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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 | import OAI.Algebra.Finitistic.Bar.MultibarSigns
namespace OAI
open CategoryTheory TensorProduct TrivSqZeroExt
open scoped TensorProduct
noncomputable section
namespace DerivedBar.SquareZero
universe u
variable (k D X N : 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]
[AddCommGroup N] [Module k N]
[Module (TrivSqZeroExt D X) N] [IsScalarTower k (TrivSqZeroExt D X) N]
local notation "A" => TrivSqZeroExt D X
/-- The parity operator is +1 on a D slot and -1 on an X slot. -/
def slotParity : Module.End k A := baseProjection k D X - idealProjection k D X
@[simp] theorem slotParity_base (d : D) : slotParity k D X (inl d) = inl d := by
simp [slotParity]
@[simp] theorem slotParity_ideal (x : X) : slotParity k D X (inr x) = -inr x := by
simp [slotParity]
/-- Explicit bar faces with a sign change on crossing a D slot and no sign
change on crossing an X slot. This is defined independently of count transport. -/
def multibarAugmentedDifferential : (n : ℕ) → term k A N (n+1) →ₗ[k] term k A N n
| 0 => ((action k A N).restrictScalars k).comp
(TensorProduct.map (slotParity k D X) LinearMap.id)
| n+1 => (((action k A (term k A N (n+1))).restrictScalars k) -
TensorProduct.map LinearMap.id (multibarAugmentedDifferential n)).comp
(TensorProduct.map (slotParity k D X) LinearMap.id)
@[simp] theorem multibarAugmentedDifferential_zero (a : A) (v : N) :
multibarAugmentedDifferential k D X N 0 (a ⊗ₜ[k] v) = slotParity k D X a • v := rfl
theorem multibarAugmentedDifferential_succ (n : ℕ) (a : A) (v : term k A N (n+1)) :
multibarAugmentedDifferential k D X N (n+1) (a ⊗ₜ[k] v) =
slotParity k D X a • v -
slotParity k D X a ⊗ₜ[k] multibarAugmentedDifferential k D X N n v := rfl
/-- The independent two-sided differential. The first face uses D's right
A-action by projection. All remaining faces use the explicit slot parity recursion. -/
def multibarTwoSidedDifferential (n : ℕ) :
D ⊗[k] term k A N (n+1) →ₗ[D] D ⊗[k] term k A N n :=
(action k D (D ⊗[k] term k A N n)).comp
(AlgebraTensorModule.lTensor D D
(FreeBaseChange.coefficient (fstHom k D X) (term k A N n))) -
AlgebraTensorModule.lTensor D D (multibarAugmentedDifferential k D X N n)
/-- Literal endpoint-minus-tail formula; no signed assembly occurs in the definition. -/
theorem multibarTwoSidedDifferential_tmul (n : ℕ) (d : D) (v : term k A N (n+1)) :
multibarTwoSidedDifferential k D X N n (d ⊗ₜ[k] v) =
d • FreeBaseChange.coefficient (fstHom k D X) (term k A N n) v -
d ⊗ₜ[k] multibarAugmentedDifferential k D X N n v := rfl
/-- A zero-length internal D block has no multiplication face: X squared is zero.
