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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 88 89 90 91 92 93 94 | import OAI.Algebra.Finitistic.Dimension.PDBarQuotientAction
import OAI.Algebra.Finitistic.Dimension.PDBarDetection
import OAI.Algebra.Finitistic.Bar.TensorModel
namespace OAI
noncomputable section
open CategoryTheory
open scoped TensorProduct
namespace LittleFinitistic.PDBarQuotient
universe u
variable {k A D : Type u} [Field k] [Ring A] [Algebra k A] [Ring D] [Algebra k D]
variable (f : A →ₐ[k] D)
variable (V : Type u) [AddCommGroup V] [Module k V] [Module A V] [IsScalarTower k A V]
/-- The restricted scalar convention in the detector agrees with the original
compatible k action by actual tensor maps in both directions. -/
def scalarEquiv :
(PDTensor.tensorFunctor k A (RightBase f)).obj (ModuleCat.of A V) ≃ₗ[k]
BalancedTensor.Tensor k A (RightBase f) V :=
LinearEquiv.ofLinearMap
(PDTensor.mapRight (k := k) (M := PDTensor.ScalarModule k A (ModuleCat.of A V))
(N := V) (RightBase f) LinearMap.id)
(PDTensor.mapRight (k := k) (M := V)
(N := PDTensor.ScalarModule k A (ModuleCat.of A V)) (RightBase f) LinearMap.id)
(by apply BalancedTensor.ext; intros; rfl)
(by apply BalancedTensor.ext; intros; rfl)
def nativeScalarEquiv : DerivedBar.Balanced.tensor f V ≃ₗ[k]
(ModuleCat.restrictScalars (algebraMap k D)).obj
(ModuleCat.of D (DerivedBar.Balanced.tensor f V)) where
toFun := id
invFun := id
left_inv := fun _ => rfl
right_inv := fun _ => rfl
map_add' := fun _ _ => rfl
map_smul' c z := (IsScalarTower.algebraMap_smul D c z).symm
def objectIso :
(PDTensor.tensorFunctor k A (RightBase f)).obj (ModuleCat.of A V) ≅
(ModuleCat.restrictScalars (algebraMap k D)).obj
(ModuleCat.of D (DerivedBar.Balanced.tensor f V)) :=
(((scalarEquiv f V).trans (equiv f V)).trans (nativeScalarEquiv f V)).toModuleIso
variable {V} {W : Type u} [AddCommGroup W] [Module k W] [Module A W] [IsScalarTower k A W]
theorem objectIso_natural (h : V →ₗ[A] W) :
(PDTensor.tensorFunctor k A (RightBase f)).map (ModuleCat.ofHom h) ≫ (objectIso f W).hom =
(objectIso f V).hom ≫ (ModuleCat.restrictScalars (algebraMap k D)).map
(ModuleCat.ofHom (DerivedBar.Balanced.map f V W h)) := by
apply ModuleCat.hom_ext
apply BalancedTensor.ext
intro d v
rfl
variable (N : Type u) [AddCommGroup N] [Module k N] [Module A N] [IsScalarTower k A N]
/-- The detector's tensor of the actual bar resolution is the bar worker's
balanced complex, after restricting only its exterior scalars. -/
def barQuotientIso :
((PDTensor.tensorFunctor k A (RightBase f)).mapHomologicalComplex _).obj
(DerivedBar.projectiveResolution k A N).complex ≅
((ModuleCat.restrictScalars (algebraMap k D)).mapHomologicalComplex _).obj
(DerivedBar.tensorBarComplex k A D N f) :=
HomologicalComplex.Hom.isoOfComponents
(fun n => objectIso f (DerivedBar.term k A N (n+1))) (by
intro i j hij
change j+1=i at hij
subst i
simpa only [DerivedBar.tensorBarComplex, DerivedBar.projectiveResolution, DerivedBar.complex, ChainComplex.of_d, Functor.mapHomologicalComplex_obj_d] using (objectIso_natural f (DerivedBar.differential k A N (j+1))).symm)
/-- Comparison to the literal two-sided bar complex used in the splitting. -/
def twoSidedIso :
((PDTensor.tensorFunctor k A (RightBase f)).mapHomologicalComplex _).obj
(DerivedBar.projectiveResolution k A N).complex ≅
((ModuleCat.restrictScalars (algebraMap k D)).mapHomologicalComplex _).obj
(DerivedBar.twoSidedComplex k A D N f) :=
barQuotientIso f N ≪≫
((ModuleCat.restrictScalars (algebraMap k D)).mapHomologicalComplex _).mapIso
(DerivedBar.tensorBarIso k A D N f)
variable [FiniteDimensional k A] [FiniteDimensional k N]
/-- Tor computed by the literal two-sided bar, with no assumed quotient comparison. -/
def torIsoTwoSided (n : ℕ) :
PDTensor.Tor (k := k) (RightBase f) (ModuleCat.of A N) n ≅
(((ModuleCat.restrictScalars (algebraMap k D)).mapHomologicalComplex _).obj
(DerivedBar.twoSidedComplex k A D N f)).homology n :=
(DerivedBar.projectiveResolution k A N).isoLeftDerivedObj
(PDTensor.tensorFunctor k A (RightBase f)) n ≪≫
(HomologicalComplex.homologyFunctor _ _ n).mapIso (twoSidedIso f N)
end LittleFinitistic.PDBarQuotient
end
end OAI
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