| import Mathlib | |
| import OAI.AlgebraicGeometry.Seshadri.Jets.PolynomialJets | |
| namespace OAI | |
| section | |
| noncomputable section | |
| section | |
| namespace MaximalSeshadri.QuadraticJets | |
| noncomputable section | |
| open MvPolynomial IsLocalRing | |
| attribute [local instance] RingHom.ker_isPrime | |
| lemma pair_square_eq (A : Type*) [CommRing A] (x y : A) : | |
| (Ideal.span ({x,y} : Set A)) ^ 2 = Ideal.span ({x^2,x*y,y^2} : Set A) := by | |
| rw [pow_two, Ideal.span_pair_mul_span_pair] | |
| congr 1 | |
| ext t | |
| simp only [Set.mem_insert_iff, Set.mem_singleton_iff, pow_two, mul_comm y x] | |
| tauto | |
| lemma mem_pair_square {A : Type*} [CommRing A] (x y f : A) | |
| (hf : f ∈ (Ideal.span ({x,y} : Set A)) ^ 2) : | |
| ∃ a b c : A, f = a*x^2 + b*x*y + c*y^2 := by | |
| rw [pair_square_eq, Ideal.mem_span_insert] at hf | |
| obtain ⟨a, g, hg, hfg⟩ := hf | |
| obtain ⟨b,c,hg⟩ := Ideal.mem_span_pair.mp hg | |
| refine ⟨a,b,c,?_⟩ | |
| rw [← hg] at hfg | |
| simpa only [mul_add, add_assoc, mul_assoc] using hfg | |
| variable {K A : Type*} [Field K] [CommRing A] [Algebra K A] | |
| [Algebra (MvPolynomial (Fin 2) K) A] [IsScalarTower K (MvPolynomial (Fin 2) K) A] | |
| theorem etale_coordinates_generate_local [Algebra.Etale (MvPolynomial (Fin 2) K) A] | |
| (ρ : A →ₐ[K] K) | |
| (hρ : ∀ i : Fin 2, ρ (algebraMap (MvPolynomial (Fin 2) K) A (X i)) = 0) : | |
| maximalIdeal (Localization.AtPrime (RingHom.ker ρ)) = Ideal.span | |
| (Set.range fun i : Fin 2 => algebraMap A (Localization.AtPrime (RingHom.ker ρ)) | |
| (algebraMap (MvPolynomial (Fin 2) K) A (X i))) := by | |
| let R := MvPolynomial (Fin 2) K | |
| let q := RingHom.ker ρ | |
| let p := q.under R | |
| have hp : p = idealOfVars (Fin 2) K := by | |
| rw [MaximalSeshadri.AlgebraicJets.idealOfVars_eq_ker_constantCoeff] | |
| change RingHom.ker (ρ.toRingHom.comp (algebraMap R A)) = _ | |
| have heq : ρ.toRingHom.comp (algebraMap R A) = constantCoeff := by | |
| apply MvPolynomial.ringHom_ext | |
| · intro c | |
| simp only [RingHom.comp_apply, constantCoeff_C] | |
| change ρ (algebraMap R A (algebraMap K R c)) = c | |
| rw [← IsScalarTower.algebraMap_apply K R A] | |
| exact ρ.commutes c | |
| · intro i | |
| simp only [RingHom.comp_apply, constantCoeff_X] | |
| change ρ (algebraMap (MvPolynomial (Fin 2) K) A (X i)) = 0 | |
| exact hρ i | |
| rw [heq] | |
| let Rp := Localization.AtPrime p | |
| let Aq := Localization.AtPrime q | |
| let : Algebra Rp Aq := Localization.AtPrime.algebraOfLiesOver p q | |
| let : Algebra.EssFiniteType Rp Aq := Algebra.EssFiniteType.of_comp R Rp Aq | |
| let : Algebra.FormallyEtale R Aq := Algebra.FormallyEtale.comp R A Aq | |
| let : Algebra.FormallyEtale Rp Aq := Algebra.FormallyEtale.of_restrictScalars (R := R) | |
| have hmax : (maximalIdeal Rp).map (algebraMap Rp Aq) = maximalIdeal Aq := | |
| Algebra.FormallyUnramified.map_maximalIdeal | |
| rw [← hmax, ← IsLocalization.AtPrime.map_eq_maximalIdeal p Rp, Ideal.map_map, | |
| ← IsScalarTower.algebraMap_eq R Rp Aq, hp, idealOfVars, Ideal.map_span] | |
| congr 1 | |
| rw [← Set.range_comp] | |
| congr 1 | |
| theorem etale_quadratic_local [Algebra.Etale (MvPolynomial (Fin 2) K) A] | |
| (ρ : A →ₐ[K] K) | |
| (hρ : ∀ i : Fin 2, ρ (algebraMap (MvPolynomial (Fin 2) K) A (X i)) = 0) | |
| (f : A) (hf : f ∈ (RingHom.ker ρ)^2) : | |
| let B := Localization.AtPrime (RingHom.ker ρ) | |
