openAI-math-dataset / lean /OAI /NumberTheory /PiExponent /Ampleness /AffineFunctionFieldCompatibility.lean
| import Mathlib.RingTheory.Localization.LocalizationLocalization | |
| import OAI.NumberTheory.PiExponent.Geometry.CurveFunctionFieldCompatibility | |
| namespace OAI | |
| noncomputable section | |
| namespace PiExponent.CurveNormalizationModel | |
| open CategoryTheory AlgebraicGeometry | |
| universe u | |
| variable (A E : Type u) [CommRing A] [IsDomain A] [Field E] | |
| [Algebra A E] [IsFractionRing A E] | |
| theorem isFractionRing_of_intermediate_domain | |
| (B : Type u) [CommRing B] [IsDomain B] [Algebra A B] | |
| [Algebra B E] [IsScalarTower A B E] | |
| (hB : Function.Injective (algebraMap B E)) : IsFractionRing B E where | |
| map_units b := isUnit_iff_ne_zero.mpr (by | |
| intro h | |
| exact (mem_nonZeroDivisors_iff_ne_zero.mp b.2) (hB (by simpa using h))) | |
| surj z := by | |
| obtain ⟨⟨a, s⟩, hs⟩ := IsLocalization.surj (nonZeroDivisors A) z | |
| have hn : algebraMap A B s ≠ 0 := by | |
| intro he | |
| have he' : algebraMap A E s = 0 := by | |
| rw [IsScalarTower.algebraMap_apply A B E, he, map_zero] | |
| exact (mem_nonZeroDivisors_iff_ne_zero.mp s.2) | |
| ((IsFractionRing.injective A E) (by simp at he')) | |
| refine ⟨⟨algebraMap A B a, ⟨algebraMap A B s, mem_nonZeroDivisors_iff_ne_zero.mpr hn⟩⟩, ?_⟩ | |
| simpa only [← IsScalarTower.algebraMap_apply A B E] using hs | |
| exists_of_eq {a b} h := ⟨1, by simpa using hB h⟩ | |
| def affineFunctionFieldEquiv : (Spec (.of A)).functionField ≃+* E := by | |
| letI : Algebra A (Spec (.of A)).functionField := | |
| AlgebraicGeometry.instAlgebraCarrierFunctionFieldSpec (.of A) | |
| letI : IsFractionRing A (Spec (.of A)).functionField := | |
| functionField_isFractionRing_of_affine (.of A) | |
| exact (IsLocalization.algEquiv (nonZeroDivisors A) (Spec (.of A)).functionField E).toRingEquiv | |
| @[simp] theorem affineFunctionFieldEquiv_toStalk (a : A) : | |
| affineFunctionFieldEquiv A E (StructureSheaf.toStalk A | |
| (genericPoint (Spec (.of A))) a) = algebraMap A E a := by | |
| let : Algebra A (Spec (.of A)).functionField := | |
| AlgebraicGeometry.instAlgebraCarrierFunctionFieldSpec (.of A) | |
| let : IsFractionRing A (Spec (.of A)).functionField := | |
| functionField_isFractionRing_of_affine (.of A) | |
| exact (IsLocalization.algEquiv (nonZeroDivisors A) | |
| (Spec (.of A)).functionField E).commutes a | |
| theorem affineFunctionFieldEquiv_stalk (q : PrimeSpectrum A) | |
| [Algebra (Localization.AtPrime q.asIdeal) E] | |
| [IsScalarTower A (Localization.AtPrime q.asIdeal) E] | |
| (a : (Spec (.of A)).presheaf.stalk q) : | |
| affineFunctionFieldEquiv A E | |
| ((Spec (.of A)).presheaf.stalkSpecializes (genericPoint_specializes q) a) = | |
| algebraMap (Localization.AtPrime q.asIdeal) E ((Spec.stalkIso (.of A) q).hom a) := by | |
| let : Algebra A ((Spec (.of A)).presheaf.stalk q) := StructureSheaf.stalkAlgebra A q | |
| let : IsLocalization.AtPrime ((Spec (.of A)).presheaf.stalk q) q.asIdeal := | |
