| import OAI.NumberTheory.PiExponent.Geometry.CurveStalkGluing |
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| namespace OAI |
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| noncomputable section |
| namespace PiExponent.BirationalOpenIso |
| open AlgebraicGeometry CategoryTheory TopologicalSpace |
| universe u |
| variable {X Y : Scheme.{u}} [IsIntegral X] [IsIntegral Y] |
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| theorem exists_open_iso_of_generic_inverse (f : Y ⟶ X) |
| [LocallyOfFiniteType f] [IsSeparated f] |
| (e : Y.functionField ≃+* X.functionField) |
| (he : Spec.map (CommRingCat.ofHom e.toRingHom) ≫ |
| Y.fromSpecStalk (genericPoint Y) ≫ f = X.fromSpecStalk (genericPoint X)) : |
| ∃ U : X.Opens, genericPoint X ∈ U ∧ IsIso (f ∣_ U) := by |
| let η : Spec X.functionField ⟶ Y := |
| Spec.map (CommRingCat.ofHom e.toRingHom) ≫ Y.fromSpecStalk (genericPoint Y) |
| have hηf : η ≫ f = X.fromSpecStalk (genericPoint X) ≫ 𝟙 X := by |
| simpa only [η, Category.assoc, Category.comp_id] using! he |
| obtain ⟨U, hU, g, hη, hg⟩ := spread_out_of_isGermInjective' (𝟙 X) f η hηf |
| simp only [Category.comp_id] at hg |
| let V : Y.Opens := f ⁻¹ᵁ U |
| have hrange : Set.range g ⊆ Set.range V.ι := by |
| rintro _ ⟨x, rfl⟩ |
| refine ⟨⟨g x, ?_⟩, rfl⟩ |
| change f (g x) ∈ U |
| have hpoint := congrArg (fun k : U.toScheme ⟶ X => k x) hg |
| change f (g x) = x.1 at hpoint |
| rw [hpoint] |
| exact x.2 |
| let s : U.toScheme ⟶ V.toScheme := IsOpenImmersion.lift V.ι g hrange |
| have hs : s ≫ V.ι = g := IsOpenImmersion.lift_fac _ _ _ |
| have hsr : s ≫ (f ∣_ U) = 𝟙 U.toScheme := by |
| apply (cancel_mono U.ι).mp |
| rw [Category.assoc, morphismRestrict_ι, ← Category.assoc, hs, hg, Category.id_comp] |
| let : IsDominant (Y.fromSpecStalk (genericPoint Y)) := |
| CurveStalkGluing.isDominant_fromSpecFunctionField |
| have hηdom : IsDominant η := by |
| let : IsIso (CommRingCat.ofHom e.toRingHom) := |
| (ConcreteCategory.isIso_iff_bijective _).mpr e.bijective |
| dsimp only [η] |
| infer_instance |
| have : IsDominant (U.fromSpecStalkOfMem (genericPoint X) hU ≫ g) := by |
| rw [← hη] |
| exact hηdom |
| have : IsDominant g := IsDominant.of_comp (U.fromSpecStalkOfMem (genericPoint X) hU) g |
| have : IsDominant (s ≫ V.ι) := by rw [hs]; infer_instance |
| have : IsDominant s := IsDominant.of_comp_of_isOpenImmersion s V.ι |
| have hbase : ((f ∣_ U) ≫ s) ≫ (f ∣_ U) = (𝟙 V.toScheme) ≫ (f ∣_ U) := by |
| rw [Category.assoc, hsr, Category.comp_id, Category.id_comp] |
| have hdense : s ≫ ((f ∣_ U) ≫ s) = s ≫ 𝟙 V.toScheme := by |
| rw [← Category.assoc, hsr, Category.id_comp, Category.comp_id] |
| have hrs : (f ∣_ U) ≫ s = 𝟙 V.toScheme := |
| ext_of_isDominant_of_isSeparated (f ∣_ U) hbase s hdense |
| exact ⟨U, hU, ⟨⟨s, hrs, hsr⟩⟩⟩ |
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| end PiExponent.BirationalOpenIso |
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| end |
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| end OAI |
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