]

Invertible twists for degree-one generated rings: For a commutative graded ring $S$ generated over $S_{0}$ by $S_{1}$ , all twists $O_{Proj S} (n)$ are invertible, with $O (m) \otimes O (n) ≅ O (m + n)$ ; the empty Proj is allowed.

[F3]

Twisting sheaf on Proj and Relative Proj of a graded quasi-coherent algebra: The relative twist $O (1)$ of a relative Proj is the sheaf whose sections over the chart $D_{+} (f)$ are the degree-one part $S (1)_{(f)}$ of the localised graded algebra; it is the sheafification of the degree-one part, and on an affine base it restricts to the absolute twist $O_{Proj} (1)$ .

[F4]

Invertible sheaves and Effective cartier divisor: A sheaf is invertible when it is locally free of rank one; an effective Cartier divisor on a scheme is given by local equations that are regular sections, i.e. nonzerodivisors on the stalks.

[F5]

Invertible sheaf of cartier divisor and Effective Cartier divisors give a short exact sequence: For an effective Cartier divisor $D$ the sheaf $O_{X} (- D)$ is the ideal sheaf $I_{D} \subseteq O_{X}$ , and there is a short exact sequence $0 \to O_{X} (- D) \to O_{X} \to i_{*} O_{D} \to 0$ .

Proof

1.1F2F3given

The Rees algebra is generated in degree one. Thus its positive twist is invertible on each affine base, and these restrictions give an invertible sheaf $O (1)$ on the relative Proj. There are two natural maps from $π^{*} I$ : the structural multiplication map $ψ$ to $O_{Bl}$ , whose image is $I O_{Bl}$ , and the degree-one map $γ$ to $O (1)$ .

2.1F1F3step 1.1

On a standard chart $B = A [I / a]$ , $O (1)$ is free with frame $a t$ . For $b \in I$ and $c \in B$ , the maps are $γ (b \otimes c) = c (b / a) (a t)$ and $ψ (b \otimes c) = c b = a c (b / a)$ . The first is surjective since $γ (a \otimes 1) = a t$ . Since $a$ is a nonzerodivisor in $B$ , these formulas give $\ker γ = \ker ψ$ , including for sums of tensors. Consequently $γ$ factors uniquely through the image $I B = a B$ of $ψ$ and induces an isomorphism $I B \to B (a t)$ taking $a$ to $a t$ . This does not assert that $I \otimes_{A} B$ is free or torsion-free.

3.1F1F4F5step 2.1∎

The equality of the kernels is local and therefore global. The induced isomorphisms are restrictions of this unique factorization, so agree on overlaps. Hence $I O_{Bl} ≅ O (1)$ canonically. The ideal cuts out $E$ and is locally generated by the nonzerodivisor $a$ , so $E$ is effective Cartier. Its ideal is $O (- E)$ , and dualizing the isomorphism gives $O (E) ≅ O (- 1)$ . Empty charts and the empty blowup satisfy the same assertions.

Remarks

Assertion 2 identifies the inverse image ideal with the twist $O (1)$ as an invertible sheaf, which is what makes $E$ Cartier even when the centre $Z$ is neither reduced nor Cartier in $X$ .
Combining assertion 3 with Effective Cartier divisors give a short exact sequence gives the short exact sequence $0 \to O (1) \to O_{Bl} \to i_{*} O_{E} \to 0$ on the blowup, a form used in cohomological computations.

Depends on

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Sources

The Stacks Project, Divisors, Sections 31.33-31.36 (Blowing up; Strict transform; Admissible blowups; Blowing up and flatness) (standard reference, not scraped)
The Stacks Project, Commutative Algebra, Section 10.70 (Blow up algebras) (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)