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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 | import OAI.Algebra.Finitistic.Bar.TwoSidedBarFunctor
namespace OAI
open CategoryTheory TensorProduct
open scoped TensorProduct
noncomputable section
namespace DerivedBar
universe u
variable (k D : Type u) [Field k] [Ring D] [Algebra k D]
variable (V W : Type u) [AddCommGroup V] [Module k V] [Module D V] [IsScalarTower k D V]
[AddCommGroup W] [Module k W] [Module D W] [IsScalarTower k D W]
/-- Each iterated free bar term preserves addition of maps. -/
theorem termMap_add (f g : V →ₗ[D] W) (n : ℕ) :
termMap k D V W (f+g) n = termMap k D V W f n + termMap k D V W g n := by
induction n with
| zero => rfl
| succ n ih =>
apply AlgebraTensorModule.ext
intro d v
change d ⊗ₜ[k] termMap k D V W (f+g) n v =
d ⊗ₜ[k] termMap k D V W f n v + d ⊗ₜ[k] termMap k D V W g n v
rw [ih, LinearMap.add_apply, TensorProduct.tmul_add]
local instance barAdditiveScalarModule (N : ModuleCat.{u} D) : Module k N :=
BimoduleTensor.scalarModule k D N
local instance barAdditiveScalarTower (N : ModuleCat.{u} D) : IsScalarTower k D N :=
IsScalarTower.of_algebraMap_smul (fun _ _ => rfl)
instance barFunctor_additive : (barFunctor k D).Additive where
map_add := by
intro V W f g
apply HomologicalComplex.Hom.ext
funext n
apply ModuleCat.hom_ext
exact termMap_add k D V W f.hom g.hom (n+1)
namespace BimoduleTensor
variable (X : Type u) [AddCommGroup X] [Module k X] [Module D X] [Module Dᵐᵒᵖ X]
[SMulCommClass D Dᵐᵒᵖ X] [IsScalarTower k D X] [IsScalarTower k Dᵐᵒᵖ X]
instance twoSidedBarFunctor_additive : (twoSidedBarFunctor k D X).Additive where
map_add := by
intro V W f g
apply HomologicalComplex.Hom.ext
funext n
apply ModuleCat.hom_ext
apply AlgebraTensorModule.ext
intro x v
change x ⊗ₜ[k] termMap k D V W (f.hom+g.hom) n v =
x ⊗ₜ[k] termMap k D V W f.hom n v + x ⊗ₜ[k] termMap k D V W g.hom n v
rw [termMap_add, LinearMap.add_apply, TensorProduct.tmul_add]
end BimoduleTensor
end DerivedBar
end
end OAI
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