openAI-math-dataset / lean /OAI /NumberTheory /PiExponent /Approximation /ClosedPullbackUnitRestriction.lean
| import OAI.NumberTheory.PiExponent.Approximation.ClosedPullbackUnit | |
| import OAI.NumberTheory.PiExponent.Approximation.OpenPullbackUnit | |
| import OAI.NumberTheory.PiExponent.LocalAlgebra.IdealModule | |
| namespace OAI | |
| namespace PiExponentSeshadri.ClosedPullbackUnitRestriction | |
| noncomputable section | |
| open CategoryTheory AlgebraicGeometry TopologicalSpace | |
| variable {X Y : Scheme.{0}} (f : Y ⟶ X) | |
| lemma restrict_unit_isIso_iff (U : X.Opens) (M : X.Modules) : | |
| IsIso ((Scheme.Modules.restrictFunctor U.ι).map | |
| ((Scheme.Modules.pullbackPushforwardAdjunction f).unit.app M)) ↔ | |
| IsIso ((Scheme.Modules.pullbackPushforwardAdjunction (f ∣_ U)).unit.app | |
| (M.restrict U.ι)) := by | |
| let g := (Scheme.Modules.restrictFunctor U.ι).map | |
| ((Scheme.Modules.pullbackPushforwardAdjunction f).unit.app M) | |
| let e := OpenBaseChange.iso f U ((Scheme.Modules.pullback f).obj M) | |
| let k := (Scheme.Modules.pushforward (f ∣_ U)).map | |
| ((OpenBaseChange.leftSquare f U).hom.app M) | |
| calc | |
| IsIso g ↔ IsIso (g ≫ e.hom) := (isIso_comp_right_iff g e.hom).symm | |
| _ ↔ IsIso (((Scheme.Modules.pullbackPushforwardAdjunction (f ∣_ U)).unit.app | |
| (M.restrict U.ι)) ≫ k) := by rw [OpenBaseChange.unit_compatibility] | |
| _ ↔ _ := isIso_comp_right_iff _ k | |
| theorem unit_isIso_of_affine (M : X.Modules) | |
| (h : ∀ U : X.affineOpens, | |
| IsIso ((Scheme.Modules.pullbackPushforwardAdjunction (f ∣_ U.1)).unit.app | |
| (M.restrict U.1.ι))) : | |
| IsIso ((Scheme.Modules.pullbackPushforwardAdjunction f).unit.app M) := by | |
| apply IdealModule.isIso_of_affine_app | |
| intro U | |
| have hr := (restrict_unit_isIso_iff f U.1 M).mpr (h U) | |
| have ht := Scheme.Modules.Hom.isIso_iff_isIso_app.mp hr ⊤ | |
| change IsIso (((Scheme.Modules.pullbackPushforwardAdjunction f).unit.app M).app | |
| (U.1.ι ''ᵁ ⊤)) at ht | |
| have he : U.1.ι ''ᵁ ⊤ = U.1 := | |
| (Scheme.Hom.image_top_eq_opensRange U.1.ι).trans (Scheme.Opens.opensRange_ι U.1) | |
| rw [he] at ht | |
| exact ht | |
| lemma morphismRestrict_surjective (hf : Function.Surjective f) (U : X.Opens) : | |
| Function.Surjective (f ∣_ U) := by | |
| intro x | |
| obtain ⟨y, hy⟩ := hf x.1 | |
| refine ⟨⟨y, ?_⟩, ?_⟩ | |
| · change f y ∈ U | |
| simpa only [hy] using x.2 | |
| · apply Subtype.ext | |
| exact (morphismRestrict_base_coe f U _).trans hy | |
| theorem unit_isIso_of_affine_pushforward [IsClosedImmersion f] | |
| (hf : Function.Surjective f) (M : X.Modules) | |
| (h : ∀ U : X.affineOpens, ∃ N : (f ⁻¹ᵁ U.1).toScheme.Modules, | |
| Nonempty (M.restrict U.1.ι ≅ (Scheme.Modules.pushforward (f ∣_ U.1)).obj N)) : | |
| IsIso ((Scheme.Modules.pullbackPushforwardAdjunction f).unit.app M) := by | |
| apply unit_isIso_of_affine f M | |
| intro U | |
| obtain ⟨N, ⟨e⟩⟩ := h U | |
| exact ClosedPullbackUnit.unit_isIso_of_pushforward (f ∣_ U.1) | |
| (morphismRestrict_surjective f hf U.1) e | |
| end | |
| end PiExponentSeshadri.ClosedPullbackUnitRestriction | |
| end OAI | |