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e3e0a0d | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 | import Mathlib
import OAI.RingTheory.Multiplicity.CechCarrierOperators
import OAI.RingTheory.Multiplicity.ReesRootCech
namespace OAI
noncomputable section
namespace Lech.CechNormalization
open CategoryTheory CategoryTheory.Limits HomologicalComplex Set CarrierOperators
universe u
variable {R : Type u} [CommRing R] {ι : Type} [Fintype ι] [LinearOrder ι]
(D : FiniteModuleCech.Diagram R ι)
abbrev fullCochains (p : ℕ) := Sections D (fullCarrier p)
abbrev fullD (p : ℕ) := lift D (deltaOp p)
abbrev fullSort (p : ℕ) := lift D (sortOp p)
abbrev fullExpand (p : ℕ) := lift D (expandOp p)
abbrev fullH (p : ℕ) := lift D (contractionOp p)
abbrev fullN (p : ℕ) := lift D (normalizeOp p)
omit [Fintype ι] in
lemma fullD_square (p : ℕ) (f : fullCochains D p) : fullD D (p+1) (fullD D p f)=0 := by
change ((lift D (deltaOp (p+1))).comp (lift D (deltaOp p))) f=0
rw [←lift_comp]
funext a
exact congrFun (FiniteCoverCech.delta_square R (D.obj (FiniteCoverCech.intersection a)) p
(toFixed D (fullCarrier p) (FiniteCoverCech.intersection a) f)) a
omit [Fintype ι] in
lemma sortedD_eq (p : ℕ) : lift D (sortedDeltaOp p)=FiniteModuleCech.d D p := by
apply LinearMap.ext
intro f
funext s
change (∑ i : Fin (p+1),(-1:ℤ)^i.val •
toFixed D (sortedCarrier p) s.val f (AlternatingCech.delete s i))=_
apply Finset.sum_congr rfl
intro i hi
congr 1
change (if h : (AlternatingCech.delete s i).val⊆s.val then _ else _)=_
rw [dite_eq_left (AlternatingCech.delete_subset s i)]
omit [Fintype ι] in
lemma fullSort_d (p : ℕ) :
(fullSort D (p+1)).comp (fullD D p)=(FiniteModuleCech.d D p).comp (fullSort D p) := by
rw [←sortedD_eq]
exact (by
rw [←lift_comp,←lift_comp]
exact lift_congr D (fun M => sort_delta M p))
lemma fullExpand_d (p : ℕ) :
(fullExpand D (p+1)).comp (FiniteModuleCech.d D p)=(fullD D p).comp (fullExpand D p) := by
rw [←sortedD_eq]
rw [←lift_comp,←lift_comp]
exact lift_congr D (fun M => expand_delta M p)
lemma fullSort_expand (p : ℕ) : (fullSort D p).comp (fullExpand D p)=LinearMap.id := by
rw [←lift_id D (sortedCarrier p)]
rw [←lift_comp]
exact lift_congr D (fun M => sort_expand M p)
lemma fullExpand_sort (p : ℕ) : (fullExpand D p).comp (fullSort D p)=fullN D p := by
rw [←lift_comp]
exact lift_congr D (fun M => expand_sort M p)
lemma fullH_zero : fullH D 0=0 := by
apply LinearMap.ext
intro f
funext a
rfl
lemma fullH_identity (p : ℕ) :
(fullH D (p+1)).comp (fullD D (p+1))+(fullD D p).comp (fullH D p)=
LinearMap.id-fullN D (p+1) := by
change (lift D (contractionOp (p+1))).comp (lift D (deltaOp (p+1)))+
(lift D (deltaOp p)).comp (lift D (contractionOp p))=LinearMap.id-lift D (normalizeOp (p+1))
rw [←lift_comp,←lift_comp,←lift_add,←lift_id D (fullCarrier (p+1)),←lift_sub]
apply lift_congr D
