File size: 3,137 Bytes
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 | import OAI.Algebra.Finitistic.Bar.DerivedBarCoproduct
import OAI.Algebra.Finitistic.Bar.BalancedTensorFunctor
import Mathlib.Algebra.Homology.Embedding.ExtendHomotopy
namespace OAI
open CategoryTheory CategoryTheory.Limits TrivSqZeroExt
noncomputable section
namespace DerivedBar
universe u
variable (D : Type u) [Ring D]
local instance derivedTensorBarSplittingHasDerivedCategory :
HasDerivedCategory.{u+1} (ModuleCat.{u} D) :=
HasDerivedCategory.standard (ModuleCat.{u} D)
/-- Extension of the nonnegative homotopy category followed by the genuine
integer derived localization. -/
abbrev homotopyToIntegerDerived : HomotopyCategory (ModuleCat.{u} D) (.down ℕ) ⥤
DerivedCategory (ModuleCat.{u} D) :=
ComplexShape.embeddingDownNat.extendHomotopyFunctor (ModuleCat.{u} D) ⋙ DerivedCategory.Qh
def homotopyToIntegerDerivedIso (K : ChainComplex (ModuleCat.{u} D) ℕ) :
(homotopyToIntegerDerived D).obj ((HomotopyCategory.quotient _ _).obj K) ≅
(integerQ D).obj K :=
(DerivedCategory.quotientCompQhIso (ModuleCat.{u} D)).app ((integerExtension D).obj K)
variable {k A : Type u} [Field k] [Ring A] [Algebra k A] [Algebra k D]
(f : A →ₐ[k] D)
/-- The actual left-derived balanced base-change functor on modules, viewed
in the integer derived category. -/
abbrev derivedBaseChange : ModuleCat.{u} A ⥤ DerivedCategory (ModuleCat.{u} D) :=
(Balanced.tensorFunctor f).leftDerivedToHomotopyCategory ⋙ homotopyToIntegerDerived D
def derivedBaseChangeBarIso (N : Type u) [AddCommGroup N] [Module k N]
[Module A N] [IsScalarTower k A N] :
(derivedBaseChange D f).obj (ModuleCat.of A N) ≅
(integerQ D).obj (tensorBarComplex k A D N f) :=
(homotopyToIntegerDerived D).mapIso (Balanced.tensorBarComputesLeftDerived f N) ≪≫
homotopyToIntegerDerivedIso D _
end DerivedBar
namespace DerivedBar.ParityBridge
universe u
variable (k D X : Type u) [Field k] [Ring D] [Algebra k D]
[AddCommGroup X] [Module k X] [Module D X] [Module Dᵐᵒᵖ X]
[SMulCommClass D Dᵐᵒᵖ X] [IsScalarTower k D X] [IsScalarTower k Dᵐᵒᵖ X]
local instance (V : ModuleCat D) : Module k V := BimoduleTensor.scalarModule k D V
local instance (V : ModuleCat D) : IsScalarTower k D V :=
IsScalarTower.of_algebraMap_smul (fun _ _ => rfl)
variable (N : ModuleCat D)
local notation "A" => TrivSqZeroExt D X
local instance : Module A N := inflatedModule D X N
local instance : IsScalarTower k A N := IsScalarTower.of_algebraMap_smul
(fun a v => IsScalarTower.algebraMap_smul D a v)
local instance : HasDerivedCategory.{u+1} (ModuleCat.{u} D) :=
HasDerivedCategory.standard (ModuleCat.{u} D)
/-- The actual left-derived D tensor over A is the actual categorical
coproduct of genuine shifted derived iterates. This proves the manuscript's
bar-splitting proposition; projective-dimension detection is a separate result. -/
def derivedTensorBarSplitting :
(derivedBaseChange D (fstHom k D X)).obj (ModuleCat.of A N) ≅
colimit (shiftedIterateDiagram k D X N) :=
derivedBaseChangeBarIso D (fstHom k D X) N ≪≫ derivedBarSplitting k D X N
end DerivedBar.ParityBridge
end
end OAI
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