| import OAI.NumberTheory.PiExponent.Polynomials.FrameCoefficients |
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| namespace OAI |
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| namespace PiExponentSeshadri.Geometry |
| noncomputable section |
| open CategoryTheory AlgebraicGeometry TopologicalSpace MonoidalCategory |
| open scoped AlgebraicGeometry |
| open PiExponentSeshadri.Frames |
| variable {X : Scheme} |
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| 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] |
|
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| 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) |
|
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| def endPower (f : structureSheaf X ⟶ structureSheaf X) : ℕ → (structureSheaf X ⟶ structureSheaf X) |
| | 0 => 𝟙 _ |
| | n + 1 => f ≫ endPower f n |
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| 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 |
|
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| 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] |
|
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| def globalPowerFrame {M : X.Modules} (e : M ≅ structureSheaf X) (n : ℕ) : |
| modulePow X M n ≅ structureSheaf X := |
| (modulePowFunctor n).mapIso e ≪≫ unitPowerIso n |
|
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| 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 |
|
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| 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 |
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| end OAI |
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