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a72760f | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 | import OAI.NumberTheory.PiExponent.Approximation.SectionPullback
namespace OAI
namespace PiExponentSeshadri.Geometry
noncomputable section
open AlgebraicGeometry CategoryTheory CategoryTheory.Limits TopologicalSpace
open PiExponentSeshadri.Frames
variable {X Y : Scheme.{0}}
lemma endValue_conjugation (b : O Y ≅ O Y) (a : O Y ⟶ O Y) :
endValue (b.hom ≫ a ≫ b.inv) = endValue a := by
rw [endValue_comp, endValue_comp]
have hb : endValue b.hom * endValue b.inv = 1 := by
rw [← endValue_comp, b.hom_inv_id, endValue_id]
calc
endValue b.hom * (endValue a * endValue b.inv) =
endValue a * (endValue b.hom * endValue b.inv) := by ring
_ = endValue a := by rw [hb, mul_one]
lemma exists_frame_natIso (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) (s : O X ⟶ M) :
∃ eF : F.obj M ≅ O Y,
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, ?_⟩
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 _
def pullbackRestrictNatIso (f : Y ⟶ X) (U : X.Opens) :
Scheme.Modules.pullback f ⋙ Scheme.Modules.restrictFunctor (f ⁻¹ᵁ U).ι ≅
Scheme.Modules.restrictFunctor U.ι ⋙ Scheme.Modules.pullback (f ∣_ U) :=
Functor.isoWhiskerLeft (Scheme.Modules.pullback f)
(Scheme.Modules.restrictFunctorIsoPullback (f ⁻¹ᵁ U).ι) ≪≫
Scheme.Modules.pullbackComp (f ⁻¹ᵁ U).ι f ≪≫
Scheme.Modules.pullbackCongr (morphismRestrict_ι f U).symm ≪≫
(Scheme.Modules.pullbackComp (f ∣_ U) U.ι).symm ≪≫
Functor.isoWhiskerRight (Scheme.Modules.restrictFunctorIsoPullback U.ι).symm
(Scheme.Modules.pullback (f ∣_ U))
theorem exists_restricted_pullback_frame (f : Y ⟶ X) (U : X.Opens)
{M : X.Modules} (e : M.restrict U.ι ≅ O U.toScheme) (s : O X ⟶ M) :
∃ eF : ((Scheme.Modules.pullback f).obj M).restrict (f ⁻¹ᵁ U).ι ≅
O (f ⁻¹ᵁ U).toScheme,
coefficient eF (restrictSection (f ⁻¹ᵁ U).ι (pullbackSection f s)) =
(f ∣_ U).appTop (coefficient e (restrictSection U.ι s)) := by
let F : X.Modules ⥤ (f ⁻¹ᵁ U).toScheme.Modules :=
Scheme.Modules.pullback f ⋙ Scheme.Modules.restrictFunctor (f ⁻¹ᵁ U).ι
let G : X.Modules ⥤ (f ⁻¹ᵁ U).toScheme.Modules :=
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 F G (pullbackRestrictNatIso f U) uF uG
(pullbackFrame (f ∣_ U) e) s
refine ⟨eF, ?_⟩
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
let P : X.Modules ⥤ Y.Modules := Scheme.Modules.pullback f
let R : Y.Modules ⥤ (f ⁻¹ᵁ U).toScheme.Modules :=
Scheme.Modules.restrictFunctor (f ⁻¹ᵁ U).ι
let S : X.Modules ⥤ U.toScheme.Modules := Scheme.Modules.restrictFunctor U.ι
let Q : U.toScheme.Modules ⥤ (f ⁻¹ᵁ U).toScheme.Modules :=
Scheme.Modules.pullback (f ∣_ U)
let p : P.obj (O X) ≅ O Y := pullbackUnitIso f
let r : R.obj (O Y) ≅ O (f ⁻¹ᵁ U).toScheme :=
Scheme.Modules.restrictUnitIso (f ⁻¹ᵁ U).ι
let a : S.obj (O X) ≅ O U.toScheme := Scheme.Modules.restrictUnitIso U.ι
let q : Q.obj (O U.toScheme) ≅ O (f ⁻¹ᵁ U).toScheme := pullbackUnitIso (f ∣_ U)
have hl : restrictSection (f ⁻¹ᵁ U).ι (pullbackSection f s) = uF.inv ≫ F.map s := by
simp only [restrictSection, pullbackSection]
change r.inv ≫ R.map (p.inv ≫ P.map s) = (r.inv ≫ R.map p.inv) ≫ R.map (P.map s)
exact (congrArg (fun k => r.inv ≫ k) (R.map_comp p.inv (P.map s))).trans
(Category.assoc r.inv (R.map p.inv) (R.map (P.map s))).symm
have hr : pullbackSection (f ∣_ U) (restrictSection U.ι s) = uG.inv ≫ G.map s := by
change q.inv ≫ Q.map (a.inv ≫ S.map s) = (q.inv ≫ Q.map a.inv) ≫ Q.map (S.map s)
exact (congrArg (fun k => q.inv ≫ k) (Q.map_comp a.inv (S.map s))).trans
(Category.assoc q.inv (Q.map a.inv) (Q.map (S.map s))).symm
exact (congrArg (coefficient eF) hl).trans
(he.trans (congrArg (coefficient (pullbackFrame (f ∣_ U) e)) hr).symm)
rw [he', coefficient_pullback]
end
end PiExponentSeshadri.Geometry
end OAI
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