File size: 6,261 Bytes
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 | import OAI.NumberTheory.PiExponent.Geometry.CurveChartFunctionField
import OAI.NumberTheory.PiExponent.Geometry.CurveNormalizationOverlap
namespace OAI
noncomputable section
namespace PiExponent.CurveNormalizationModel
open CategoryTheory AlgebraicGeometry
open scoped Polynomial nonZeroDivisors
open PiExponent.CurveZeroPole
universe u
variable {F E : Type u} [Field F] [Field E] [Algebra F E]
variable (f : E) (hf : Transcendental F f)
variable [FiniteDimensional (IntermediateField.adjoin F {f}) E]
abbrev parameterChartFractionRing : IsFractionRing (parameterChart f hf) E :=
let := parameterAlgebra f hf
let := parameterPolynomialAlgebra f hf
let := parameter_scalarTower f hf
let := parameter_finite f hf
inferInstance
abbrev parameterChartLocalAlgebra (q : PrimeSpectrum (parameterChart f hf)) :
Algebra (Localization.AtPrime q.asIdeal) E :=
let := parameterChartFractionRing f hf
IsLocalization.localizationAlgebraOfSubmonoidLe (Localization.AtPrime q.asIdeal) E
q.asIdeal.primeCompl (nonZeroDivisors (parameterChart f hf))
q.asIdeal.primeCompl_le_nonZeroDivisors
theorem parameterChartLocal_scalarTower (q : PrimeSpectrum (parameterChart f hf)) :
letI := parameterChartLocalAlgebra f hf q
IsScalarTower (parameterChart f hf) (Localization.AtPrime q.asIdeal) E := by
let := parameterChartFractionRing f hf
exact IsLocalization.localization_isScalarTower_of_submonoid_le
(Localization.AtPrime q.asIdeal) E q.asIdeal.primeCompl
(nonZeroDivisors (parameterChart f hf)) q.asIdeal.primeCompl_le_nonZeroDivisors
theorem parameterLaurentChartFractionRing : IsFractionRing (parameterLaurentChart f hf) E := by
let := parameterChartFractionRing f hf
let : Algebra (parameterChart f hf) (parameterLaurentChart f hf) :=
(parameterChartOverlapHom f hf).toAlgebra
have : IsScalarTower (parameterChart f hf) (parameterLaurentChart f hf) E :=
IsScalarTower.of_algebraMap_eq' rfl
exact isFractionRing_of_intermediate_domain (parameterChart f hf) E
(parameterLaurentChart f hf) Subtype.val_injective
theorem parameterCurveFunctionFieldEquiv_eq_infinity :
letI : FiniteDimensional (IntermediateField.adjoin F {f⁻¹}) E :=
(adjoin_inverse_eq (F := F) f).symm ▸ inferInstance
letI := parameterChartFractionRing f⁻¹ (transcendental_inverse f hf)
(parameterCurveFunctionFieldEquiv f hf).toRingHom =
affineChartFunctionFieldMap (parameterChart f⁻¹ (transcendental_inverse f hf)) E
(infinityChartInclusion f hf) := by
let : FiniteDimensional (IntermediateField.adjoin F {f⁻¹}) E :=
(adjoin_inverse_eq (F := F) f).symm ▸ inferInstance
let := parameterChartFractionRing f hf
let := parameterChartFractionRing f⁻¹ (transcendental_inverse f hf)
let := parameterLaurentChartFractionRing f hf
let : Algebra (parameterChart f hf) (parameterLaurentChart f hf) :=
(parameterChartOverlapHom f hf).toAlgebra
let : Algebra (parameterChart f⁻¹ (transcendental_inverse f hf))
(parameterLaurentChart f hf) :=
((reciprocalLaurentChartEquiv f hf).symm.toRingHom.comp
(parameterChartOverlapHom f⁻¹ (transcendental_inverse f hf))).toAlgebra
have : IsScalarTower (parameterChart f hf) (parameterLaurentChart f hf) E :=
IsScalarTower.of_algebraMap_eq' rfl
have : IsScalarTower (parameterChart f⁻¹ (transcendental_inverse f hf))
(parameterLaurentChart f hf) E := IsScalarTower.of_algebraMap_eq' rfl
have : IsOpenImmersion (Spec.map (CommRingCat.ofHom
(algebraMap (parameterChart f hf) (parameterLaurentChart f hf)))) :=
zeroOverlap_isOpenImmersion f hf
have : IsOpenImmersion (Spec.map (CommRingCat.ofHom
(algebraMap (parameterChart f⁻¹ (transcendental_inverse f hf))
(parameterLaurentChart f hf)))) := infinityOverlap_isOpenImmersion f hf
change affineChartFunctionFieldMap (parameterChart f hf) E (zeroChartInclusion f hf) = _
exact affineChartFunctionFieldMap_eq_of_overlap (parameterChart f hf) E
(parameterChart f⁻¹ (transcendental_inverse f hf)) (parameterLaurentChart f hf)
(zeroChartInclusion f hf) (infinityChartInclusion f hf)
(parameterCurve_overlap_compatibility f hf)
theorem zeroChartStalk_functionField_compatibility
(q : PrimeSpectrum (parameterChart f hf))
(a : (parameterCurve f hf).presheaf.stalk (zeroChartInclusion f hf q)) :
letI := parameterChartLocalAlgebra f hf q
parameterCurveFunctionFieldEquiv f hf
(algebraMap ((parameterCurve f hf).presheaf.stalk (zeroChartInclusion f hf q))
(parameterCurve f hf).functionField a) =
algebraMap (Localization.AtPrime q.asIdeal) E ((zeroChartStalkIso f hf q).hom a) := by
let := parameterChartFractionRing f hf
let := parameterChartLocalAlgebra f hf q
let := parameterChartLocal_scalarTower f hf q
exact affineChartFunctionFieldMap_stalk (parameterChart f hf) E
(zeroChartInclusion f hf) q a
theorem infinityChartStalk_functionField_compatibility
(q : PrimeSpectrum (parameterChart f⁻¹ (transcendental_inverse f hf)))
(a : (parameterCurve f hf).presheaf.stalk (infinityChartInclusion f hf q)) :
letI : FiniteDimensional (IntermediateField.adjoin F {f⁻¹}) E :=
(adjoin_inverse_eq (F := F) f).symm ▸ inferInstance
letI := parameterChartLocalAlgebra f⁻¹ (transcendental_inverse f hf) q
parameterCurveFunctionFieldEquiv f hf
(algebraMap ((parameterCurve f hf).presheaf.stalk (infinityChartInclusion f hf q))
(parameterCurve f hf).functionField a) =
algebraMap (Localization.AtPrime q.asIdeal) E ((infinityChartStalkIso f hf q).hom a) := by
let : FiniteDimensional (IntermediateField.adjoin F {f⁻¹}) E :=
(adjoin_inverse_eq (F := F) f).symm ▸ inferInstance
let := parameterChartFractionRing f⁻¹ (transcendental_inverse f hf)
let := parameterChartLocalAlgebra f⁻¹ (transcendental_inverse f hf) q
let := parameterChartLocal_scalarTower f⁻¹ (transcendental_inverse f hf) q
change (parameterCurveFunctionFieldEquiv f hf).toRingHom _ = _
rw [parameterCurveFunctionFieldEquiv_eq_infinity]
exact affineChartFunctionFieldMap_stalk (parameterChart f⁻¹ (transcendental_inverse f hf)) E
(infinityChartInclusion f hf) q a
end PiExponent.CurveNormalizationModel
end
end OAI
|