openAI-math-dataset / lean /OAI /NumberTheory /PiExponent /Geometry /CurveGeometricFunctionField.lean
| import Mathlib.RingTheory.Spectrum.Prime.Topology | |
| import OAI.NumberTheory.PiExponent.Ampleness.AffineFunctionFieldCompatibility | |
| import OAI.NumberTheory.PiExponent.Geometry.CurveParameterFinite | |
| import OAI.NumberTheory.PiExponent.LocalAlgebra.AffineTranscendenceDimension | |
| namespace OAI | |
| noncomputable section | |
| namespace PiExponent.CurveGeometricFunctionField | |
| open CategoryTheory AlgebraicGeometry | |
| variable (k A E : Type*) [Field k] [CommRing A] [IsDomain A] [Field E] | |
| [Algebra k A] [Algebra k E] [Algebra A E] [IsScalarTower k A E] | |
| [IsFractionRing A E] | |
| omit [IsDomain A] in | |
| theorem fractionField_essFiniteType [Algebra.FiniteType k A] : | |
| Algebra.EssFiniteType k E := by | |
| let : Algebra.EssFiniteType A E := | |
| Algebra.EssFiniteType.of_isLocalization E (nonZeroDivisors A) | |
| exact Algebra.EssFiniteType.comp k A E | |
| theorem fractionField_trdeg_eq_one [Algebra.FiniteType k A] | |
| (hdim : ringKrullDim A = 1) : Algebra.trdeg k E = 1 := by | |
| let : FaithfulSMul A E := | |
| (faithfulSMul_iff_algebraMap_injective A E).mpr (IsFractionRing.injective A E) | |
| have hd : (Algebra.trdeg k A).toNat = 1 := by | |
| rw [PiExponentJets.W24.finiteType_domain_krullDim_eq_trdeg k A] at hdim | |
| exact_mod_cast hdim | |
| have he := PiExponentJets.W24.localization_trdeg_toNat_eq k A E | |
| (nonZeroDivisors A) | |
| exact Cardinal.toNat_eq_one.mp (he.symm.trans hd) | |
| def coordinates {ι : Type*} (φ : MvPolynomial ι k →ₐ[k] A) : ι → E := | |
| fun i => algebraMap A E (φ (MvPolynomial.X i)) | |
| omit [IsDomain A] in | |
| theorem adjoin_coordinates_eq_top {ι : Type*} | |
| (φ : MvPolynomial ι k →ₐ[k] A) (hφ : Function.Surjective φ) : | |
| IntermediateField.adjoin k (Set.range (coordinates k A E φ)) = ⊤ := by | |
| let x := coordinates k A E φ | |
| let K := IntermediateField.adjoin k (Set.range x) | |
| have hψ : (IsScalarTower.toAlgHom k A E).comp φ = MvPolynomial.aeval x := by | |
| apply MvPolynomial.algHom_ext | |
| intro i | |
| simp [x, coordinates] | |
| have hmem (a : A) : algebraMap A E a ∈ K := by | |
| obtain ⟨p, rfl⟩ := hφ a | |
| apply IntermediateField.algebra_adjoin_le_adjoin k (Set.range x) | |
| rw [← MvPolynomial.aeval_range] | |
| refine ⟨p, ?_⟩ | |
| exact (congrArg (fun f : MvPolynomial ι k →ₐ[k] E => f p) hψ).symm | |
| apply top_le_iff.mp | |
| intro z _ | |
| obtain ⟨a, b, _, rfl⟩ := IsFractionRing.div_surjective A z | |
| exact K.div_mem (hmem a) (hmem b) | |
| theorem of_surjective_coordinates {ι : Type*} [Finite ι] | |
| (φ : MvPolynomial ι k →ₐ[k] A) (hφ : Function.Surjective φ) | |
| (hdim : ringKrullDim A = 1) : | |
| Algebra.EssFiniteType k E ∧ Algebra.trdeg k E = 1 ∧ | |
| IntermediateField.adjoin k (Set.range (coordinates k A E φ)) = ⊤ := by | |
| let : Algebra.FiniteType k A := Algebra.FiniteType.of_surjective φ hφ | |
