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Blowing up I and I^d agree

Statement

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 $X$ (Quasi-coherent ideal sheaves) and let $d \geq 1$ . Then there is a canonical isomorphism of $X$ -schemes ${Bl}_{I^{d}} X ⟶ {Bl}_{I} X;$ more precisely $R (I^{d})$ is the Veronese regrading $R (I)^{(d)}$ of the Rees algebra sheaf, the degree- $n$ piece of $R (I^{d})$ being $I^{d n}$ , and the canonical identification $Proj R (I) = Proj R (I)^{(d)}$ of Proj is invariant under Veronese regrading glues over $X$ .

Facts & Assumptions

Given: A scheme $X$ , a quasi-coherent ideal sheaf $I$ of finite type, its powers $I^{n}$ , the Rees algebra sheaves $R (I) = ⨁_{n \geq 0} I^{n}$ and $R (I^{d}) = ⨁_{n \geq 0} I^{d n}$ (Rees algebra sheaf of a finite type ideal), and the blowups ${Bl}_{I} X = {Proj}_{X} R (I)$ and ${Bl}_{I^{d}} X = {Proj}_{X} R (I^{d})$ of Blowup of a scheme along an ideal sheaf.

[F1]

Rees algebra sheaf of a finite type ideal: The Rees algebra sheaf of a quasi-coherent ideal sheaf is the graded $O_{X}$ -algebra $⨁_{n \geq 0} I^{n}$ with degree- $n$ piece $I^{n}$ , and $R (I)^{(d)} : = ⨁_{n \geq 0} R (I)_{d n} = ⨁_{n \geq 0} I^{d n}$ is its Veronese regrading; the graded pieces are quasi-coherent.

[F2]

Proj is invariant under Veronese regrading: For a commutative nonnegatively graded ring $S$ and $d \geq 1$ there is a canonical isomorphism $Proj S ≅ Proj S^{(d)}$ mapping the chart $D_{+} (f)$ of $Proj S$ , for homogeneous $f \in S_{+}$ of positive degree, to the chart $D_{+} (f^{d})$ of $Proj S^{(d)}$ with the same coordinate ring $S_{(f)} = S_{(f^{d})}^{(d)}$ ; it is the identity for $d = 1$ and sends the empty Proj to the empty Proj.

[F3]

Affine blowup standard charts and overlaps: For $I = (f_{0}, \dots, f_{r})$ the standard opens $D_{+} (f_{i} t) = Spec A [I / f_{i}]$ cover ${Bl}_{I} Spec A$ with the stated overlap identifications, and the presentation is independent of the chosen generating family.

[F4]

Relative Proj of a graded quasi-coherent algebra: The relative Proj ${Proj}_{X} A$ of a quasi-coherent graded $O_{X}$ -algebra is constructed by gluing the spectra of the degree-zero localisations over affine opens of $X$ , compatibly with restriction to smaller affine opens.

[F5]

Blowups restrict to open subschemes of the base: For an open subscheme $U ↪ X$ there is a canonical isomorphism ${Bl}_{I ∣_{U}} U \to {Bl}_{I} X \times_{X} U$ , and the blowup is determined up to canonical isomorphism by its restrictions to an open cover.

Proof

1.1F1

The graded $O_{X}$ -algebras $R (I^{d})$ and $R (I)^{(d)}$ are canonically isomorphic: their degree- $n$ pieces are $(I^{d})^{n} = I^{d n}$ in both cases, the multiplications are the multiplication of $O_{X}$ , and the identifications are compatible with restriction to open subschemes.

1.2F2

On an affine open $U = Spec A \subseteq X$ with $I ∣_{U} = \tilde{I}$ and $R (I) = ⨁_{n \geq 0} I^{n} t^{n}$ , [F2] applied to the graded ring $S = R (I)$ gives a canonical isomorphism $Proj R (I) ≅ Proj R (I)^{(d)}$ that maps a chart $D_{+} (f)$ , for homogeneous $f \in R (I)_{+}$ of positive degree, to $D_{+} (f^{d})$ with the same coordinate ring $R (I)_{(f)} = R (I)_{(f^{d})}^{(d)}$ .

2.1F1F3step 1.1step 1.2

The isomorphism of step 1.2 identifies the standard charts of the two blowups: for $a \in I$ the chart $D_{+} (a t)$ of $Proj R (I)$ corresponds to $D_{+} ((a t)^{d}) = D_{+} (a^{d} t^{d})$ in $Proj R (I)^{(d)}$ , and under step 1.1 this is the standard chart $D_{+} (a^{d} t^{'})$ of $Proj R (I^{d})$ , with coordinate ring $A [I / a] = (R (I))_{(a t)} = R (I^{d})_{(a^{d} t^{'})} = A [I^{d} / a^{d}]$ via the identification $a t \leftrightarrow a^{d} t^{'}$ ; the charts $D_{+} (a t)$ cover ${Bl}_{I} U$ for any generating family by [F3], and their images cover ${Bl}_{I^{d}} U$ .

3.1F2F4step 2.1

The chartwise identifications are canonical: on the overlap of two charts they are the identity of the common localisation of $R (I)$ , and on restriction to a smaller affine open $V \subseteq U$ the identification for $V$ is the restriction of the identification for $U$ , because both sides are computed by the same graded localisations and the Veronese isomorphism of [F2] is natural in the graded ring; hence the identifications are compatible with the gluing data of the standard charts.

4.1F4F5step 1.1step 3.1∎

The identifications of step 3.1 glue over an affine cover of $X$ to an isomorphism of $X$ -schemes ${Proj}_{X} R (I) \to {Proj}_{X} R (I^{d})$ by [F4], and over arbitrary open subschemes the restriction compatibility of [F5] gives the same isomorphism; since $R (I^{d}) ≅ R (I)^{(d)}$ by step 1.1, this proves the stated canonical isomorphism ${Bl}_{I^{d}} X \to {Bl}_{I} X$ , which is the identity when $d = 1$ .

Remarks

The identification matches the relative twists $O_{{Bl}_{I^{d}} X} (1)$ with $O_{{Bl}_{I} X} (d)$ under the Veronese isomorphism, as recorded in Proj is invariant under Veronese regrading.
Together with Invariance of the blowup under invertible (fractional) rescaling of the ideal this shows that the blowup depends on the ideal sheaf only up to the equivalence generated by invertible rescaling and positive powers.

Depends on

Used by

Blowing up I and I² give the same scheme Example

Dependency tree · two levels

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Sources

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)
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)