| import OAI.Algebra.Finitistic.Bimodule.BimoduleTensorAcross |
| import OAI.Algebra.Finitistic.Bimodule.BimoduleTotalDifferential |
|
|
| namespace OAI |
| noncomputable section |
| open CategoryTheory |
| namespace LittleFinitistic.BimoduleTotal |
| open ExternalTensor.Section4 |
| universe u |
| variable {k B : Type u} [Field k] [Ring B] [Algebra k B] |
| variable {P P' P'' : BiComplex k B} (f : P ⟶ P') |
| variable (Z : CochainComplex (ModuleCat.{u} B) ℤ) |
|
|
| /-- The actual tensor map induced by a bimodule-complex morphism in each bidegree. -/ |
| def leftTermMap (a q : ℤ) : Term k B P Z a q →ₗ[B] Term k B P' Z a q := |
| tensorAcross (f.f a) (𝟙 (Z.X q)) |
|
|
| theorem leftTermMap_horizontal (n m a b : ℤ) (h : n+1=m) (hab : a+1=b) : |
| (leftTermMap f Z b (m + -b)).comp (horizontalBetween k B P Z n m a b h hab) = |
| (horizontalBetween k B P' Z n m a b h hab).comp (leftTermMap f Z a (n + -a)) := by |
| unfold horizontalBetween leftTermMap |
| rw [← tensorAcross_eq_termMap, ← tensorAcross_eq_termMap, |
| tensorAcross_comp,tensorAcross_comp] |
| simp only [Category.comp_id,Category.id_comp] |
| rw [f.comm] |
|
|
| theorem leftTermMap_entry (n m a b : ℤ) (h : n+1=m) : |
| (leftTermMap f Z b (m + -b)).comp (entryBetween k B P Z n m a b h) = |
| (entryBetween k B P' Z n m a b h).comp (leftTermMap f Z a (n + -a)) := by |
| unfold entryBetween |
| split_ifs with hab |
| · exact leftTermMap_horizontal f Z n m a b h hab |
| · simp |
|
|
| theorem leftTermMap_vertical (n m a : ℤ) (x : Term k B P Z a (n + -a)) : |
| leftTermMap f Z a (m + -a) |
| (((ComplexTensor.column k B (right k B P) Z a).d n m).hom x) = |
| ((ComplexTensor.column k B (right k B P') Z a).d n m).hom |
| (leftTermMap f Z a (n + -a) x) := by |
| let L := (leftTermMap f Z a (m + -a)).toAddMonoidHom.comp |
| ((ComplexTensor.column k B (right k B P) Z a).d n m).hom.toAddMonoidHom |
| let R := ((ComplexTensor.column k B (right k B P') Z a).d n m).hom.toAddMonoidHom.comp |
| (leftTermMap f Z a (n + -a)).toAddMonoidHom |
| induction x using BalancedTensor.induction_on with |
| | hz => exact L.map_zero.trans R.map_zero.symm |
| | ht x y => |
| rw [ComplexTensor.column_d,ComplexTensor.column_d] |
| change leftTermMap f Z a (m + -a) |
| ((-a).negOnePow • BalancedTensor.mk k B (BimoduleBalancedTensor.Input k B (P.X a)) |
| (ComplexTensor.leftTerm k B Z (m + -a)) x ((Z.d (n + -a) (m + -a)).hom y)) = |
| (-a).negOnePow • BalancedTensor.mk k B (BimoduleBalancedTensor.Input k B (P'.X a)) |
| (ComplexTensor.leftTerm k B Z (m + -a)) ((f.f a).hom x) ((Z.d (n + -a) (m + -a)).hom y) |
| simp only [Units.smul_def, map_zsmul] |
| rfl |
| | ha x y hx hy => |
| exact (L.map_add x y).trans |
| ((congrArg₂ (· + ·) hx hy).trans (R.map_add x y).symm) |
|
|
| variable {l : ℕ} (p : Fin l → ℤ) (hp : StrictMono p) |
|
|
| def leftTotalMapF (n : ℤ) : TotalTerm k B P p Z n →ₗ[B] TotalTerm k B P' p Z n where |
| toFun x i := leftTermMap f Z (p i) (n + -(p i)) (x i) |
| map_add' x y := by ext i; exact map_add _ _ _ |
| map_smul' b x := by |
| funext i |
| exact (leftTermMap f Z (p i) (n + -(p i))).map_smul b (x i) |
|
|
| /-- Tensoring an actual bimodule chain map gives an actual B-linear total chain map. -/ |
| def leftTotalMap : total k B P p hp Z ⟶ total k B P' p hp Z where |
| f n := ModuleCat.ofHom (leftTotalMapF f Z p n) |
| comm' n m h := by |
| apply ModuleCat.hom_ext |
| apply LinearMap.ext |
| intro x |
| funext j |
| change ((total k B P' p hp Z).d n m).hom (leftTotalMapF f Z p n x) j = |
| leftTermMap f Z (p j) (m + -(p j)) (((total k B P p hp Z).d n m).hom x j) |
| erw [total_d_apply k B P' Z p hp n m h, total_d_apply k B P Z p hp n m h, |
| map_add, map_sum] |
| congr 1 |
| · exact (leftTermMap_vertical f Z n m (p j) (x j)).symm |
| · apply Finset.sum_congr rfl |
| intro i _ |
| exact (LinearMap.congr_fun (leftTermMap_entry f Z n m (p i) (p j) h) (x i)).symm |
|
|
| @[simp] theorem leftTotalMap_apply (n : ℤ) (x : TotalTerm k B P p Z n) (i : Fin l) : |
| ((leftTotalMap f Z p hp).f n).hom x i = leftTermMap f Z (p i) (n + -(p i)) (x i) := rfl |
| end LittleFinitistic.BimoduleTotal |
|
|
| end |
|
|
| end OAI |
|
|
