def Lec9Blowups.blowupAlgebra (A : Type u_1) [CommRing A] (𝔪 : Ideal A) : Subalgebra A (Polynomial A)

The blowup (Rees) algebra ⨁_{n ≥ 0} 𝔪ⁿ tⁿ ⊆ A[t] of the ideal 𝔪 ⊆ A (Lec 9, Def 20).

theorem Lec9Blowups.blowupAlgebra.monomial_mem {A : Type u_1} [CommRing A] {𝔪 : Ideal A} {n : ℕ} {a : A} : (Polynomial.monomial n) a ∈ blowupAlgebra A 𝔪 ↔ a ∈ 𝔪 ^ n

A monomial a · tⁿ lies in the blowup algebra of 𝔪 iff a ∈ 𝔪ⁿ.

theorem Lec9Blowups.mem_blowupAlgebra_iff {A : Type u_1} [CommRing A] {𝔪 : Ideal A} (f : Polynomial A) : f ∈ blowupAlgebra A 𝔪 ↔ ∀ (i : ℕ), f.coeff i ∈ 𝔪 ^ i

A polynomial lies in the blowup algebra of 𝔪 iff each of its coefficients lies in the corresponding power of 𝔪.

theorem Lec9Blowups.map_mem_blowupAlgebra {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] (φ : R →+* A) (𝔪 : Ideal A) (f : Polynomial R) (hf : f ∈ blowupAlgebra R (Ideal.comap φ 𝔪)) : Polynomial.map φ f ∈ blowupAlgebra A 𝔪

A ring map φ : R → A carries the blowup algebra of φ⁻¹(𝔪) into the blowup algebra of 𝔪.

noncomputable def Lec9Blowups.blowupMapRestrict {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] (φ : R →+* A) (𝔪 : Ideal A) : ↥(blowupAlgebra R (Ideal.comap φ 𝔪)) →+* ↥(blowupAlgebra A 𝔪)

The induced ring map between blowup algebras coming from φ : R → A.

theorem Lec9Blowups.blowupMapRestrict_val {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] (φ : R →+* A) (𝔪 : Ideal A) (f : ↥(blowupAlgebra R (Ideal.comap φ 𝔪))) : ↑((blowupMapRestrict φ 𝔪) f) = Polynomial.map φ ↑f

The induced map between blowup algebras is the polynomial map of φ applied to representatives.

theorem Lec9Blowups.blowupMapRestrict_surjective {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] (φ : R →+* A) (hφ : Function.Surjective ⇑φ) (𝔪 : Ideal A) : Function.Surjective ⇑(blowupMapRestrict φ 𝔪)

If φ : R → A is surjective, the induced map on blowup algebras is surjective.

noncomputable def Lec9Blowups.blowupQuotientEquiv {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] (φ : R →+* A) (hφ : Function.Surjective ⇑φ) (𝔪 : Ideal A) : ↥(blowupAlgebra R (Ideal.comap φ 𝔪)) ⧸ RingHom.ker (blowupMapRestrict φ 𝔪) ≃+* ↥(blowupAlgebra A 𝔪)

For a surjection φ : R → A, the blowup algebra of 𝔪 in A is isomorphic to the blowup algebra of φ⁻¹(𝔪) in R modulo the kernel of the induced map.

noncomputable def Lec9Blowups.blowup_independent_of_embedding {R₁ : Type u_1} {R₂ : Type u_2} {A : Type u_3} [CommRing R₁] [CommRing R₂] [CommRing A] (φ₁ : R₁ →+* A) (hφ₁ : Function.Surjective ⇑φ₁) (φ₂ : R₂ →+* A) (hφ₂ : Function.Surjective ⇑φ₂) (𝔪 : Ideal A) : ↥(blowupAlgebra R₁ (Ideal.comap φ₁ 𝔪)) ⧸ RingHom.ker (blowupMapRestrict φ₁ 𝔪) ≃+* ↥(blowupAlgebra R₂ (Ideal.comap φ₂ 𝔪)) ⧸ RingHom.ker (blowupMapRestrict φ₂ 𝔪)

Lec 9, Prop 13: the blowup of A along 𝔪 is independent of the chosen surjective presentation of A.

noncomputable def Lec9Blowups.blowup_intrinsic (k : Type u_1) [Field k] {R₁ : Type u_2} {R₂ : Type u_3} {A : Type u_4} [CommRing R₁] [CommRing R₂] [CommRing A] [Algebra k A] [Algebra k R₁] [Algebra k R₂] (φ₁ : R₁ →+* A) (hφ₁ : Function.Surjective ⇑φ₁) (φ₂ : R₂ →+* A) (hφ₂ : Function.Surjective ⇑φ₂) (𝔪 : Ideal A) (_h𝔪 : 𝔪.IsMaximal) : ↥(blowupAlgebra R₁ (Ideal.comap φ₁ 𝔪)) ⧸ RingHom.ker (blowupMapRestrict φ₁ 𝔪) ≃+* ↥(blowupAlgebra R₂ (Ideal.comap φ₂ 𝔪)) ⧸ RingHom.ker (blowupMapRestrict φ₂ 𝔪)

Lec 9, Prop 13 (intrinsic blowup): the blowup at a maximal ideal of a k-algebra A is independent of the chosen surjective presentation by a k-algebra R.

noncomputable def Lec9Blowups.blowup_intrinsic_equiv (k : Type u_1) [Field k] {R : Type u_2} {A : Type u_3} [CommRing R] [CommRing A] [Algebra k A] [Algebra k R] (φ : R →+* A) (hφ : Function.Surjective ⇑φ) (𝔪 : Ideal A) (_h𝔪 : 𝔪.IsMaximal) : ↥(blowupAlgebra R (Ideal.comap φ 𝔪)) ⧸ RingHom.ker (blowupMapRestrict φ 𝔪) ≃+* ↥(blowupAlgebra A 𝔪)

The intrinsic blowup of A at a maximal ideal 𝔪 is naturally isomorphic to the blowup of any k-algebra presentation.