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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 | import OAI.NumberTheory.PiExponent.Polynomials.FrameCoefficients
namespace OAI
namespace PiExponentSeshadri.Geometry
noncomputable section
open CategoryTheory AlgebraicGeometry TopologicalSpace MonoidalCategory
open scoped AlgebraicGeometry
open PiExponentSeshadri.Frames
variable {X : Scheme}
lemma tensor_unit_end {C : Type*} [Category C] [MonoidalCategory C]
(f g : 𝟙_ C ⟶ 𝟙_ C) :
(f ⊗ₘ g) ≫ (λ_ (𝟙_ C)).hom = (λ_ (𝟙_ C)).hom ≫ (f ≫ g) := by
rw [MonoidalCategory.tensorHom_def, Category.assoc, leftUnitor_naturality, ← Category.assoc,
unitors_equal, rightUnitor_naturality, Category.assoc]
lemma moduleTensorUnit_scalar (f g : structureSheaf X ⟶ structureSheaf X) :
moduleTensorMap f g ≫ (moduleTensorUnit (structureSheaf X)).hom =
(moduleTensorUnit (structureSheaf X)).hom ≫ (f ≫ g) := by
let : MonoidalCategory (PresheafOfModules X.ringCatSheaf.obj) :=
PresheafOfModulesOfCommRing.monoidalCategory (R := X.presheaf)
let F : PresheafOfModules X.ringCatSheaf.obj ⥤ X.Modules :=
PresheafOfModules.sheafification (𝟙 X.ringCatSheaf.obj)
let G : X.Modules ⥤ PresheafOfModules X.ringCatSheaf.obj :=
SheafOfModules.forget X.ringCatSheaf ⋙
PresheafOfModules.restrictScalars (𝟙 X.ringCatSheaf.obj)
let ε : G ⋙ F ⟶ 𝟭 X.Modules :=
(PresheafOfModules.sheafificationAdjunction (𝟙 X.ringCatSheaf.obj)).counit
have h : (f.val ⊗ₘ g.val) ≫ (λ_ (structureSheaf X).val).hom =
(λ_ (structureSheaf X).val).hom ≫ (f ≫ g).val := by
let f₀ : 𝟙_ (PresheafOfModules X.ringCatSheaf.obj) ⟶
𝟙_ (PresheafOfModules X.ringCatSheaf.obj) := f.val
let g₀ : 𝟙_ (PresheafOfModules X.ringCatSheaf.obj) ⟶
𝟙_ (PresheafOfModules X.ringCatSheaf.obj) := g.val
exact tensor_unit_end f₀ g₀
change F.map (f.val ⊗ₘ g.val) ≫ (F.map (λ_ (structureSheaf X).val).hom ≫ ε.app (structureSheaf X)) =
(F.map (λ_ (structureSheaf X).val).hom ≫ ε.app (structureSheaf X)) ≫ (f ≫ g)
erw [← Category.assoc, ← F.map_comp, h, F.map_comp, Category.assoc, Category.assoc]
congr 1
exact ε.naturality (f ≫ g)
def endPower (f : structureSheaf X ⟶ structureSheaf X) : ℕ → (structureSheaf X ⟶ structureSheaf X)
| 0 => 𝟙 _
| n + 1 => f ≫ endPower f n
lemma endValue_endPower (f : structureSheaf X ⟶ structureSheaf X) (n : ℕ) :
endValue (endPower f n) = endValue f ^ n := by
induction n with
| zero =>
change (1 : Γ(X, ⊤)) = endValue f ^ 0
exact (pow_zero _).symm
| succ n hn =>
calc
endValue (endPower f (n + 1)) = endValue f * endValue (endPower f n) :=
endValue_comp f (endPower f n)
_ = endValue f * endValue f ^ n := congrArg (endValue f * ·) hn
_ = _ := (pow_succ' _ _).symm
lemma moduleTensorUnit_natural {M N : X.Modules} (g : M ⟶ N) :
moduleTensorMap (𝟙 (structureSheaf X)) g ≫ (moduleTensorUnit N).hom =
(moduleTensorUnit M).hom ≫ g := by
let : MonoidalCategory (PresheafOfModules X.ringCatSheaf.obj) :=
PresheafOfModulesOfCommRing.monoidalCategory (R := X.presheaf)
let F : PresheafOfModules X.ringCatSheaf.obj ⥤ X.Modules :=
PresheafOfModules.sheafification (𝟙 X.ringCatSheaf.obj)
let G : X.Modules ⥤ PresheafOfModules X.ringCatSheaf.obj :=
