| import OAI.NumberTheory.PiExponent.Ampleness.AffineFunctionFieldCompatibility | |
| 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] | |
| def affineChartFunctionFieldMap {X : Scheme.{u}} [IsIntegral X] | |
| (i : Spec (.of A) ⟶ X) [IsOpenImmersion i] : X.functionField →+* E := | |
| (affineFunctionFieldEquiv A E).toRingHom.comp (functionFieldRestriction i).hom | |
| theorem affineChartFunctionFieldMap_stalk {X : Scheme.{u}} [IsIntegral X] | |
| (i : Spec (.of A) ⟶ X) [IsOpenImmersion i] (q : PrimeSpectrum A) | |
| [Algebra (Localization.AtPrime q.asIdeal) E] | |
| [IsScalarTower A (Localization.AtPrime q.asIdeal) E] | |
| (a : X.presheaf.stalk (i q)) : | |
| affineChartFunctionFieldMap A E i | |
| (X.presheaf.stalkSpecializes (genericPoint_specializes (i q)) a) = | |
| algebraMap (Localization.AtPrime q.asIdeal) E | |
| ((Spec.stalkIso (.of A) q).hom (i.stalkMap q a)) := by | |
| have h := congrArg (fun h => h a) (functionFieldRestriction_stalk i q) | |
| change functionFieldRestriction i | |
| (X.presheaf.stalkSpecializes (genericPoint_specializes (i q)) a) = | |
| (Spec (.of A)).presheaf.stalkSpecializes (genericPoint_specializes q) | |
| (i.stalkMap q a) at h | |
| change affineFunctionFieldEquiv A E (_) = _ | |
| rw [h] | |
| exact affineFunctionFieldEquiv_stalk A E q _ | |
| private theorem restriction_congr {U X : Scheme.{u}} [IsIntegral U] [IsIntegral X] | |
| {i j : U ⟶ X} [IsOpenImmersion i] [IsOpenImmersion j] (h : i = j) : | |
| functionFieldRestriction i = functionFieldRestriction j := by | |
| subst j | |
| rfl | |
| theorem affineChartFunctionFieldMap_eq_of_overlap | |
| (B G : Type u) [CommRing B] [IsDomain B] [CommRing G] [IsDomain G] | |
| [Algebra B E] [IsFractionRing B E] [Algebra G E] [IsFractionRing G E] | |
| [Algebra A G] [Algebra B G] [IsScalarTower A G E] [IsScalarTower B G E] | |
| {X : Scheme.{u}} [IsIntegral X] | |
| (iA : Spec (.of A) ⟶ X) (iB : Spec (.of B) ⟶ X) | |
| [IsOpenImmersion iA] [IsOpenImmersion iB] | |
| [IsOpenImmersion (Spec.map (CommRingCat.ofHom (algebraMap A G)))] | |
| [IsOpenImmersion (Spec.map (CommRingCat.ofHom (algebraMap B G)))] | |
| (h : Spec.map (CommRingCat.ofHom (algebraMap A G)) ≫ iA = | |
| Spec.map (CommRingCat.ofHom (algebraMap B G)) ≫ iB) : | |
| affineChartFunctionFieldMap A E iA = affineChartFunctionFieldMap B E iB := by | |
| have hr : functionFieldRestriction iA ≫ | |
| functionFieldRestriction (Spec.map (CommRingCat.ofHom (algebraMap A G))) = | |
| functionFieldRestriction iB ≫ | |
| functionFieldRestriction (Spec.map (CommRingCat.ofHom (algebraMap B G))) := by | |
| rw [← functionFieldRestriction_comp, ← functionFieldRestriction_comp] | |
| exact restriction_congr h | |
| apply RingHom.ext | |
| intro z | |
| have hz := congrArg (fun g => affineFunctionFieldEquiv G E (g z)) hr | |
| change affineFunctionFieldEquiv G E | |
| (functionFieldRestriction (Spec.map (CommRingCat.ofHom (algebraMap A G))) | |
| (functionFieldRestriction iA z)) = | |
| affineFunctionFieldEquiv G E | |
| (functionFieldRestriction (Spec.map (CommRingCat.ofHom (algebraMap B G))) | |
| (functionFieldRestriction iB z)) at hz | |
| have hA := RingHom.congr_fun (affineFunctionFieldEquiv_naturality A E G) | |
| (functionFieldRestriction iA z) | |
| have hB := RingHom.congr_fun (affineFunctionFieldEquiv_naturality B E G) | |
| (functionFieldRestriction iB z) | |
| exact hA.symm.trans (hz.trans hB) | |
| end PiExponent.CurveNormalizationModel | |
| end | |
| end OAI | |