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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.

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

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources