openAI-math-dataset / lean /OAI /NumberTheory /PiExponent /Geometry /CanonicalRecoveryProperties.lean
| import OAI.NumberTheory.PiExponent.Ampleness.ExceptionalRecoveryMap | |
| import OAI.NumberTheory.PiExponent.LocalAlgebra.IdealPullbackOpenSquare | |
| namespace OAI | |
| noncomputable section | |
| namespace PiExponent.CanonicalRecoveryProperties | |
| open AlgebraicGeometry CategoryTheory TopologicalSpace | |
| open PiExponentSeshadri.Geometry | |
| open GeometrySupport | |
| def ComapBijective {X Y : Scheme.{0}} (f : Y ⟶ X) [QuasiCompact f] | |
| (I : X.IdealSheafData) (U : X.Opens) : Prop := | |
| Function.Bijective ((IdealPullbackMap.comap f I).app U) | |
| def OrdinaryBijective {X Y : Scheme.{0}} (f : Y ⟶ X) [QuasiCompact f] | |
| (I : X.IdealSheafData) (E : LineBundle Y) | |
| (ι : E.sheaf ⟶ structureSheaf Y) (hE : PresentsPullbackIdeal I f E ι) | |
| (n : ℕ) (U : X.Opens) : Prop := | |
| Function.Bijective ((ExceptionalRecoveryMap.ordinaryMap f I E ι hE n).app U) | |
| theorem ordinary_iff_comap {X Y : Scheme.{0}} (f : Y ⟶ X) [QuasiCompact f] | |
| (I : X.IdealSheafData) (E : LineBundle Y) | |
| (ι : E.sheaf ⟶ structureSheaf Y) (hE : PresentsPullbackIdeal I f E ι) | |
| (n : ℕ) (U : X.Opens) : | |
| OrdinaryBijective f I E ι hE n U ↔ ComapBijective f (I^n) U := by | |
| have hb := ConcreteCategory.bijective_of_isIso | |
| ((PresentedIdealIso.powerIso I f E ι hE n).inv.app (f ⁻¹ᵁ U)) | |
| change Function.Bijective | |
| (((PresentedIdealIso.powerIso I f E ι hE n).inv.app (f ⁻¹ᵁ U)) ∘ | |
| ((IdealPullbackMap.comap f (I^n)).app U)) ↔ _ | |
| exact hb.of_comp_iff' _ | |
| theorem comap_open_square {X Y X' Y' : Scheme.{0}} | |
| (f : Y ⟶ X) [QuasiCompact f] | |
| (j : X' ⟶ X) [IsOpenImmersion j] | |
| (k : Y' ⟶ Y) [IsOpenImmersion k] | |
| (f' : Y' ⟶ X') [QuasiCompact f'] (h : k ≫ f = f' ≫ j) | |
| (I : X.IdealSheafData) (U : X.Opens) (U' : X'.Opens) (V' : Y'.Opens) | |
| (hj : j ''ᵁ U' = U) (hk : k ''ᵁ V' = f ⁻¹ᵁ U) | |
| (hv : V' = f' ⁻¹ᵁ U') : | |
| ComapBijective f I U ↔ ComapBijective f' (I.comap j) U' := | |
| IdealSectionRestriction.comap_app_bijective_iff_open_square f j k f' h I U U' V' hj hk hv | |
| end PiExponent.CanonicalRecoveryProperties | |
| end | |
| end OAI | |