The trivial blow-up: blowing up an invertible ideal is an isomorphism #
When the centre is already an effective Cartier divisor, the blow-up map is an isomorphism [Kol07, Warning 20]; [Hau14, Remark 4.5]; [TSPA, Tag 0807].
The argument #
A morphism π : B ⟶ X is a blow-up of X along the ideal sheaf I [Hau14, Definition 4.4] if
π is admissible — the inverse image ideal I·𝒪_B = I.comap π is invertible — and
universal: every admissible f : Y ⟶ X factors through π by a unique g : Y ⟶ B. Suppose
I itself is invertible. Then the identity of X is admissible (its inverse image of I is I),
so it factors as σ ≫ π = 𝟙 X for a unique σ : X ⟶ B. The morphism π is admissible by
hypothesis, and both π ≫ σ and 𝟙 B are factorizations of π through π
((π ≫ σ) ≫ π = π ≫ 𝟙 X = π); by uniqueness they coincide, π ≫ σ = 𝟙 B. So σ is a two-sided
inverse of π. This is isIso_of_isInvertible_of_universal, stated for any morphism with the
universal property; Hironaka.Scheme.BlowUp.Glue.Trivial applies it to the blow-up.
The affine model of a principal centre #
For Spec R and the centre (g), the affine blow-up is covered by the single chart of g (the
generating set {g}), whose ring R[(g)/g] ⊆ R_g consists of the constants: an element y has
y·gⁿ = x with x ∈ (g)ⁿ = (gⁿ), say x = r·gⁿ, and gⁿ is a unit of R_g, so y = r
(affineBlowUpAlgebra.surjective_algebraMap_span_singleton). Hence R[(g)/g] is the image of
R in R_g, i.e. R/T_g with T_g = ker (R → R_g) = ⋃ₖ Ann(gᵏ) the g-power torsion
(affineBlowUpAlgebra.ker_algebraMap_span_singleton,
affineBlowUpAlgebra.mem_ker_algebraMap_span_singleton_iff), and B_{(g)} Spec R = Spec (R/T_g).
Hauser's Example 4.31 describes the blow-up along a zero divisor g with hg = 0 as the
subscheme V(h): it lies in V(h)
(affineBlowUpAlgebra.algebraMap_span_singleton_eq_zero_of_mul_eq_zero) and equals it exactly
when T_g = (h). When g is a nonzerodivisor, R → R_g is injective, so R → R[(g)/g] is
bijective (affineBlowUpAlgebra.bijective_algebraMap_span_singleton) and the blow-up map — the
inverse of the chart followed by Spec (R ≅ R[(g)/g]) — is an isomorphism
(affineBlowUp.isIso_π_span_singleton): B_{(g)} Spec R = Spec R [Hau14, Example 4.29]. The
same example's first assertion, that the Rees algebra R[gt] ⊂ R[t] is isomorphic as a graded
R-algebra to the polynomial ring R[v] with deg v = 1, is reesAlgebra.polynomialEquiv:
v ↦ gt is surjective because the Rees algebra of (g) is generated by gt (Mathlib's
adjoin_monomial_eq_reesAlgebra), injective because the n-th coefficient of the image of p is
pₙ·gⁿ (Mathlib's comp_C_mul_X_coeff) and gⁿ is a nonzerodivisor, and graded because it
sends the monomial r vⁿ to r gⁿ tⁿ (reesAlgebra.ofPolynomial_monomial_mem_grading).
A blow-up along an invertible ideal is an isomorphism [TSPA, Tag 0807];
[Hau14, Remark 4.5 and Definition 4.4]; [Kol07, Warning 20]. If π : B ⟶ X is admissible for
I (I.comap π invertible) and universal among
admissible morphisms (every f : Y ⟶ X with I.comap f invertible factors uniquely through π),
and I is invertible, then π is an isomorphism: its inverse is the unique lift of 𝟙 X, and
π ≫ σ = 𝟙 B because both are lifts of the admissible morphism π.
The chart ring R[(g)/g] ⊆ R_g of the principal centre (g) at its generator consists of the
constants: y·gⁿ = r·gⁿ in R_g forces y = r [Hau14, Examples 4.29 and 4.31].
The kernel of R → R[(g)/g] is the kernel of R → R_g (the values agree, R[(g)/g] being a
subring of R_g).
The kernel of R → R[(g)/g] is the g-power torsion T_g = ⋃ₖ Ann(gᵏ), so
R[(g)/g] ≅ R/T_g and B_{(g)} Spec R = Spec (R/T_g) (compare [Hau14, Example 4.31]).
For h·g = 0 the element h dies in the chart ring R[(g)/g], so the blow-up
B_{(g)} Spec R lies in V(h) [Hau14, Example 4.31] (it equals V(h) exactly when T_g = (h),
as in [Hau14, Example 4.32]).
For a nonzerodivisor g, R[(g)/g] = R [Hau14, Example 4.29]: the structure map is
bijective (injective because R → R_g is, surjective by
affineBlowUpAlgebra.surjective_algebraMap_span_singleton).
The chart of g is all of the affine blow-up along (g): the generating set {g}.
The chart of g is a surjective open immersion, hence an isomorphism.
B_{(g)} Spec R = Spec R for a nonzerodivisor g [Hau14, Example 4.29]; the trivial
blow-up of [Kol07, Warning 20] in the affine model: the blow-up map is an isomorphism, being the
inverse of the chart of g followed by Spec (R ≅ R[(g)/g]).
v ↦ g t: the R-algebra map from the polynomial ring R[v] to the Rees algebra of the
principal ideal (g) [Hau14, Example 4.29].
Equations
Instances For
In R[t], the image of p(v) is p(g t), the substitution v ↦ g t.
v ↦ g t is injective for a nonzerodivisor g: the n-th coefficient of p(g t) is
pₙ gⁿ.
v ↦ g t is surjective: the Rees algebra of (g) is generated over R by g t
(Mathlib's adjoin_monomial_eq_reesAlgebra, with monomial 1 (r g) = C r * monomial 1 g).
For a nonzerodivisor g, the Rees algebra R[gt] is isomorphic to the polynomial ring
R[v], by v ↦ g t [Hau14, Example 4.29].
Equations
Instances For
The isomorphism is graded ("deg r = 0 for r ∈ R and deg v = 1"): the monomial r vⁿ
goes to r gⁿ tⁿ, an element of degree n of the Rees algebra.