| 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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