The displayed surviving term consists only of the later blocks' faces. -/
theorem multibarAugmented_empty_block (n : ℕ) (x y : X) (v : term k A N n) :
multibarAugmentedDifferential k D X N (n+1)
((inr x : A) ⊗ₜ[k] ((inr y : A) ⊗ₜ[k] v)) =
(inr x : A) ⊗ₜ[k] multibarAugmentedDifferential k D X N n
((inr y : A) ⊗ₜ[k] v) := by
rw [multibarAugmentedDifferential_succ, slotParity_ideal]
have ha : (-inr x : A) • ((inr y : A) ⊗ₜ[k] v) = 0 := by
change ((-inr x : A) * inr y) ⊗ₜ[k] v = 0
simp only [neg_mul, inr_mul_inr, neg_zero, TensorProduct.zero_tmul]
rw [ha, zero_sub, TensorProduct.neg_tmul, neg_neg]
/-- At the inflated module endpoint, an X action vanishes. -/
theorem multibarAugmented_ideal_endpoint
(hN : ∀ (x : X) (v : N), (inr x : TrivSqZeroExt D X) • v = (0 : N))
(x : X) (v : N) :
multibarAugmentedDifferential k D X N 0 ((inr x : A) ⊗ₜ[k] v) = 0 := by
rw [multibarAugmentedDifferential_zero, slotParity_ideal, _root_.neg_smul, hN, neg_zero]
omit [IsScalarTower k A N] in
/-- Rescaling commutes with the D-action on each word. -/
theorem signedProjection_base_action_word {n r : ℕ} {v : term k A N n}
(hv : Word k D X N n r v) (d : D) :
signedProjection k D X N n r ((inl d : A) • v) =
(inl d : A) • signedProjection k D X N n r v := by
cases hv with
| nil v => rfl
| @base n r v b hv =>
rw [TensorProduct.smul_tmul', smul_eq_mul, inl_mul_inl,
signedProjection_base, signedProjection_base]
have ha : (inl d : A) • ((inl b : A) ⊗ₜ[k] signedProjection k D X N n r v) =
(inl (d*b) : A) ⊗ₜ[k] signedProjection k D X N n r v := by
change ((inl d : A) * inl b) ⊗ₜ[k] _ = _
rw [inl_mul_inl]
rw [smul_comm (inl d : A) ((-1 : k)^r), ha]
| @ideal n r v x hv =>
rw [TensorProduct.smul_tmul', smul_eq_mul, inl_mul_inr,
signedProjection_ideal_succ, signedProjection_ideal_succ,
TensorProduct.smul_tmul', smul_eq_mul, inl_mul_inr]
omit [IsScalarTower k A N] in
/-- Multiplying a D slot by X removes its r inversions with the remaining X slots. -/
theorem signedProjection_ideal_action_word
(hN : ∀ (x : X) (v : N), (inr x : TrivSqZeroExt D X) • v = (0 : N)) {n r : ℕ} {v : term k A N n}
(hv : Word k D X N n r v) (x : X) :
signedProjection k D X N n (r+1) ((inr x : A) • v) =
(-1 : k)^r • ((inr x : A) • signedProjection k D X N n r v) := by
cases hv with
| nil v => simp [signedProjection, hN]
| @base n r v d hv =>
rw [TensorProduct.smul_tmul', smul_eq_mul, inr_mul_inl,
signedProjection_ideal_succ, signedProjection_base,
smul_comm (inr x : A) ((-1 : k)^r), smul_smul, ← mul_pow]
simp only [neg_one_mul, neg_neg, one_pow, one_smul,
TensorProduct.smul_tmul', smul_eq_mul, inr_mul_inl]
| @ideal n r v y hv =>
rw [TensorProduct.smul_tmul', smul_eq_mul, inr_mul_inr, TensorProduct.zero_tmul,
map_zero, signedProjection_ideal_succ, TensorProduct.smul_tmul', smul_eq_mul,
inr_mul_inr, TensorProduct.zero_tmul, smul_zero]
/-- Exact sign identity between the literal augmented faces and the ordinary bar
on every homogeneous word. The differential was defined independently above. -/
theorem multibarAugmented_sign_word
(hN : ∀ (x : X) (v : N), (inr x : TrivSqZeroExt D X) • v = (0 : N)) (n r : ℕ) (v : term k A N (n+1))
(hv : Word k D X N (n+1) r v) :
multibarAugmentedDifferential k D X N n (signedProjection k D X N (n+1) r v) =
(-1 : k)^r • signedProjection k D X N n r (differential k A N n v) := by
induction n generalizing r with
| zero =>
cases hv with
| base d hv =>
cases hv
simp [signedProjection, multibarAugmentedDifferential,
