openAI-math-dataset / lean /OAI /NumberTheory /PiExponent /Approximation /CanonicalAffineRecovery.lean
| import OAI.NumberTheory.PiExponent.Geometry.CanonicalRecoveryProperties | |
| import OAI.NumberTheory.PiExponent.LocalAlgebra.AffineIdealSections | |
| import OAI.NumberTheory.PiExponent.LocalAlgebra.IdealSectionRestriction | |
| import OAI.NumberTheory.PiExponent.LocalAlgebra.PresentedIdealIso | |
| namespace OAI | |
| noncomputable section | |
| namespace PiExponent.CanonicalAffineRecovery | |
| open AlgebraicGeometry CategoryTheory TopologicalSpace Opposite | |
| open PiExponentSeshadri.Geometry PiExponentSeshadri.IdealModule PiExponentSeshadri.IdealPullback | |
| open GeometrySupport | |
| private theorem bijective_iff_of_square {A B C D : Type*} | |
| (p : A → B) (q : C → D) (a : A → C) (b : B → D) | |
| (ha : Function.Bijective a) (hb : Function.Bijective b) | |
| (h : ∀ x, b (p x) = q (a x)) : Function.Bijective p ↔ Function.Bijective q := by | |
| have he : b ∘ p = q ∘ a := funext h | |
| rw [← hb.of_comp_iff' p, he] | |
| exact Function.Bijective.of_comp_iff q ha | |
| theorem bijective_iff_comap {R : Type} [CommRing R] {Y : Scheme.{0}} | |
| (I : Ideal R) (f : Y ⟶ Spec (CommRingCat.of R)) [QuasiCompact f] | |
| (L : LineBundle Y) (ι : L.sheaf ⟶ structureSheaf Y) | |
| (hL : PresentsPullbackIdeal (specIdeal I) f L ι) | |
| (r : I → Γ(L.sheaf,⊤)) | |
| (hr : ∀ a : I, ι.app ⊤ (r a) = | |
| f.appTop ((Scheme.ΓSpecIso (CommRingCat.of R)).inv a.val)) : | |
| Function.Bijective r ↔ | |
| Function.Bijective ((IdealPullbackMap.comap f (specIdeal I)).app ⊤) := by | |
| let t := PresentedIdealIso.toIdealModule (specIdeal I) f L ι hL | |
| have ht : Function.Bijective (t.app ⊤) := ConcreteCategory.bijective_of_isIso _ | |
| have hdiag (a : I) : | |
| t.app ⊤ (r a) = | |
| IdealSectionRestriction.pullback f (specIdeal I) ⊤ ⊤ (by simp) | |
| ((AffineIdealSections.equiv I).symm a) := by | |
| apply IdealSectionRestriction.inclusion_injective ((specIdeal I).comap f) ⊤ | |
| have h₁ := congrArg (fun g => g.app ⊤ (r a)) | |
| (PresentedIdealIso.toIdealModule_inclusion (specIdeal I) f L ι hL) | |
| have h₂ := IdealSectionRestriction.pullback_inclusion f (specIdeal I) ⊤ ⊤ | |
| (by simp) ((AffineIdealSections.equiv I).symm a) | |
| have htop : f.appLE ⊤ ⊤ (by simp) = f.appTop := f.appLE_eq_app | |
| rw [htop] at h₂ | |
| have h₃ := AffineIdealSections.equiv_symm_inclusion I a | |
| exact (h₁.trans (hr a)).trans | |
| ((congrArg f.appTop h₃).symm.trans h₂.symm) | |
| exact (bijective_iff_of_square r | |
| (IdealSectionRestriction.pullback f (specIdeal I) ⊤ ⊤ (by simp)) | |
| (AffineIdealSections.equiv I).symm (t.app ⊤) | |
| (AffineIdealSections.equiv I).symm.bijective ht hdiag).trans | |
| (IdealSectionRestriction.pullback_bijective_iff_comap f (specIdeal I) ⊤ ⊤ | |
| (by simp) (by simp)) | |
| theorem power_bijective_iff_comap {R : Type} [CommRing R] {Y : Scheme.{0}} | |
| (I : Ideal R) (f : Y ⟶ Spec (CommRingCat.of R)) [QuasiCompact f] | |
| (L : LineBundle Y) (ι : L.sheaf ⟶ structureSheaf Y) | |
| (hL : PresentsPullbackIdeal (specIdeal I) f L ι) (n : ℕ) | |
| (r : ↥(I^n : Ideal R) → Γ((L.pow n).sheaf,⊤)) | |
| (hr : ∀ a : ↥(I^n : Ideal R), (idealPowerInclusion L ι n).app ⊤ (r a) = | |
| f.appTop ((Scheme.ΓSpecIso (CommRingCat.of R)).inv a.val)) : | |
| Function.Bijective r ↔ CanonicalRecoveryProperties.ComapBijective f (specIdeal (I^n)) ⊤ := by | |
| have hp : PresentsPullbackIdeal (specIdeal (I^n)) f (L.pow n) (idealPowerInclusion L ι n) := by | |
| rw [specIdeal_pow] | |
| exact PresentedIdealIso.idealPower_presents (specIdeal I) f L ι hL n | |
| exact bijective_iff_comap (I^n) f (L.pow n) (idealPowerInclusion L ι n) hp r hr | |
| end PiExponent.CanonicalAffineRecovery | |
| end | |
| end OAI | |