| 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 |
|
|
