| import OAI.Algebra.Finitistic.Ordinary.OrdinaryBimoduleTotalFunctor |
| import OAI.Algebra.Finitistic.Ordinary.OrdinaryBimoduleResolutionProjective |
| import Mathlib.Algebra.Homology.DerivedCategory.Basic |
|
|
| namespace OAI |
| noncomputable section |
| open CategoryTheory |
| open scoped TensorProduct |
| namespace LittleFinitistic.BimoduleComplexTensor |
| universe u |
| variable (k S A : Type u) [Field k] [Ring S] [Ring A] [Algebra k S] [Algebra k A] |
| variable (Q : CochainComplex (ModuleCat.{u} (S ⊗[k] Aᵐᵒᵖ)) ℤ) |
| variable {l : ℕ} (p : Fin l → ℤ) (hp : StrictMono p) |
| variable [HasDerivedCategory.{u+1} (ModuleCat.{u} A)] |
| variable [HasDerivedCategory.{u+1} (ModuleCat.{u} S)] |
| variable [∀ i, Module.Projective Aᵐᵒᵖ (ComplexTensor.rightTerm k A (rightComplex k S A Q) (p i))] |
|
|
| omit [HasDerivedCategory.{u+1} (ModuleCat.{u} A)] in |
| theorem total_inverts : (HomologicalComplex.quasiIso (ModuleCat.{u} A) (.up ℤ)).IsInvertedBy |
| (totalFunctor k S A Q p hp ⋙ DerivedCategory.Q) := by |
| intro Z Z' f hf |
| let : QuasiIso f := hf |
| let := totalFunctor_quasiIso k S A Q p hp f |
| exact (inferInstance : IsIso (DerivedCategory.Q.map ((totalFunctor k S A Q p hp).map f))) |
|
|
| /-- Localization of the actual tensor total, retaining the outside action. -/ |
| def derivedTotal : DerivedCategory (ModuleCat.{u} A) ⥤ DerivedCategory (ModuleCat.{u} S) := |
| Localization.lift (totalFunctor k S A Q p hp ⋙ DerivedCategory.Q) |
| (total_inverts k S A Q p hp) DerivedCategory.Q |
|
|
| def derivedTotalComparison : |
| DerivedCategory.Q ⋙ derivedTotal k S A Q p hp ≅ totalFunctor k S A Q p hp ⋙ DerivedCategory.Q := |
| Localization.fac (totalFunctor k S A Q p hp ⋙ DerivedCategory.Q) |
| (total_inverts k S A Q p hp) DerivedCategory.Q |
| end LittleFinitistic.BimoduleComplexTensor |
| namespace LittleFinitistic.OrdinarySimulation.TensorResolution |
| universe u |
| variable (k B : Type u) [Field k] [Ring B] [Algebra k B] (l : ℕ) |
| variable [HasDerivedCategory.{u+1} (ModuleCat.{u} (B ⊗[k] ChainAlgebra k l))] |
| variable [HasDerivedCategory.{u+1} (ModuleCat.{u} B)] |
|
|
| /-- The actual Q_T tensor lift with the source's left B-action retained. -/ |
| def derivedBimoduleTensor : DerivedCategory (ModuleCat.{u} (B ⊗[k] ChainAlgebra k l)) ⥤ |
| DerivedCategory (ModuleCat.{u} B) := |
| BimoduleComplexTensor.derivedTotal k B (B ⊗[k] ChainAlgebra k l) (integerBimoduleComplex k B l) |
| (finiteDegrees l) (finiteDegrees_strictMono l) |
|
|
| def derivedBimoduleTensorComparison : |
| DerivedCategory.Q ⋙ derivedBimoduleTensor k B l ≅ |
| BimoduleComplexTensor.totalFunctor k B (B ⊗[k] ChainAlgebra k l) (integerBimoduleComplex k B l) |
| (finiteDegrees l) (finiteDegrees_strictMono l) ⋙ DerivedCategory.Q := |
| BimoduleComplexTensor.derivedTotalComparison k B (B ⊗[k] ChainAlgebra k l) (integerBimoduleComplex k B l) |
| (finiteDegrees l) (finiteDegrees_strictMono l) |
| end LittleFinitistic.OrdinarySimulation.TensorResolution |
|
|
| end |
|
|
| end OAI |
|
|
