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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 | import OAI.Algebra.Finitistic.Bar.IntegerCountRestriction
namespace OAI
open CategoryTheory CategoryTheory.Limits TensorProduct TrivSqZeroExt
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 (V : ModuleCat D) : Module k V := BimoduleTensor.scalarModule k D V
local instance (V : ModuleCat D) : IsScalarTower k D V :=
IsScalarTower.of_algebraMap_smul (fun _ _ => rfl)
local instance (r n : ℕ) : DecidableEq (Index r n) := Classical.decEq _
local instance (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 : Module A N := inflatedModule D X N
local instance : 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)
theorem shapeCount_le {n r : ℕ} (p : Multibar.Indexed.Shape n r) : r≤n := by
induction p <;> omega
theorem signedBlock_isZero_of_lt (n r : ℕ) (h : n<r) : IsZero ((K r).X n) := by
have : IsEmpty (Multibar.Indexed.Shape n r) :=
⟨fun shape => (Nat.not_le_of_lt h) (shapeCount_le shape)⟩
change IsZero (ModuleCat.of D (⨁ p : Multibar.Indexed.Shape n r, Multibar.Indexed.outerTensor k D X N p))
exact ModuleCat.isZero_of_subsingleton _
/-- The actual -r-shifted count complex is strictly zero in positive cochain
degrees, including the range cut off by the lower count bound. -/
theorem deshiftedCount_isZero_positive (r : ℕ) (j : ℤ) (hj : 0<j) :
IsZero ((deshiftedCount k D X N r).X j) := by
change IsZero (((K r).extend E).X (j + -(r:ℤ)))
by_cases hle : j + -(r:ℤ) ≤ 0
· let n := (-(j + -(r:ℤ))).toNat
have hn : (E).f n=j + -(r:ℤ) := by
change -(n:ℤ)=j + -(r:ℤ)
dsimp [n]
rw [Int.toNat_of_nonneg (by omega)]
omega
have hnr : n<r := by
change -(n:ℤ)=j + -(r:ℤ) at hn
omega
exact (signedBlock_isZero_of_lt k D X N n r hnr).of_iso ((K r).extendXIso E hn)
· exact (K r).isZero_extend_X E _ (fun i hi => by
change -(i:ℤ)=j + -(r:ℤ) at hi
omega)
instance deshiftedCount_strictSupport (r : ℕ) : (deshiftedCount k D X N r).IsStrictlySupported E where
isZero j hj := deshiftedCount_isZero_positive k D X N r j (by
by_contra h
have hle : j≤0 := by omega
apply hj (-j).toNat
change -((-j).toNat:ℤ)=j
rw [Int.toNat_of_nonneg (by omega)]
omega)
/-- Extension of the genuine recursive complex is the actual -r shift of
the count complex. Strict support, not truncation by assumption, supplies the inverse. -/
def integerIndexedDeshiftedCountIso (r : ℕ) :
(integerExtension D).obj (indexedMultibar k D X N r) ≅ deshiftedCount k D X N r :=
(integerExtension D).mapIso (indexedDeshiftedCountIso k D X N r) ≪≫
strictSupportExtensionIso E (deshiftedCount k D X N r)
/-- The manuscript's actual [r] cochain shift of the indexed recursive bar
is isomorphic to the entire signed count complex. -/
def integerIndexedCountShiftIso (r : ℕ) :
((integerExtension D).obj (indexedMultibar k D X N r))⟦(r:ℤ)⟧ ≅
(integerExtension D).obj (K r) :=
(CategoryTheory.shiftFunctor (CochainComplex C ℤ) (r:ℤ)).mapIso
(integerIndexedDeshiftedCountIso k D X N r) ≪≫
((CategoryTheory.shiftFunctorAdd' (CochainComplex C ℤ) (-(r:ℤ)) (r:ℤ) 0 (by omega)).symm.app
((integerExtension D).obj (K r))) ≪≫
(CategoryTheory.shiftFunctorZero (CochainComplex C ℤ) ℤ).app ((integerExtension D).obj (K r))
end DerivedBar.ParityBridge
end
end OAI
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