intro M
apply LinearMap.ext
exact contraction_identity R M p
def fullPositive : CochainComplex (ModuleCat.{u} R) ℕ :=
CochainComplex.of (fun p => ModuleCat.of R (fullCochains D (p+1)))
(fun p => ModuleCat.ofHom (fullD D (p+1))) (fun p => by
apply ModuleCat.hom_ext
apply LinearMap.ext
exact fullD_square D (p+1))
omit [Fintype ι] in
lemma fullPositive_d (p : ℕ) : (fullPositive D).d p (p+1)=ModuleCat.ofHom (fullD D (p+1)) := by
unfold fullPositive
exact CochainComplex.of_d (fun q => ModuleCat.of R (fullCochains D (q+1)))
(fun q => ModuleCat.ofHom (fullD D (q+1))) p
def sortMap : fullPositive D ⟶ FiniteModuleCech.positiveComplex D :=
CochainComplex.ofHom (fun p => ModuleCat.ofHom (fullSort D (p+1))) (fun p => by
rw [fullPositive_d,FiniteModuleCech.positiveComplex_d]
apply ModuleCat.hom_ext
exact (fullSort_d D (p+1)).symm)
def expandMap : FiniteModuleCech.positiveComplex D ⟶ fullPositive D :=
CochainComplex.ofHom (fun p => ModuleCat.ofHom (fullExpand D (p+1))) (fun p => by
rw [fullPositive_d,FiniteModuleCech.positiveComplex_d]
apply ModuleCat.hom_ext
exact (fullExpand_d D (p+1)).symm)
lemma expand_sortMap : expandMap D ≫ sortMap D=𝟙 _ := by
apply Hom.ext
funext p
apply ModuleCat.hom_ext
exact fullSort_expand D (p+1)
lemma sort_expandMap (p : ℕ) :
((sortMap D ≫ expandMap D).f p).hom=fullN D (p+1) := fullExpand_sort D (p+1)
def homotopyComponent (i j : ℕ) : (fullPositive D).X i ⟶ (fullPositive D).X j :=
if h : j+1=i then by subst i; exact ModuleCat.ofHom (fullH D (j+1)) else 0
lemma homotopyComponent_succ (j : ℕ) :
homotopyComponent D (j+1) j=ModuleCat.ofHom (fullH D (j+1)) := by
rw [homotopyComponent,dite_eq_left rfl]
def normalizationHomotopy : Homotopy (𝟙 (fullPositive D)) (sortMap D ≫ expandMap D) where
hom := homotopyComponent D
zero i j h := by
change ¬j+1=i at h
rw [homotopyComponent,dite_eq_right h]
comm i := by
rw [Homotopy.dNext_cochainComplex,homotopyComponent_succ,fullPositive_d]
cases i with
| zero =>
rw [Homotopy.prevD_zero_cochainComplex,add_zero]
apply ModuleCat.hom_ext
have hh := fullH_identity D 0
rw [fullH_zero,LinearMap.comp_zero,add_zero] at hh
change LinearMap.id=(fullH D 1).comp (fullD D 1)+_
rw [sort_expandMap]
exact (eq_sub_iff_add_eq.mp hh).symm
| succ i =>
rw [Homotopy.prevD_succ_cochainComplex,homotopyComponent_succ,fullPositive_d]
apply ModuleCat.hom_ext
change LinearMap.id=(fullH D (i+2)).comp (fullD D (i+2))+
(fullD D (i+1)).comp (fullH D (i+1))+_
rw [sort_expandMap]
exact (eq_sub_iff_add_eq.mp (fullH_identity D (i+1))).symm
def homotopyEquiv : HomotopyEquiv (fullPositive D) (FiniteModuleCech.positiveComplex D) where
hom := sortMap D
inv := expandMap D
homotopyHomInvId := (normalizationHomotopy D).symm
homotopyInvHomId := Homotopy.ofEq (expand_sortMap D)
end Lech.CechNormalization
end
end OAI
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