| import Mathlib |
| import OAI.AlgebraicGeometry.RelativeTriviality.ProjectiveLine |
| import OAI.AlgebraicGeometry.RelativeTriviality.IsomorphismLocus |
| import OAI.AlgebraicGeometry.RelativeTriviality.CechBaseChange |
| import OAI.AlgebraicGeometry.RelativeTriviality.Frames |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace CampanaPeternellSix.RelativeTriviality |
| open CategoryTheory AlgebraicGeometry Limits |
| universe u |
| variable (R : Type u) [CommRing R] |
| theorem isIso_near_isIso_fiber {F E : (line R).Modules} |
| (hF : ∃ q : F.LocalGeneratorsData, q.IsLocallyFreeData ∧ q.IsFiniteType) |
| (hE : ∃ q : E.LocalGeneratorsData, q.IsLocallyFreeData ∧ q.IsFiniteType) |
| (p : PrimeSpectrum R) (φ : F ⟶ E) |
| [IsIso ((Scheme.Modules.pullback (fiberProjection R p)).map φ)] : |
| ∃ r : R, r ∉ p.asIdeal ∧ |
| IsIso ((Scheme.Modules.pullback (awayProjection R r)).map φ) := by |
| have h : IsIso ((Scheme.Modules.pullback (fiberProjection R p)).map φ) := inferInstance |
| exact @RelativeCech.isIso_near_fiber_proper (.of R) (line R) (toBase R) |
| (line_toBase_universallyClosed R) F E hF hE p φ h |
| end CampanaPeternellSix.RelativeTriviality |
|
|
| namespace CampanaPeternellSix.ActualCech |
| open CategoryTheory AlgebraicGeometry Opposite |
| open RelativeLineCharts RelativeProjectiveBaseChange Scheme.Modules FiberFrame |
| universe u |
| variable {R : Type u} [CommRing R] |
| theorem exists_fiber_trivializing_map (E : (scheme R).Modules) |
| (hE : ∃ q : E.LocalGeneratorsData, q.IsLocallyFreeData ∧ q.IsFiniteType) |
| (p : PrimeSpectrum R) (n : ℕ) |
| (hf : Nonempty ((pullback (baseMap (algebraMap R p.asIdeal.ResidueField) 1)).obj E ≅ |
| SheafOfModules.free (ULift.{u} (Fin n)))) : |
| ∃ φ : SheafOfModules.free (ULift.{u} (Fin n)) ⟶ E, |
| IsIso ((pullback (baseMap (algebraMap R p.asIdeal.ResidueField) 1)).map φ) := by |
| classical |
| let S := p.asIdeal.ResidueField |
| let f := baseMap (algebraMap R S) 1 |
| let E' := (pullback f).obj E |
| let I := ULift.{u} (Fin n) |
| let N := (coefficientSheaf S E').obj.obj (op ⊤) |
| let : Module R N := Module.compHom N (algebraMap R S) |
| let : IsScalarTower R S N := IsScalarTower.of_algebraMap_smul (fun _ _ => rfl) |
| let e := Classical.choice hf |
| let s : I → N := fun i => sectionsTopEquiv E' (E'.freeHomEquiv e.inv i) |
| obtain ⟨d,hd,t,ht⟩ := RelativeCech.clear_finite_sections_denominator p.asIdeal |
| (actualUnit (S := S) E ⊤ ⊤ (by intro z _; trivial)) |
| (global_sections_base_change_at_trivial_fiber E hE p n hf) I s |
| let φ : SheafOfModules.free I ⟶ E := E.freeHomEquiv.symm |
| (fun i => (sectionsTopEquiv E).symm (t i)) |
| have hd' : algebraMap R S d ≠ 0 := by |
| change algebraMap R p.asIdeal.ResidueField d ≠ 0 |
| exact fun h => hd (Ideal.algebraMap_residueField_eq_zero.mp h) |
| let a : Γ(RelativePolynomialEvaluation.space S 1,⊤)ˣ := Units.map (RelativeLineGlobalFunctions.scalars S).toMonoidHom |
| (Units.mk0 (algebraMap R S d) hd') |
| let ψ := e.inv ≫ (scalarIso E' a).hom |
| have hφ : (pullbackFreeIso f).inv ≫ (pullback f).map φ = ψ := by |
| apply E'.freeHomEquiv.injective |
| funext i |
| apply (sectionsTopEquiv E').injective |
| erw [← pullbackFree_top f φ i] |
| change actualUnit (S := S) E ⊤ ⊤ _ |
| (sectionsTopEquiv E (E.freeHomEquiv φ i)) = _ |
| have hti : sectionsTopEquiv E (E.freeHomEquiv φ i) = t i := by |
| have h := congrFun (E.freeHomEquiv.apply_symm_apply |
| (fun j : I => (sectionsTopEquiv E).symm (t j))) i |
| exact (congrArg (sectionsTopEquiv E) h).trans |
| ((sectionsTopEquiv E).apply_symm_apply (t i)) |
| rw [hti, ht] |
| change algebraMap R S d • s i = |
