| 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 |
|
|
