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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 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 | import OAI.Algebra.Finitistic.Bar.TensorModel
import OAI.Algebra.Finitistic.Bimodule.BimoduleComparison
namespace OAI
open CategoryTheory TensorProduct
open scoped TensorProduct
noncomputable section
namespace DerivedBar.Balanced
universe u
variable {k A D : Type u} [Field k] [Ring A] [Algebra k A] [Ring D] [Algebra k D]
(f : A →ₐ[k] D)
local instance (V : ModuleCat.{u} A) : Module k V := BimoduleTensor.scalarModule k A V
local instance (V : ModuleCat.{u} A) : IsScalarTower k A V :=
IsScalarTower.of_algebraMap_smul (fun _ _ => rfl)
def tensorFunctor : ModuleCat.{u} A ⥤ ModuleCat.{u} D where
obj V := ModuleCat.of D (tensor f V)
map h := ModuleCat.ofHom (map f _ _ h.hom)
map_id V := ModuleCat.hom_ext (map_id f V)
map_comp h g := ModuleCat.hom_ext (map_comp f _ _ _ h.hom g.hom)
instance tensorFunctor_additive : (tensorFunctor f).Additive where
map_add := by
intro V W h g
apply ModuleCat.hom_ext
apply (relations f V).linearMap_qext
apply AlgebraTensorModule.ext
intro d v
change quotient f W (d ⊗ₜ[k] (h.hom v + g.hom v)) =
quotient f W (d ⊗ₜ[k] h.hom v) + quotient f W (d ⊗ₜ[k] g.hom v)
rw [TensorProduct.tmul_add, map_add]
def tensorFunctorObjIso (V : Type u) [AddCommGroup V] [Module k V]
[Module A V] [IsScalarTower k A V] :
(tensorFunctor f).obj (ModuleCat.of A V) ≅ ModuleCat.of D (tensor f V) := by
have h := BimoduleTensor.scalarModule_eq k A V
cases h
exact Iso.refl _
theorem tensorFunctorObjIso_natural (V W : Type u)
[AddCommGroup V] [Module k V] [Module A V] [IsScalarTower k A V]
[AddCommGroup W] [Module k W] [Module A W] [IsScalarTower k A W]
(h : V →ₗ[A] W) :
(tensorFunctor f).map (ModuleCat.ofHom h) ≫ (tensorFunctorObjIso f W).hom =
(tensorFunctorObjIso f V).hom ≫ ModuleCat.ofHom (map f V W h) := by
have hV := BimoduleTensor.scalarModule_eq k A V
have hW := BimoduleTensor.scalarModule_eq k A W
cases hV
cases hW
rfl
variable (N : Type u) [AddCommGroup N] [Module k N] [Module A N] [IsScalarTower k A N]
def resolvedTensorIso :
((tensorFunctor f).mapHomologicalComplex (.down ℕ)).obj
(DerivedBar.projectiveResolution k A N).complex ≅ tensorBarComplex k A D N f :=
HomologicalComplex.Hom.isoOfComponents
(fun n => tensorFunctorObjIso f (DerivedBar.term k A N (n+1)))
(by
intro i j hij
change j+1=i at hij
subst i
simpa only [tensorBarComplex, DerivedBar.projectiveResolution,
DerivedBar.complex, ChainComplex.of_d, Functor.mapHomologicalComplex_obj_d]
using (tensorFunctorObjIso_natural f (DerivedBar.term k A N (j+1+1))
(DerivedBar.term k A N (j+1)) (DerivedBar.differential k A N (j+1))).symm)
/-- The left-derived balanced tensor functor is computed by exactly the
original tensorBarComplex, via the proved free bar projective resolution. -/
def tensorBarComputesLeftDerived :
(tensorFunctor f).leftDerivedToHomotopyCategory.obj (ModuleCat.of A N) ≅
(HomotopyCategory.quotient (ModuleCat.{u} D) (.down ℕ)).obj (tensorBarComplex k A D N f) :=
(DerivedBar.projectiveResolution k A N).isoLeftDerivedToHomotopyCategoryObj (tensorFunctor f) ≪≫
(HomotopyCategory.quotient (ModuleCat.{u} D) (.down ℕ)).mapIso (resolvedTensorIso f N)
def tensorBarHomologyIso (i : ℕ) :
((tensorFunctor f).leftDerived i).obj (ModuleCat.of A N) ≅
(tensorBarComplex k A D N f).homology i :=
(DerivedBar.projectiveResolution k A N).isoLeftDerivedObj (tensorFunctor f) i ≪≫
(HomologicalComplex.homologyFunctor (ModuleCat.{u} D) (.down ℕ) i).mapIso (resolvedTensorIso f N)
end DerivedBar.Balanced
end
end OAI
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