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b797a5b | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 | 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
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