| import OAI.AlgebraicGeometry.CharacterVarieties.Foundation.Flags |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace IntegralCharacterVarieties.NamedBandGrades |
| open scoped Classical |
| variable {R V α β : Type*} [CommRing R] [AddCommGroup V] [Module R V] |
|
|
| def gradeCoordinates (a : α → ℕ) : |
| (α → R) ≃ₗ[R] ((k : ℕ) → {i // a i=k} → R) where |
| toFun v k i := v i.val |
| invFun w i := w (a i) ⟨i,rfl⟩ |
| left_inv w := rfl |
| right_inv w := by |
| funext k i |
| rcases i with ⟨i,hi⟩ |
| subst k |
| rfl |
| map_add' _ _ := rfl |
| map_smul' _ _ := rfl |
|
|
| def sumGrades (a : α → ℕ) (b : β → ℕ) |
| (q : (k : ℕ) → ({i // a i=k} → R) ≃ₗ[R] ({i // b i=k} → R)) : |
| (α → R) ≃ₗ[R] (β → R) := |
| (gradeCoordinates a).trans ((LinearEquiv.piCongrRight q).trans (gradeCoordinates b).symm) |
|
|
| lemma grade_sumGrades (a : α → ℕ) (b : β → ℕ) |
| (q : (k : ℕ) → ({i // a i=k} → R) ≃ₗ[R] ({i // b i=k} → R)) (k : ℕ) (v : α → R) : |
| coordinateGrade R b k (sumGrades a b q v)=q k (coordinateGrade R a k v) := by |
| ext i |
| rcases i with ⟨i,hi⟩ |
| subst k |
| rfl |
|
|
| lemma prefix_sumGrades (a : α → ℕ) (b : β → ℕ) |
| (q : (k : ℕ) → ({i // a i=k} → R) ≃ₗ[R] ({i // b i=k} → R)) (k : ℕ) : |
| (coordinatePrefix R a k).map (sumGrades a b q).toLinearMap=coordinatePrefix R b k := by |
| have hh (a : α → ℕ) (b : β → ℕ) |
| (q : (k : ℕ) → ({i // a i=k} → R) ≃ₗ[R] ({i // b i=k} → R)) |
| (v : α → R) (hv : v∈coordinatePrefix R a k) : |
| sumGrades a b q v∈coordinatePrefix R b k := by |
| intro i hi |
| have hz : coordinateGrade R a (b i) v=0 := by |
| ext j |
| exact hv j.val (j.property.symm ▸ hi) |
| have he := congrFun (grade_sumGrades a b q (b i) v) ⟨i,rfl⟩ |
| rw [hz,map_zero] at he |
| exact he |
| apply le_antisymm |
| · rintro _ ⟨v,hv,rfl⟩ |
| exact hh a b q v hv |
| · intro v hv |
| let w := sumGrades b a (fun k => (q k).symm) v |
| have hw : w∈coordinatePrefix R a k := by |
| intro i hi |
| have hz : coordinateGrade R b (a i) v=0 := by |
| ext j |
| exact hv j.val (j.property.symm ▸ hi) |
| have he := congrFun (grade_sumGrades b a (fun k => (q k).symm) (a i) v) ⟨i,rfl⟩ |
| rw [hz,map_zero] at he |
| exact he |
| refine ⟨w,hw,?_⟩ |
| ext i |
| have he := congrFun (grade_sumGrades a b q (b i) w) ⟨i,rfl⟩ |
| rw [show coordinateGrade R a (b i) w=(q (b i)).symm (coordinateGrade R b (b i) v) from |
| grade_sumGrades b a (fun k => (q k).symm) (b i) v, |
| LinearEquiv.apply_symm_apply] at he |
| exact he |
|
|
| lemma sumGrades_symm (a : α → ℕ) (b : β → ℕ) |
| (q : (k : ℕ) → ({i // a i=k} → R) ≃ₗ[R] ({i // b i=k} → R)) : |
| (sumGrades a b q).symm=sumGrades b a (fun k => (q k).symm) := by |
| ext v i |
| rfl |
|
|
| / |
| ring. -/ |
| theorem namedFrame_holds (a : α → ℕ) (b : β → ℕ) |
| (e : (α → R) ≃ₗ[R] V) (f : (β → R) ≃ₗ[R] V) |
| (q : (k : ℕ) → ({i // a i=k} → R) ≃ₗ[R] ({i // b i=k} → R)) |
| (hp : ∀ k,framedPrefix a e k=framedPrefix b f k) |
| (hg : ∀ k v,v∈framedPrefix a e (k+1) → q k (framedGrade a e k v)=framedGrade b f k v) : |
| SameFramedFlag b ((sumGrades a b q).symm.trans e) f := by |
| have hp' (k : ℕ) : framedPrefix b ((sumGrades a b q).symm.trans e) k=framedPrefix a e k := by |
| change (coordinatePrefix R b k).map (e.toLinearMap.comp (sumGrades a b q).symm.toLinearMap)=_ |
| rw [Submodule.map_comp,sumGrades_symm,prefix_sumGrades] |
| rfl |
| refine ⟨fun k => (hp' k).trans (hp k),?_⟩ |
| intro k v hv |
| rw [hp'] at hv |
| change coordinateGrade R b k (sumGrades a b q (e.symm v))=framedGrade b f k v |
| rw [grade_sumGrades] |
| exact hg k v hv |
| end IntegralCharacterVarieties.NamedBandGrades |
| end |
|
|
| noncomputable section |
| namespace IntegralCharacterVarieties.NamedBandGrades |
| open scoped Classical |
| variable {R : Type*} [CommRing R] {α β : Type*} |
|
|
| def sectionGrade (a : α → ℕ) (k : ℕ) : |