| StructureSheaf.IsLocalization.to_stalk A q | |
| have heq : (affineFunctionFieldEquiv A E).toRingHom.comp | |
| ((Spec (.of A)).presheaf.stalkSpecializes (genericPoint_specializes q)).hom = | |
| (algebraMap (Localization.AtPrime q.asIdeal) E).comp | |
| (Spec.stalkIso (.of A) q).hom.hom := by | |
| apply IsLocalization.ringHom_ext q.asIdeal.primeCompl | |
| apply RingHom.ext | |
| intro b | |
| change affineFunctionFieldEquiv A E | |
| ((Spec (.of A)).presheaf.stalkSpecializes (genericPoint_specializes q) | |
| (StructureSheaf.toStalk A q b)) = | |
| algebraMap (Localization.AtPrime q.asIdeal) E | |
| ((Spec.stalkIso (.of A) q).hom (StructureSheaf.toStalk A q b)) | |
| have hs : (Spec (.of A)).presheaf.stalkSpecializes (genericPoint_specializes q) | |
| (StructureSheaf.toStalk A q b) = | |
| StructureSheaf.toStalk A (genericPoint (Spec (.of A))) b := by | |
| exact congrArg (fun h => h b) | |
| (StructureSheaf.toStalk_stalkSpecializes (R := A) (genericPoint_specializes q)) | |
| have ht : (Spec.stalkIso (.of A) q).hom (StructureSheaf.toStalk A q b) = | |
| algebraMap A (Localization.AtPrime q.asIdeal) b := | |
| (StructureSheaf.stalkIso A q).symm.commutes b | |
| rw [hs, ht, affineFunctionFieldEquiv_toStalk] | |
| exact IsScalarTower.algebraMap_apply A (Localization.AtPrime q.asIdeal) E b | |
| exact RingHom.congr_fun heq a | |
| theorem affineFunctionFieldEquiv_naturality | |
| (B : Type u) [CommRing B] [IsDomain B] [Algebra A B] | |
| [Algebra B E] [IsFractionRing B E] [IsScalarTower A B E] | |
| [IsOpenImmersion (Spec.map (CommRingCat.ofHom (algebraMap A B)))] : | |
| (affineFunctionFieldEquiv B E).toRingHom.comp | |
| (functionFieldRestriction (Spec.map (CommRingCat.ofHom (algebraMap A B)))).hom = | |
| (affineFunctionFieldEquiv A E).toRingHom := by | |
| let : Algebra A (Spec (.of A)).functionField := | |
| AlgebraicGeometry.instAlgebraCarrierFunctionFieldSpec (.of A) | |
| let : IsFractionRing A (Spec (.of A)).functionField := | |
| functionField_isFractionRing_of_affine (.of A) | |
| apply IsLocalization.ringHom_ext (nonZeroDivisors A) | |
| apply RingHom.ext | |
| intro a | |
| change affineFunctionFieldEquiv B E | |
| (functionFieldRestriction (Spec.map (CommRingCat.ofHom (algebraMap A B))) | |
| (StructureSheaf.toStalk A (genericPoint (Spec (.of A))) a)) = | |
| affineFunctionFieldEquiv A E | |
| (StructureSheaf.toStalk A (genericPoint (Spec (.of A))) a) | |
| have h := congrArg (fun h => h a) | |
| (functionFieldRestriction_toStalk (CommRingCat.ofHom (algebraMap A B))) | |
| change functionFieldRestriction (Spec.map (CommRingCat.ofHom (algebraMap A B))) | |
| (StructureSheaf.toStalk A (genericPoint (Spec (.of A))) a) = | |
| StructureSheaf.toStalk B (genericPoint (Spec (.of B))) (algebraMap A B a) at h | |
| rw [h, affineFunctionFieldEquiv_toStalk, affineFunctionFieldEquiv_toStalk] | |
| exact (IsScalarTower.algebraMap_apply A B E a).symm | |
| end PiExponent.CurveNormalizationModel | |
| end | |
| end OAI | |