| import Mathlib |
| import OAI.AlgebraicGeometry.Seshadri.Divisors.SectionNaturality |
| import OAI.AlgebraicGeometry.Seshadri.Sheaves.FramedPullback |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| |
| section |
|
|
| namespace MaximalSeshadri.Geometry |
| noncomputable section |
| open AlgebraicGeometry CategoryTheory CategoryTheory.Limits TopologicalSpace |
| open MaximalSeshadri.Frames |
|
|
| variable {X Y : Scheme.{0}} |
|
|
| lemma exists_frame_natIso_all (F G : X.Modules ⥤ Y.Modules) (v : F ≅ G) |
| (uF : F.obj (O X) ≅ O Y) (uG : G.obj (O X) ≅ O Y) |
| {M : X.Modules} (e : G.obj M ≅ O Y) : |
| ∃ eF : F.obj M ≅ O Y, ∀ s : O X ⟶ M, |
| coefficient eF (uF.inv ≫ F.map s) = coefficient e (uG.inv ≫ G.map s) := by |
| let b : O Y ≅ O Y := uF.symm ≪≫ v.app (O X) ≪≫ uG |
| refine ⟨v.app M ≪≫ e ≪≫ b.symm, ?_⟩ |
| intro s |
| have h : (uF.inv ≫ F.map s) ≫ (v.app M).hom = |
| b.hom ≫ (uG.inv ≫ G.map s) := by |
| simp only [b, Iso.trans_hom, Iso.symm_hom, Category.assoc] |
| simp only [Iso.hom_inv_id_assoc] |
| exact congrArg (fun q => uF.inv ≫ q) (v.hom.naturality s) |
| change endValue (((uF.inv ≫ F.map s) ≫ (v.app M).hom) ≫ e.hom ≫ b.inv) = _ |
| rw [h] |
| change endValue (b.hom ≫ (uG.inv ≫ G.map s ≫ e.hom) ≫ b.inv) = _ |
| exact endValue_conjugation b _ |
|
|
| theorem exists_restricted_pullback_frame_all (f : Y ⟶ X) (U : X.Opens) |
| {M : X.Modules} (e : M.restrict U.ι ≅ O U.toScheme) : |
| ∃ eF : ((Scheme.Modules.pullback f).obj M).restrict (f ⁻¹ᵁ U).ι ≅ |
| O (f ⁻¹ᵁ U).toScheme, ∀ s : O X ⟶ M, |
| coefficient eF (restrictSection (f ⁻¹ᵁ U).ι (pullbackSection f s)) = |
| (f ∣_ U).appTop (coefficient e (restrictSection U.ι s)) := by |
| let F := Scheme.Modules.pullback f ⋙ Scheme.Modules.restrictFunctor (f ⁻¹ᵁ U).ι |
| let G := Scheme.Modules.restrictFunctor U.ι ⋙ Scheme.Modules.pullback (f ∣_ U) |
| let uF : F.obj (O X) ≅ O (f ⁻¹ᵁ U).toScheme := |
| (Scheme.Modules.restrictFunctor (f ⁻¹ᵁ U).ι).mapIso (pullbackUnitIso f) ≪≫ |
| Scheme.Modules.restrictUnitIso (f ⁻¹ᵁ U).ι |
| let uG : G.obj (O X) ≅ O (f ⁻¹ᵁ U).toScheme := |
| (Scheme.Modules.pullback (f ∣_ U)).mapIso (Scheme.Modules.restrictUnitIso U.ι) ≪≫ |
| pullbackUnitIso (f ∣_ U) |
| obtain ⟨eF,he⟩ := exists_frame_natIso_all F G (pullbackRestrictNatIso f U) uF uG |
| (pullbackFrame (f ∣_ U) e) |
| refine ⟨eF, ?_⟩ |
| intro s |
| have he' : coefficient eF (restrictSection (f ⁻¹ᵁ U).ι (pullbackSection f s)) = |
| coefficient (pullbackFrame (f ∣_ U) e) |
| (pullbackSection (f ∣_ U) (restrictSection U.ι s)) := by |
| have restrict_pullback : restrictSection (f ⁻¹ᵁ U).ι (pullbackSection f s) = |
| uF.inv ≫ F.map s := by |
| exact (congrArg (fun sectionMap => |
| (Scheme.Modules.restrictUnitIso (f ⁻¹ᵁ U).ι).inv ≫ sectionMap) |
| ((Scheme.Modules.restrictFunctor (f ⁻¹ᵁ U).ι).map_comp |
| (pullbackUnitIso f).inv ((Scheme.Modules.pullback f).map s))).trans |
| (Category.assoc _ _ _).symm |
| have pullback_restrict : pullbackSection (f ∣_ U) (restrictSection U.ι s) = |
| uG.inv ≫ G.map s := by |
| exact (congrArg (fun sectionMap => (pullbackUnitIso (f ∣_ U)).inv ≫ sectionMap) |
| ((Scheme.Modules.pullback (f ∣_ U)).map_comp |
| (Scheme.Modules.restrictUnitIso U.ι).inv |
| ((Scheme.Modules.restrictFunctor U.ι).map s))).trans |
| (Category.assoc _ _ _).symm |