| sectionsTopEquiv E' (E'.freeHomEquiv (e.inv ≫ (scalarIso E' a).hom) i) |
| erw [SheafOfModules.freeHomEquiv_comp_apply, sectionsTopEquiv_naturality] |
| change algebraMap R S d • s i = (scalarEnd E' ↑a).app ⊤ (s i) |
| erw [scalarEnd_app_top] |
| change ((RelativePolynomialEvaluation.space S 1).presheaf.map (𝟙 (op ⊤)) |
| (RelativeLineGlobalFunctions.scalars S (algebraMap R S d))) • |
| (show Γ(E',⊤) from s i) = |
| (RelativeLineGlobalFunctions.scalars S (algebraMap R S d)) • |
| (show Γ(E',⊤) from s i) |
| simp |
|
|
| refine ⟨φ, ?_⟩ |
| have : IsIso ψ := IsIso.comp_isIso' e.isIso_inv (scalarIso E' a).isIso_hom |
| have : IsIso ((pullbackFreeIso f).inv ≫ (pullback f).map φ) := by rw [hφ]; infer_instance |
| let : IsIso (pullbackFreeIso f (I := I)).inv := (pullbackFreeIso f (I := I)).isIso_inv |
| exact IsIso.of_isIso_comp_left (pullbackFreeIso f (I := I)).inv ((pullback f).map φ) |
| end CampanaPeternellSix.ActualCech |
|
|
| namespace CampanaPeternellSix.RelativeTriviality |
| open CategoryTheory AlgebraicGeometry Limits |
| open Scheme.Modules |
| universe u |
| variable (R : Type u) [CommRing R] |
| theorem trivial_near_trivial_fiber (E : (line R).Modules) |
| (hE : ∃ q : E.LocalGeneratorsData, q.IsLocallyFreeData ∧ q.IsFiniteType) |
| (p : PrimeSpectrum R) (n : ℕ) |
| (hf : Nonempty ((Scheme.Modules.pullback (fiberProjection R p)).obj E ≅ |
| SheafOfModules.free (ULift.{u} (Fin n)))) : |
| ∃ r : R, r ∉ p.asIdeal ∧ Nonempty |
| ((Scheme.Modules.pullback (awayProjection R r)).obj E ≅ |
| SheafOfModules.free (ULift.{u} (Fin n))) := by |
| classical |
| let S := p.asIdeal.ResidueField |
| let g := RelativeProjectiveBaseChange.baseMap (algebraMap R S) 1 |
| have hc : IsPullback g (RelativePolynomialEvaluation.toBase S 1) (toBase R) |
| (Spec.map (CommRingCat.ofHom (algebraMap R S))) := |
| RelativeProjectiveBaseChange.baseMap_isPullback _ |
| let e := hc.isoPullback |
| have heg : e.hom ≫ fiberProjection R p = g := hc.isoPullback_hom_fst |
| have hge : e.inv ≫ g = fiberProjection R p := hc.isoPullback_inv_fst |
| have hfg : Nonempty ((Scheme.Modules.pullback g).obj E ≅ |
| SheafOfModules.free (ULift.{u} (Fin n))) := by |
| refine ⟨(pullbackCongr heg.symm).app E ≪≫ |
| (pullbackComp e.hom (fiberProjection R p)).symm.app E ≪≫ |
| (Scheme.Modules.pullback e.hom).mapIso (Classical.choice hf) ≪≫ |
| FiberFrame.pullbackFreeIso e.hom⟩ |
| obtain ⟨φ, hφ⟩ := ActualCech.exists_fiber_trivializing_map E hE p n hfg |
| have : IsIso ((Scheme.Modules.pullback g).map φ) := hφ |
| have hpb : IsIso ((Scheme.Modules.pullback e.inv).map |
| ((Scheme.Modules.pullback g).map φ)) := Functor.map_isIso _ _ |
| have : IsIso ((Scheme.Modules.pullback (e.inv ≫ g)).map φ) := |
| (NatIso.isIso_map_iff (pullbackComp e.inv g) φ).mp hpb |
| have hfφ : IsIso ((Scheme.Modules.pullback (fiberProjection R p)).map φ) := |
| (NatIso.isIso_map_iff (pullbackCongr hge) φ).mp inferInstance |
| let I := ULift.{u} (Fin n) |
| let F : (line R).Modules := SheafOfModules.free I |
| let q := (SheafOfModules.free.generatingSections (R := (line R).ringCatSheaf) I).localGeneratorsData |
| have hq : q.IsLocallyFreeData := inferInstance |
| have ht : q.IsFiniteType := by |
| constructor |
| intro U |
| constructor |
| change Finite I |
| infer_instance |
| obtain ⟨r,hr,hiso⟩ := @isIso_near_isIso_fiber R _ F E ⟨q,hq,ht⟩ hE p φ hfφ |
| refine ⟨r,hr,⟨(@asIso _ _ _ _ ((Scheme.Modules.pullback (awayProjection R r)).map φ) hiso).symm ≪≫ |
| FiberFrame.pullbackFreeIso (awayProjection R r)⟩⟩ |
| end CampanaPeternellSix.RelativeTriviality |
|
|
| end |
|
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| end OAI |
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