| import OAI.Algebra.Finitistic.Ordinary.OrdinaryTensorBimap |
| import OAI.Algebra.Finitistic.Ordinary.OrdinaryShiftSign |
|
|
| namespace OAI |
| noncomputable section |
| open CategoryTheory |
| namespace LittleFinitistic.ComplexTensor |
| open BalancedTensor |
| universe u |
| variable (k A : Type u) [Field k] [Ring A] [Algebra k A] |
| variable (Q : CochainComplex (ModuleCat.{u} Aᵐᵒᵖ) ℤ) |
| variable (Z : CochainComplex (ModuleCat.{u} A) ℤ) |
|
|
| abbrev rightTerm (p : ℤ) := ScalarLeftModule k Aᵐᵒᵖ (Q.X p) |
| abbrev leftTerm (q : ℤ) := ScalarLeftModule k A (Z.X q) |
| abbrev term (p q : ℤ) := Tensor k A (rightTerm k A Q p) (leftTerm k A Z q) |
|
|
| / |
| def map {p p' q q' : ℤ} (f : Q.X p ⟶ Q.X p') (g : Z.X q ⟶ Z.X q') : |
| term k A Q Z p q →ₗ[k] term k A Q Z p' q' := |
| bimap (k := k) |
| (M := rightTerm k A Q p) (M' := rightTerm k A Q p') |
| (N := leftTerm k A Z q) (N' := leftTerm k A Z q') f.hom g.hom |
|
|
| @[simp] theorem map_comp {p p' p'' q q' q'' : ℤ} |
| (f : Q.X p ⟶ Q.X p') (f' : Q.X p' ⟶ Q.X p'') |
| (g : Z.X q ⟶ Z.X q') (g' : Z.X q' ⟶ Z.X q'') : |
| (map k A Q Z f' g').comp (map k A Q Z f g) = map k A Q Z (f ≫ f') (g ≫ g') := |
| bimap_comp _ _ _ _ |
|
|
| @[simp] theorem map_zero_left {p p' q q' : ℤ} (g : Z.X q ⟶ Z.X q') : |
| map k A Q Z (0 : Q.X p ⟶ Q.X p') g = 0 := bimap_zero_left _ |
|
|
| @[simp] theorem map_zero_right {p p' q q' : ℤ} (f : Q.X p ⟶ Q.X p') : |
| map k A Q Z f (0 : Z.X q ⟶ Z.X q') = 0 := bimap_zero_right _ |
|
|
| / |
| def column (p : ℤ) : CochainComplex (ModuleCat.{u} k) ℤ := |
| (((fixedTensorFunctor k A (rightTerm k A Q p)).mapHomologicalComplex (.up ℤ)).obj Z)⟦-p⟧ |
|
|
| @[simp] theorem column_d (p n m : ℤ) : |
| ((column k A Q Z p).d n m).hom = |
| (-p).negOnePow • map k A Q Z (𝟙 (Q.X p)) (Z.d (n + -p) (m + -p)) := by |
| apply BalancedTensor.ext |
| intro x y |
| rfl |
|
|
| / |
| def horizontal (n p p' : ℤ) (h : p+1=p') : |
| term k A Q Z p (n + -p) →ₗ[k] term k A Q Z p' ((n+1) + -p') := |
| map k A Q Z (Q.d p p') (eqToHom (congrArg Z.X (by omega))) |
|
|
| theorem horizontal_square (n p p' p'' : ℤ) (h : p+1=p') (h' : p'+1=p'') : |
| (horizontal k A Q Z (n+1) p' p'' h').comp (horizontal k A Q Z n p p' h) = 0 := by |
| unfold horizontal |
| rw [map_comp,Q.d_comp_d,map_zero_left] |
| end LittleFinitistic.ComplexTensor |
|
|
| end |
|
|
| end OAI |
|
|