| let x := algebraMap A B (algebraMap (MvPolynomial (Fin 2) K) A (X 0)) | |
| let y := algebraMap A B (algebraMap (MvPolynomial (Fin 2) K) A (X 1)) | |
| ∃ a b c : B, algebraMap A B f = a*x^2 + b*x*y + c*y^2 := by | |
| dsimp only | |
| apply mem_pair_square | |
| have hm := etale_coordinates_generate_local ρ hρ | |
| have hr (g : Fin 2 → Localization.AtPrime (RingHom.ker ρ)) : | |
| Set.range g = {g 0,g 1} := by | |
| ext t | |
| simp only [Set.mem_range, Fin.exists_fin_two, Set.mem_insert_iff, Set.mem_singleton_iff] | |
| tauto | |
| rw [hr] at hm | |
| rw [← hm, ← IsLocalization.AtPrime.map_eq_maximalIdeal (RingHom.ker ρ), ← Ideal.map_pow] | |
| exact Ideal.mem_map_of_mem _ hf | |
| theorem etale_quadratic_neighborhood [Algebra.Etale (MvPolynomial (Fin 2) K) A] | |
| (ρ : A →ₐ[K] K) | |
| (hρ : ∀ i : Fin 2, ρ (algebraMap (MvPolynomial (Fin 2) K) A (X i)) = 0) | |
| (f : A) (hf : f ∈ (RingHom.ker ρ)^2) : | |
| let x := algebraMap (MvPolynomial (Fin 2) K) A (X 0) | |
| let y := algebraMap (MvPolynomial (Fin 2) K) A (X 1) | |
| ∃ d a b c : A, ρ d ≠ 0 ∧ d*f = a*x^2 + b*x*y + c*y^2 := by | |
| dsimp only | |
| let B := Localization.AtPrime (RingHom.ker ρ) | |
| let x := algebraMap (MvPolynomial (Fin 2) K) A (X 0) | |
| let y := algebraMap (MvPolynomial (Fin 2) K) A (X 1) | |
| have hmem : algebraMap A B f ∈ | |
| ((Ideal.span ({x,y} : Set A))^2).map (algebraMap A B) := by | |
| rw [Ideal.map_pow, Ideal.map_span] | |
| simp only [Set.image_insert_eq, Set.image_singleton] | |
| rw [pair_square_eq, Ideal.mem_span_insert] | |
| obtain ⟨a,b,c,heq⟩ := etale_quadratic_local ρ hρ f hf | |
| refine ⟨a, b*(algebraMap A B x * algebraMap A B y) + | |
| c*(algebraMap A B y)^2, Ideal.mem_span_pair.mpr ⟨b,c,rfl⟩, ?_⟩ | |
| simpa only [add_assoc, mul_assoc] using heq | |
| obtain ⟨d, hd, hdf⟩ := (IsLocalization.algebraMap_mem_map_algebraMap_iff | |
| (RingHom.ker ρ).primeCompl B _ f).mp hmem | |
| obtain ⟨a,b,c,heq⟩ := mem_pair_square x y _ hdf | |
| exact ⟨d,a,b,c,hd,heq⟩ | |
| theorem etale_linear_neighborhood [Algebra.Etale (MvPolynomial (Fin 2) K) A] | |
| (ρ : A →ₐ[K] K) | |
| (hρ : ∀ i : Fin 2, ρ (algebraMap (MvPolynomial (Fin 2) K) A (X i)) = 0) | |
| (f : A) (hf : ρ f = 0) : | |
| let x := algebraMap (MvPolynomial (Fin 2) K) A (X 0) | |
| let y := algebraMap (MvPolynomial (Fin 2) K) A (X 1) | |
| ∃ d a b : A, ρ d ≠ 0 ∧ d*f = a*x + b*y := by | |
| let B := Localization.AtPrime (RingHom.ker ρ) | |
| let x := algebraMap (MvPolynomial (Fin 2) K) A (X 0) | |
| let y := algebraMap (MvPolynomial (Fin 2) K) A (X 1) | |
| have heq : (Ideal.span ({x,y} : Set A)).map (algebraMap A B) = maximalIdeal B := by | |
| rw [etale_coordinates_generate_local ρ hρ, Ideal.map_span] | |
| simp only [Set.image_insert_eq, Set.image_singleton] | |
| congr 1 | |
| ext t | |
| simp only [Set.mem_range, Fin.exists_fin_two, Set.mem_insert_iff, Set.mem_singleton_iff] | |
| tauto | |
| have hmem : algebraMap A B f ∈ (Ideal.span ({x,y} : Set A)).map (algebraMap A B) := by | |
| rw [heq, ← IsLocalization.AtPrime.map_eq_maximalIdeal (RingHom.ker ρ)] | |
| exact Ideal.mem_map_of_mem _ hf | |
| obtain ⟨d,hd,hdf⟩ := (IsLocalization.algebraMap_mem_map_algebraMap_iff | |
| (RingHom.ker ρ).primeCompl B _ f).mp hmem | |
| obtain ⟨a,b,heq⟩ := Ideal.mem_span_pair.mp hdf | |
| exact ⟨d,a,b,hd,heq.symm⟩ | |
| end | |
| end MaximalSeshadri.QuadraticJets | |
| end | |
| end | |
| end | |
| end OAI | |