| ({i // a i=k} → R) →ₗ[R] (α → R) where |
| toFun v i := if h : a i=k then v ⟨i,h⟩ else 0 |
| map_add' v w := by ext i; by_cases h : a i=k <;> simp [h] |
| map_smul' c v := by ext i; by_cases h : a i=k <;> simp [h] |
|
|
| / |
| def gradeMap (a : α → ℕ) (b : β → ℕ) |
| (e : (α → R) ≃ₗ[R] (β → R)) (k : ℕ) : |
| ({i // a i=k} → R) →ₗ[R] ({i // b i=k} → R) := |
| (coordinateGrade R b k).comp (e.toLinearMap.comp (sectionGrade a k)) |
|
|
| end IntegralCharacterVarieties.NamedBandGrades |
| end |
|
|
| noncomputable section |
| namespace IntegralCharacterVarieties.NamedBandGrades |
| open scoped Classical |
| variable {R : Type*} [CommRing R] {α β : Type*} |
|
|
| @[simp] lemma grade_section (a : α → ℕ) (k : ℕ) (v : {i // a i=k} → R) : |
| coordinateGrade R a k (sectionGrade a k v)=v := by |
| ext i |
| simp [coordinateGrade,sectionGrade,i.property] |
|
|
| lemma section_prefix (a : α → ℕ) (k : ℕ) (v : {i // a i=k} → R) : |
| sectionGrade a k v ∈ coordinatePrefix R a (k+1) := by |
| intro i hi |
| have hn : a i≠k := by omega |
| simp [sectionGrade,hn] |
|
|
| lemma grade_prefix_zero (a : α → ℕ) (k : ℕ) (v : α → R) |
| (hv : v∈coordinatePrefix R a k) : coordinateGrade R a k v=0 := by |
| ext i |
| exact hv i.val (le_of_eq i.property.symm) |
|
|
| lemma subtract_section_prefix (a : α → ℕ) (k : ℕ) (v : α → R) |
| (hv : v∈coordinatePrefix R a (k+1)) : |
| v-sectionGrade a k (coordinateGrade R a k v) ∈ coordinatePrefix R a k := by |
| intro i hi |
| by_cases he : a i=k |
| · simp [sectionGrade,coordinateGrade,he] |
| · have hg : k+1≤a i := by omega |
| simp [sectionGrade,coordinateGrade,he,hv i hg] |
|
|
| lemma filtered_apply (a : α → ℕ) (b : β → ℕ) (e : (α → R) ≃ₗ[R] (β → R)) |
| (h : ∀ k,(coordinatePrefix R a k).map e.toLinearMap=coordinatePrefix R b k) |
| (k : ℕ) {v : α → R} (hv : v∈coordinatePrefix R a k) : |
| e v∈coordinatePrefix R b k := by |
| rw [←h k] |
| exact ⟨v,hv,rfl⟩ |
|
|
| lemma filtered_symm (a : α → ℕ) (b : β → ℕ) (e : (α → R) ≃ₗ[R] (β → R)) |
| (h : ∀ k,(coordinatePrefix R a k).map e.toLinearMap=coordinatePrefix R b k) : |
| ∀ k,(coordinatePrefix R b k).map e.symm.toLinearMap=coordinatePrefix R a k := by |
| intro k |
| rw [←h k,←Submodule.map_comp] |
| simp |
|
|
| lemma gradeMap_naturality (a : α → ℕ) (b : β → ℕ) |
| (e : (α → R) ≃ₗ[R] (β → R)) |
| (h : ∀ k,(coordinatePrefix R a k).map e.toLinearMap=coordinatePrefix R b k) |
| (k : ℕ) (v : α → R) (hv : v∈coordinatePrefix R a (k+1)) : |
| gradeMap a b e k (coordinateGrade R a k v)=coordinateGrade R b k (e v) := by |
| have hh := grade_prefix_zero b k (e (v-sectionGrade a k (coordinateGrade R a k v))) |
| (filtered_apply a b e h k (subtract_section_prefix a k v hv)) |
| rw [map_sub,map_sub,sub_eq_zero] at hh |
| exact hh.symm |
|
|
| lemma gradeMap_left_inv (a : α → ℕ) (b : β → ℕ) |
| (e : (α → R) ≃ₗ[R] (β → R)) |
| (h : ∀ k,(coordinatePrefix R a k).map e.toLinearMap=coordinatePrefix R b k) |
| (k : ℕ) (v : {i // a i=k} → R) : |
| gradeMap b a e.symm k (gradeMap a b e k v)=v := by |
| have hn := gradeMap_naturality b a e.symm (filtered_symm a b e h) k |
| (e (sectionGrade a k v)) (filtered_apply a b e h _ (section_prefix a k v)) |
| change gradeMap b a e.symm k (coordinateGrade R b k (e (sectionGrade a k v)))=v |
| rw [hn,LinearEquiv.symm_apply_apply,grade_section] |
|
|
| / |
| the coefficient ring. -/ |
| def gradeEquiv (a : α → ℕ) (b : β → ℕ) (e : (α → R) ≃ₗ[R] (β → R)) |
| (h : ∀ k,(coordinatePrefix R a k).map e.toLinearMap=coordinatePrefix R b k) |
| (k : ℕ) : ({i // a i=k} → R) ≃ₗ[R] ({i // b i=k} → R) := |
| { gradeMap a b e k with |
| invFun := gradeMap b a e.symm k |
| left_inv := gradeMap_left_inv a b e h k |
| right_inv := gradeMap_left_inv b a e.symm (filtered_symm a b e h) k } |
|
|
| end IntegralCharacterVarieties.NamedBandGrades |
| end |
|
|
| end OAI |
|
|