| exact (congrArg (coefficient eF) restrict_pullback).trans |
| ((he s).trans (congrArg (coefficient (pullbackFrame (f ∣_ U) e)) |
| pullback_restrict).symm) |
| rw [he', coefficient_pullback] |
|
|
| end |
| end MaximalSeshadri.Geometry |
|
|
| namespace MaximalSeshadri.Projective |
| noncomputable section |
| open AlgebraicGeometry CategoryTheory TopologicalSpace |
| open MaximalSeshadri.Frames MaximalSeshadri.Geometry |
| attribute [local instance] MvPolynomial.gradedAlgebra |
|
|
| variable {K σ : Type} [CommRing K] {X Y : Scheme.{0}} |
|
|
| lemma pullback_isoOpen_le {M : X.Modules} (s : O X ⟶ M) (f : Y ⟶ X) : |
| f ⁻¹ᵁ SectionOpens.isoOpen s ≤ SectionOpens.isoOpen (pullbackSection f s) := by |
| let U := SectionOpens.isoOpen s |
| obtain ⟨e,he⟩ := exists_restricted_pullback_frame_all f U (sectionFrame s) |
| have hn : coefficient e (restrictSection (f ⁻¹ᵁ U).ι (pullbackSection f s)) = 1 := by |
| rw [he,sectionFrame_normalized,map_one] |
| have hp : (f ⁻¹ᵁ U).ι ⁻¹ᵁ SectionOpens.isoOpen (pullbackSection f s) = ⊤ := by |
| rw [preimage_isoOpen (pullbackSection f s) (f ⁻¹ᵁ U).ι e, hn] |
| exact (isUnit_iff_basicOpen_top _).mp isUnit_one |
| intro x hx |
| have h : (⟨x,hx⟩ : (f ⁻¹ᵁ U).toScheme) ∈ |
| (f ⁻¹ᵁ U).ι ⁻¹ᵁ SectionOpens.isoOpen (pullbackSection f s) := by rw [hp]; trivial |
| exact h |
| |
| lemma pullback_sections_cover {M : X.Modules} (s : σ → (O X ⟶ M)) |
| (hs : (⨆ i, SectionOpens.isoOpen (s i)) = ⊤) (f : Y ⟶ X) : |
| (⨆ i, SectionOpens.isoOpen (pullbackSection f (s i))) = ⊤ := by |
| apply top_le_iff.mp |
| rw [← Scheme.Hom.preimage_top f, ← hs, Scheme.Hom.preimage_iSup] |
| exact iSup_mono fun i => pullback_isoOpen_le (s i) f |
|
|
| theorem sectionsMorphism_pullback {M : X.Modules} (k : K →+* Γ(X,⊤)) |
| (s : σ → (O X ⟶ M)) (hs : (⨆ i, SectionOpens.isoOpen (s i)) = ⊤) |
| (f : Y ⟶ X) : |
| f ≫ sectionsMorphism k s hs = |
| sectionsMorphism (f.appTop.hom.comp k) (fun i => pullbackSection f (s i)) |
| (pullback_sections_cover s hs f) := by |
| let t (i : σ) := pullbackSection f (s i) |
| let ht := pullback_sections_cover s hs f |
| have hc : (⨆ i, f ⁻¹ᵁ SectionOpens.isoOpen (s i)) = ⊤ := by |
| rw [← Scheme.Hom.preimage_iSup,hs,Scheme.Hom.preimage_top] |
| apply (Y.openCoverOfIsOpenCover (fun i => f ⁻¹ᵁ SectionOpens.isoOpen (s i)) hc).hom_ext |
| intro i |
| change σ at i |
| let U := SectionOpens.isoOpen (s i) |
| obtain ⟨e,he⟩ := exists_restricted_pullback_frame_all f U (sectionFrame (s i)) |
| have hi : coefficient e (restrictSection (f ⁻¹ᵁ U).ι (t i)) = 1 := by |
| rw [he,sectionFrame_normalized,map_one] |
| change (f ⁻¹ᵁ U).ι ≫ (f ≫ sectionsMorphism k s hs) = |
| (f ⁻¹ᵁ U).ι ≫ sectionsMorphism (f.appTop.hom.comp k) t ht |
| rw [← Category.assoc, ← morphismRestrict_ι f U, Category.assoc, |
| sectionsMorphism_local, coordinatesMap_natural, |
| sectionsMorphism_frame _ t ht (f ⁻¹ᵁ U).ι e i hi] |
| congr 1 |
| · rw [← RingHom.comp_assoc, ← CommRingCat.hom_comp, ← Scheme.Hom.comp_appTop, |
| morphismRestrict_ι f U, Scheme.Hom.comp_appTop] |
| rfl |
| · exact (funext fun j => he (s j)).symm |
| end |
| end MaximalSeshadri.Projective |
| end |
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|
|
| end |
| end |
|
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| end OAI |
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