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216fb31 | 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 | import OAI.Algebra.Finitistic.Bar.GlobalIndexedComparison
import OAI.Algebra.Finitistic.Bar.IntegerDerivedMultibar
import OAI.Algebra.Finitistic.Bar.StrictSupportExtension
namespace OAI
open CategoryTheory CategoryTheory.Limits TensorProduct TrivSqZeroExt HomologicalComplex
open scoped TensorProduct DirectSum
noncomputable section
namespace DerivedBar.ParityBridge
open DerivedBar.Multibar DerivedBar.RecursiveTotal
universe u
variable (k D X : Type u) [Field k] [Ring D] [Algebra k D]
[AddCommGroup X] [Module k X] [Module D X] [Module Dᵐᵒᵖ X]
[SMulCommClass D Dᵐᵒᵖ X] [IsScalarTower k D X] [IsScalarTower k Dᵐᵒᵖ X]
local instance integerCountRestrictionScalarModule (V : ModuleCat D) : Module k V :=
BimoduleTensor.scalarModule k D V
local instance integerCountRestrictionScalarTower (V : ModuleCat D) : IsScalarTower k D V :=
IsScalarTower.of_algebraMap_smul (fun _ _ => rfl)
local instance integerCountRestrictionIndexDecidableEq (r n : ℕ) :
DecidableEq (Index r n) := Classical.decEq _
local instance integerCountRestrictionShapeDecidableEq (n r : ℕ) :
DecidableEq (Multibar.Indexed.Shape n r) := Classical.decEq _
variable (N : ModuleCat D)
local notation "A" => TrivSqZeroExt D X
local notation "XB" => BimoduleTensor.twoSidedBarFunctor k D X
local notation "Z" => Functor.obj (ChainComplex.single₀ (ModuleCat D)) N
local notation "B" => (fun _ : ℕ => XB)
local notation "c" => ComplexShape.down ℕ
local instance integerCountRestrictionInflatedModule : Module A N := inflatedModule D X N
local instance integerCountRestrictionInflatedScalarTower : IsScalarTower k A N :=
IsScalarTower.of_algebraMap_smul
(fun a v => IsScalarTower.algebraMap_smul D a v)
local notation "C" => ModuleCat D
local notation "E" => ComplexShape.embeddingDownNat
local notation "K" => Multibar.Indexed.signedBlockComplex k D X N (inflated_ideal_zero D X N)
/-- The actual integer cochain count complex shifted by -r. -/
def deshiftedCount (r : ℕ) : CochainComplex C ℤ :=
((integerExtension D).obj (K r))⟦-(r:ℤ)⟧
def deshiftedCountRestriction (r : ℕ) : ChainComplex C ℕ :=
(deshiftedCount k D X N r).restriction E
def deshiftedCountXIso (r n : ℕ) : (deshiftedCountRestriction k D X N r).X n ≅
indexedModule k D X N r n :=
(K r).extendXIso E
(show (E).f (r+n) = (E).f n + -(r:ℤ) by change -((r+n:ℕ):ℤ) = -(n:ℤ) + -(r:ℤ); push_cast; omega)
theorem units_hom_apply (V W : ModuleCat.{u} D) (s : ℤˣ) (f : V ⟶ W) (v : V) :
(s • f).hom v = s • f.hom v := by
change ((s:ℤ) • f).hom v = (s:ℤ) • f.hom v
simp only [ModuleCat.hom_smul, LinearMap.smul_apply]
/-- The standard unit-valued cochain shift sign is the proved count sign. -/
theorem deshiftedCount_sign (r n : ℕ) :
(-(r:ℤ)).negOnePow • (K r).d (r+(n+1)) (r+n) = indexedDifferential k D X N r n := by
have hd : (K r).d (r+(n+1)) (r+n) =
ModuleCat.ofHom (Multibar.Indexed.blockDifferential k D X N (inflated_ideal_zero D X N) (r+n) r) := by
change ChainComplex.of.d
(fun m => ModuleCat.of D (⨁ p : Multibar.Indexed.Shape m r, Multibar.Indexed.outerTensor k D X N p))
(fun m => ModuleCat.ofHom (Multibar.Indexed.blockDifferential k D X N
(inflated_ideal_zero D X N) m r)) ((r+n)+1) (r+n) = _
exact ChainComplex.of_d _ _ (r+n)
rw [hd]
apply ModuleCat.hom_ext
apply LinearMap.ext
intro v
erw [units_hom_apply]
erw [globalIndexedComparison]
change (-(r:ℤ)).negOnePow •
Multibar.Indexed.blockDifferential k D X N (inflated_ideal_zero D X N) (r+n) r v =
(-1:k)^r •
Multibar.Indexed.blockDifferential k D X N (inflated_ideal_zero D X N) (r+n) r v
have hu : (r:ℤ).negOnePow = (-1:ℤˣ)^r := by
apply Units.ext
simpa using Int.coe_negOnePow_natCast r
rw [Int.negOnePow_neg, hu]
exact totalSign_smul k (⨁ p : Multibar.Indexed.Shape (r+n) r, Multibar.Indexed.outerTensor k D X N p) r
(Multibar.Indexed.blockDifferential k D X N (inflated_ideal_zero D X N) (r+n) r v)
theorem deshiftedCount_d (r n : ℕ) :
(deshiftedCountRestriction k D X N r).d (n+1) n =
(deshiftedCountXIso k D X N r (n+1)).hom ≫ indexedDifferential k D X N r n ≫
(deshiftedCountXIso k D X N r n).inv := by
change (-(r:ℤ)).negOnePow • ((K r).extend E).d ((E).f (n+1) + -(r:ℤ)) ((E).f n + -(r:ℤ)) = _
rw [extend_d_eq (K r) E
(show (E).f (r+(n+1))=(E).f (n+1) + -(r:ℤ) by change -((r+(n+1):ℕ):ℤ) = -((n+1:ℕ):ℤ) + -(r:ℤ); push_cast; omega)
(show (E).f (r+n)=(E).f n + -(r:ℤ) by change -((r+n:ℕ):ℤ) = -(n:ℤ) + -(r:ℤ); push_cast; omega)]
rw [← deshiftedCount_sign]
erw [CategoryTheory.Linear.units_smul_comp, CategoryTheory.Linear.comp_units_smul]
rfl
/-- A genuine chain isomorphism using the standard integer shift differential,
not a newly stipulated shift or transported count differential. -/
def indexedDeshiftedCountIso (r : ℕ) : indexedMultibar k D X N r ≅
deshiftedCountRestriction k D X N r :=
HomologicalComplex.Hom.isoOfComponents
(fun n => (deshiftedCountXIso k D X N r n).symm) (by
intro i j hij
change j+1=i at hij
subst i
rw [deshiftedCount_d]
simp only [Iso.symm_hom]
erw [Iso.inv_hom_id_assoc]
simp only [indexedMultibar, ChainComplex.of_d])
end DerivedBar.ParityBridge
end
end OAI
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