| import OAI.Algebra.Finitistic.Ordinary.OrdinaryFirstDerivedCompatibility |
| import OAI.Algebra.Finitistic.Ordinary.OrdinaryFirstComponentExact |
|
|
| namespace OAI |
| noncomputable section |
| open CategoryTheory CategoryTheory.Limits |
| namespace LittleFinitistic.BlockModule |
| universe u |
| variable (L R : Type u) [Ring L] [Ring R] |
| variable [HasDerivedCategory.{u+1} (ModuleCat.{u} L)] |
| variable [HasDerivedCategory.{u+1} (ModuleCat.{u} (L × R))] |
| |
| /-- The exact derived first-component functor on the whole product-ring derived category. -/ |
| abbrev derivedFirstComponent := (ProductModules.firstFunctor.{u,u} L R).mapDerivedCategory |
|
|
| def firstComplexComponentIso : firstComplexFunctor L R ⋙ |
| (ProductModules.firstFunctor L R).mapHomologicalComplex (.up ℤ) ≅ |
| 𝟭 (CochainComplex (ModuleCat.{u} L) ℤ) := |
| Functor.mapHomologicalComplexCompIso (ProductModules.firstInflationComponentIso L R) (.up ℤ) ≪≫ |
| Functor.mapHomologicalComplexIdIso (ModuleCat.{u} L) (.up ℤ) |
|
|
| /-- On chain representatives, the composite is the identity localization. -/ |
| def derivedFirstComponentFactors : DerivedCategory.Q ⋙ derivedFirst L R ⋙ |
| derivedFirstComponent L R ≅ (DerivedCategory.Q : CochainComplex (ModuleCat.{u} L) ℤ ⥤ _) := |
| (Functor.associator ..).symm ≪≫ |
| Functor.isoWhiskerRight (derivedFirstComparison L R) _ ≪≫ |
| Functor.associator .. ≪≫ |
| Functor.isoWhiskerLeft _ (ProductModules.firstFunctor L R).mapDerivedCategoryFactors ≪≫ |
| (Functor.associator ..).symm ≪≫ |
| Functor.isoWhiskerRight (firstComplexComponentIso L R) _ ≪≫ Functor.leftUnitor _ |
|
|
| /-- Taking the first component recovers every derived object, naturally. -/ |
| def derivedFirstRetraction : derivedFirst L R ⋙ derivedFirstComponent L R ≅ |
| 𝟭 (DerivedCategory (ModuleCat.{u} L)) := by |
| letI : Localization.Lifting DerivedCategory.Q |
| (HomologicalComplex.quasiIso (ModuleCat.{u} L) (.up ℤ)) DerivedCategory.Q |
| (derivedFirst L R ⋙ derivedFirstComponent L R) := ⟨derivedFirstComponentFactors L R⟩ |
| exact Localization.liftNatIso DerivedCategory.Q |
| (HomologicalComplex.quasiIso (ModuleCat.{u} L) (.up ℤ)) |
| DerivedCategory.Q DerivedCategory.Q _ _ (Iso.refl _) |
|
|
| /-- The same retraction for mathlib's exact derived inflation. -/ |
| def exactFirstRetraction : (ProductModules.firstInflation L R).mapDerivedCategory ⋙ |
| derivedFirstComponent L R ≅ 𝟭 (DerivedCategory (ModuleCat.{u} L)) := |
| Functor.isoWhiskerRight (derivedFirstIso L R).symm _ ≪≫ derivedFirstRetraction L R |
|
|
| instance derivedFirst_additive : (derivedFirst L R).Additive := |
| Functor.additive_of_iso (derivedFirstIso L R).symm |
|
|
| /-- No derived object becomes zero merely by putting it in the first block. -/ |
| theorem derivedFirst_isZero_iff (Y : DerivedCategory (ModuleCat.{u} L)) : |
| IsZero ((derivedFirst L R).obj Y) ↔ IsZero Y := by |
| constructor |
| · intro h |
| exact ((derivedFirstRetraction L R).app Y).isZero_iff.mp |
| ((derivedFirstComponent L R).map_isZero h) |
| · exact (derivedFirst L R).map_isZero |
|
|
| theorem exactFirst_isZero_iff (Y : DerivedCategory (ModuleCat.{u} L)) : |
| IsZero ((ProductModules.firstInflation L R).mapDerivedCategory.obj Y) ↔ IsZero Y := |
| ((derivedFirstIso L R).app Y).isZero_iff.symm.trans (derivedFirst_isZero_iff L R Y) |
| end LittleFinitistic.BlockModule |
|
|
| end |
|
|
| end OAI |
|
|