| exact ⟨fractionField_essFiniteType k A E, | |
| fractionField_trdeg_eq_one k A E hdim, adjoin_coordinates_eq_top k A E φ hφ⟩ | |
| theorem finite_over_every_parameter [Algebra.FiniteType k A] | |
| (hdim : ringKrullDim A = 1) : | |
| ∀ f : E, Transcendental k f → | |
| FiniteDimensional (IntermediateField.adjoin k {f}) E := by | |
| let := fractionField_essFiniteType k A E | |
| exact CurveParameterFinite.finite_over_every_parameter k E | |
| (fractionField_trdeg_eq_one k A E hdim) | |
| omit [IsDomain A] in | |
| theorem ringKrullDim_eq_one_of_affine_iso | |
| (X : Scheme) (e : X ≅ Spec (.of A)) (hdim : topologicalKrullDim X = 1) : | |
| ringKrullDim A = 1 := by | |
| have h := e.hom.homeomorph.isHomeomorph.topologicalKrullDim_eq | |
| change topologicalKrullDim X = topologicalKrullDim (PrimeSpectrum A) at h | |
| rw [PrimeSpectrum.topologicalKrullDim_eq_ringKrullDim] at h | |
| exact h.symm.trans hdim | |
| theorem of_affine_curve {ι : Type*} [Finite ι] | |
| (X : Scheme) (e : X ≅ Spec (.of A)) (hdim : topologicalKrullDim X = 1) | |
| (φ : MvPolynomial ι k →ₐ[k] A) (hφ : Function.Surjective φ) : | |
| Algebra.EssFiniteType k E ∧ Algebra.trdeg k E = 1 ∧ | |
| IntermediateField.adjoin k (Set.range (coordinates k A E φ)) = ⊤ := | |
| of_surjective_coordinates k A E φ hφ | |
| (ringKrullDim_eq_one_of_affine_iso A X e hdim) | |
| @[instance_reducible] def intrinsicSectionAlgebra (X : Scheme) [IsIntegral X] : | |
| Algebra Γ(X, ⊤) X.functionField := by | |
| letI : Nonempty (⊤ : X.Opens) := ⟨⟨genericPoint X, trivial⟩⟩ | |
| exact (X.germToFunctionField ⊤).hom.toAlgebra | |
| attribute [local instance] intrinsicSectionAlgebra | |
| @[instance_reducible] def intrinsicFunctionFieldAlgebra (k : Type*) [Field k] | |
| (X : Scheme) [IsIntegral X] [Algebra k Γ(X, ⊤)] : | |
| Algebra k X.functionField := | |
| ((algebraMap Γ(X, ⊤) X.functionField).comp (algebraMap k Γ(X, ⊤))).toAlgebra | |
| theorem intrinsic_functionField (k : Type*) [Field k] | |
| (X : Scheme) [IsIntegral X] [IsAffine X] [Algebra k Γ(X, ⊤)] | |
| {ι : Type*} [Finite ι] (hdim : topologicalKrullDim X = 1) | |
| (φ : MvPolynomial ι k →ₐ[k] Γ(X, ⊤)) (hφ : Function.Surjective φ) : | |
| letI : Algebra k X.functionField := intrinsicFunctionFieldAlgebra k X | |
| Algebra.EssFiniteType k X.functionField ∧ Algebra.trdeg k X.functionField = 1 ∧ | |
| IntermediateField.adjoin k | |
| (Set.range (coordinates k Γ(X, ⊤) X.functionField φ)) = ⊤ := by | |
| let : Algebra k X.functionField := intrinsicFunctionFieldAlgebra k X | |
| let : IsScalarTower k Γ(X, ⊤) X.functionField := | |
| IsScalarTower.of_algebraMap_eq fun _ => rfl | |
| let : Nonempty (⊤ : X.Opens) := ⟨⟨genericPoint X, trivial⟩⟩ | |
| let : IsFractionRing Γ(X, ⊤) X.functionField := | |
| functionField_isFractionRing_of_isAffineOpen X (⊤ : X.Opens) (isAffineOpen_top X) | |
| exact of_affine_curve k Γ(X, ⊤) X.functionField X X.isoSpec hdim φ hφ | |
| end PiExponent.CurveGeometricFunctionField | |
| end | |
| end OAI | |