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Integrality and reducedness of blowups from the Rees charts

Statement

Assume the Axiom of Choice, inherited from the relative Proj construction (The Axiom of Choice). Let $X$ be an integral scheme (Integral schemes) and let $I$ be a nonzero quasi-coherent ideal sheaf of finite type on $X$ (Quasi-coherent ideal sheaves). Then the blowup ${Bl}_{I} X$ of Blowup of a scheme along an ideal sheaf is integral: the affine blowup algebras $A [I / a]$ are domains because $A$ is a domain and $a$ is nonzero, and they glue along localisations. More generally, if $X$ is reduced then ${Bl}_{I} X$ is reduced, because the affine blowup algebras of a reduced ring are reduced.

Facts & Assumptions

Given: An integral (respectively reduced) scheme $X$ , a nonzero quasi-coherent ideal sheaf $I$ of finite type, and for an affine open $U = Spec A \subseteq X$ with $I = Γ (U, I)$ and $a \in I$ the affine blowup algebra $A [I / a] = (R (I))_{(a)} \subseteq A_{a}$ , the degree-zero part of the localisation of $R (I) = ⨁_{n \geq 0} I^{n} t^{n}$ (Affine blowup algebras: normal form, nonzerodivisors, reducedness, domains, Rees algebra sheaf of a finite type ideal).

[F1]

Integral schemes: $X$ is integral exactly when it is nonempty and every nonempty affine open of $X$ is the spectrum of a domain; equivalently $X$ is reduced and its underlying space is irreducible.

[F2]

Affine blowup algebras: normal form, nonzerodivisors, reducedness, domains: For $a \in I$ the affine blowup algebra $A [I / a]$ has $I A [I / a] = a A [I / a]$ with $a$ a nonzerodivisor and $(A [I / a])_{a} = A_{a}$ . Its construction as the degree-zero part of the localisation of $R (I) \subseteq A [t]$ at the degree-one element $a t$ embeds $A [I / a]$ into $A_{a}$ as the subring generated by $A$ and the fractions $i / a$ , $i \in I$ ; if $A$ is a domain and $a \neq 0$ then $A [I / a]$ is a domain, and if $A$ is reduced then $A [I / a]$ is reduced.

[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 ${Bl}_{I} Spec A$ and the presentation is independent of the chosen generating family.

[F4]

A principal localization identifies its spectrum with a distinguished open: For $a \in A$ the principal open $D (a)$ is $Spec A_{a}$ and the localisation morphism is an open immersion $D (a) ↪ Spec A$ with image the complement of $V (a)$ .

[F5]

Blowups restrict to open subschemes of the base: For an open subscheme $U ↪ X$ there is a canonical isomorphism ${Bl}_{I ∣_{U}} U \to {Bl}_{I} X \times_{X} U$ ; the blowup is covered by the restrictions over an affine cover of $X$ .

[F6]

The reduction of a scheme: On $Spec A$ , the reduction is $Spec (A / \sqrt{(0)})$ . Consequently reducedness is affine-local: a nilpotent section on a reduced affine chart is zero, and these charts cover all stalks. Also a subring of a reduced ring is reduced, since its nilpotent elements are nilpotent in the larger ring and hence zero.

Proof

1.1F2F3F4F5F6

Over an affine base $U = Spec A$ , discard generators $f_{i} = 0$ , whose charts are empty. If $A$ is a domain, every remaining chart $B_{i} = A [I / f_{i}] \subset A_{f_{i}}$ is a domain; if $A$ is reduced, every chart is reduced (including empty charts). Hence the blowup is reduced in either case, by affine-local reducedness. Moreover $D (f_{i})$ in chart $i$ is $Spec A_{f_{i}}$ by the chart localization identity.

2.1F1F2F3F4step 1.1

Suppose $X$ is integral and $I \neq 0$ . Then $X ∖ Z$ is a nonempty open: a nonzero local section of the ideal in a domain remains nonzero at the generic point. Put $W = π^{- 1} (X ∖ Z)$ . On chart $i$ , the inverse-image ideal is $f_{i} B_{i}$ , so $W \cap Spec B_{i} = D (f_{i})$ . The identifications $D (f_{i}) = Spec A_{f_{i}}$ are the structural morphism and agree on intersections: after both denominators are inverted, the ratio transition maps fix $A$ and the ordinary fractions. Thus they glue to $W ≅ X ∖ Z$ . This is a nonempty irreducible open.

3.1F1F3F6step 1.1step 2.1∎

Every nonempty chart is a domain chart with a nonzero denominator, so its nonempty principal open $D (f_{i}) \subset W$ is dense. The closure of $W$ therefore contains every chart and is the whole blowup. A closure of an irreducible set is irreducible. Together with reducedness and nonemptiness, this proves integrality. If the ideal is zero, all charts are empty and the reducedness assertion still holds.

Remarks

The argument does not need $X$ to be Noetherian or $I$ to be principal anywhere; it only uses that the affine blowup algebra sits inside the localisation $A_{a}$ .
If $I = 0$ the blowup is empty and hence reduced, while irreducibility and nonemptiness fail; this is why the integral statement assumes $I \neq 0$ .

Depends on

Used by

Dependency tree · two levels

37 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)
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)