SheafOfModules.forget X.ringCatSheaf ⋙
PresheafOfModules.restrictScalars (𝟙 X.ringCatSheaf.obj)
let ε : G ⋙ F ⟶ 𝟭 X.Modules :=
(PresheafOfModules.sheafificationAdjunction (𝟙 X.ringCatSheaf.obj)).counit
change F.map ((𝟙 (structureSheaf X).val) ⊗ₘ g.val) ≫
(F.map (λ_ N.val).hom ≫ ε.app N) =
(F.map (λ_ M.val).hom ≫ ε.app M) ≫ g
have h : ((𝟙 (structureSheaf X).val) ⊗ₘ g.val) ≫ (λ_ N.val).hom =
(λ_ M.val).hom ≫ g.val := by
rw [id_tensorHom]
exact leftUnitor_naturality g.val
have hF : F.map (𝟙 (structureSheaf X).val ⊗ₘ g.val) ≫ F.map (λ_ N.val).hom =
F.map (λ_ M.val).hom ≫ F.map g.val :=
(F.map_comp _ _).symm.trans ((congrArg F.map h).trans (F.map_comp _ _))
refine (Category.assoc _ _ _).symm.trans ?_
refine (congrArg (· ≫ ε.app N) hF).trans ?_
refine (Category.assoc _ _ _).trans ?_
refine (congrArg (F.map (λ_ M.val).hom ≫ ·) (ε.naturality g)).trans ?_
exact (Category.assoc _ _ _).symm
lemma modulePowMap_unitPower (f : structureSheaf X ⟶ structureSheaf X) (n : ℕ) :
modulePowMap f n ≫ (unitPowerIso n).hom =
(unitPowerIso n).hom ≫ endPower f n := by
induction n with
| zero =>
change 𝟙 (structureSheaf X) ≫ 𝟙 (structureSheaf X) = 𝟙 (structureSheaf X) ≫ 𝟙 (structureSheaf X)
rfl
| succ n hn =>
change moduleTensorMap f (modulePowMap f n) ≫
((moduleTensorUnit _).hom ≫ (unitPowerIso n).hom) =
((moduleTensorUnit _).hom ≫ (unitPowerIso n).hom) ≫ (f ≫ endPower f n)
calc
_ = moduleTensorMap f (modulePowMap f n) ≫
moduleTensorMap (𝟙 (structureSheaf X)) (unitPowerIso n).hom ≫
(moduleTensorUnit (structureSheaf X)).hom := by
rw [moduleTensorUnit_natural]
_ = moduleTensorMap f ((unitPowerIso n).hom ≫ endPower f n) ≫
(moduleTensorUnit (structureSheaf X)).hom := by
rw [← Category.assoc, ← moduleTensorMap_comp, Category.comp_id, hn]
_ = moduleTensorMap (𝟙 (structureSheaf X)) (unitPowerIso n).hom ≫
moduleTensorMap f (endPower f n) ≫ (moduleTensorUnit (structureSheaf X)).hom := by
rw [← Category.assoc, ← moduleTensorMap_comp, Category.id_comp]
_ = (moduleTensorUnit (modulePow X (structureSheaf X) n)).hom ≫
(unitPowerIso n).hom ≫ (f ≫ endPower f n) := by
rw [moduleTensorUnit_scalar, ← Category.assoc, moduleTensorUnit_natural,
Category.assoc]
_ = _ := by simp only [Category.assoc]
def globalPowerFrame {M : X.Modules} (e : M ≅ structureSheaf X) (n : ℕ) :
modulePow X M n ≅ structureSheaf X :=
(modulePowFunctor n).mapIso e ≪≫ unitPowerIso n
lemma powerSection_coefficient {M : X.Modules} (e : M ≅ structureSheaf X)
(s : structureSheaf X ⟶ M) (n : ℕ) :
coefficient (globalPowerFrame e n) (powerSection s n) = coefficient e s ^ n := by
change endValue ((unitPowerIso n).inv ≫ modulePowMap s n ≫
(modulePowMap e.hom n ≫ (unitPowerIso n).hom)) = _
erw [← Category.assoc (modulePowMap s n), ← modulePowMap_comp,
modulePowMap_unitPower, Iso.inv_hom_id_assoc, endValue_endPower]
rfl
lemma frameChange_globalPowerFrame {M : X.Modules}
(e f : M ≅ structureSheaf X) (n : ℕ) :
(frameChange (globalPowerFrame e n) (globalPowerFrame f n) : Γ(X,⊤)) =
(frameChange e f : Γ(X,⊤)) ^ n := by
have h := powerSection_coefficient f e.inv n
exact h
end
end PiExponentSeshadri.Geometry
end OAI
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