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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 | 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)
/-- The balanced map for actual maps between terms of the two input complexes. -/
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 _
/-- The column at right degree p, including the standard total-complex sign (-1)^p. -/
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
/-- Horizontal tensor maps reindex the unchanged left factor along the total degree. -/
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
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