slotParity_base, differential]
| ideal x hv =>
cases hv
simp [signedProjection, multibarAugmentedDifferential,
slotParity_ideal, differential, hN]
| succ n ih =>
cases hv with
| @base m r v d hv =>
rw [signedProjection_base, map_smul, multibarAugmentedDifferential_succ,
slotParity_base, ih r v hv, differential_succ_tmul, map_sub,
signedProjection_base_action_word k D X N hv d, signedProjection_base,
TensorProduct.tmul_smul]
| @ideal m r v x hv =>
rw [signedProjection_ideal_succ, multibarAugmentedDifferential_succ,
slotParity_ideal, ih r v hv, differential_succ_tmul, map_sub,
signedProjection_ideal_action_word k D X N hN hv x, signedProjection_ideal_succ]
simp only [TensorProduct.neg_tmul, TensorProduct.tmul_smul, smul_sub, smul_smul]
rw [← pow_add, show r+1+r=2*r+1 by omega, pow_add, pow_mul]
simp [pow_succ]
exact _root_.neg_smul (inr x : A) (signedProjection k D X N (n+1) r v)
omit [Module A N] [IsScalarTower k A N] in
/-- The leading endpoint has the same prescribed rescaling. -/
theorem coefficient_signed_word (n r : ℕ) (v : term k A N (n+1))
(hv : Word k D X N (n+1) r v) :
FreeBaseChange.coefficient (fstHom k D X) (term k A N n)
(signedProjection k D X N (n+1) r v) =
(-1 : k)^r • twoSidedSignedProjection k D X N n r
(FreeBaseChange.coefficient (fstHom k D X) (term k A N n) v) := by
cases hv with
| base d hv =>
rw [signedProjection_base, map_smul, FreeBaseChange.coefficient_tmul,
FreeBaseChange.coefficient_tmul]
rfl
| ideal x hv =>
rw [signedProjection_ideal_succ, FreeBaseChange.coefficient_tmul,
FreeBaseChange.coefficient_tmul]
simp
/-- Equality of the independently defined two-sided faces and the shifted signed
ordinary bar differential, on every homogeneous pure tensor. -/
theorem multibarTwoSided_sign_word
(hN : ∀ (x : X) (v : N), (inr x : TrivSqZeroExt D X) • v = (0 : N))
(n r : ℕ) (d : D) (v : term k A N (n+1))
(hv : Word k D X N (n+1) r v) :
multibarTwoSidedDifferential k D X N n
(twoSidedSignedProjection k D X N (n+1) r (d ⊗ₜ[k] v)) =
(-1 : k)^r • twoSidedSignedProjection k D X N n r
(twoSidedDifferential k A D N (fstHom k D X) n (d ⊗ₜ[k] v)) := by
change multibarTwoSidedDifferential k D X N n
(d ⊗ₜ[k] signedProjection k D X N (n+1) r v) = _
rw [multibarTwoSidedDifferential_tmul, coefficient_signed_word k D X N n r v hv,
multibarAugmented_sign_word k D X N hN n r v hv, twoSidedDifferential_tmul,
map_sub, map_smul, smul_sub, TensorProduct.tmul_smul,
smul_comm d ((-1 : k)^r)]
rfl
/-- The sign comparison extends to the whole count submodule by linearity. -/
theorem multibarTwoSided_sign
(hN : ∀ (x : X) (v : N), (inr x : TrivSqZeroExt D X) • v = (0 : N))
(n r : ℕ) {z : D ⊗[k] term k A N (n+1)}
(hz : z ∈ twoSidedWeight k D X N (n+1) r) :
multibarTwoSidedDifferential k D X N n (twoSidedSignedProjection k D X N (n+1) r z) =
(-1 : k)^r • twoSidedSignedProjection k D X N n r
(twoSidedDifferential k A D N (fstHom k D X) n z) := by
have h : twoSidedWeight k D X N (n+1) r ≤
((multibarTwoSidedDifferential k D X N n).comp
(twoSidedSignedProjection k D X N (n+1) r) -
(-1 : k)^r • (twoSidedSignedProjection k D X N n r).comp
(twoSidedDifferential k A D N (fstHom k D X) n)).ker := by
apply Submodule.span_le.mpr
rintro z ⟨d,v,hv,rfl⟩
exact sub_eq_zero.mpr (multibarTwoSided_sign_word k D X N hN n r d v hv)
exact sub_eq_zero.mp (h hz)
end DerivedBar.SquareZero
end
end OAI
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