| import OAI.Algebra.Finitistic.Ordinary.OrdinaryBlockComplexes |
| import OAI.Algebra.Finitistic.Ordinary.OrdinaryRestrictionHomology |
| import Mathlib.Algebra.Homology.DerivedCategory.Basic |
|
|
| namespace OAI |
| noncomputable section |
| open CategoryTheory |
| 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))] |
|
|
| abbrev firstComplexFunctor := (ProductModules.firstInflation L R).mapHomologicalComplex (.up ℤ) |
|
|
| omit [HasDerivedCategory.{u+1} (ModuleCat.{u} L)] in |
| theorem first_inverts : (HomologicalComplex.quasiIso (ModuleCat.{u} L) (.up ℤ)).IsInvertedBy |
| (firstComplexFunctor L R ⋙ DerivedCategory.Q) := by |
| intro Z Z' f hf |
| let : QuasiIso f := hf |
| have : QuasiIso ((firstComplexFunctor L R).map f) := |
| (RestrictionHomology.quasiIso_iff (RingHom.fst L R) f).mpr inferInstance |
| exact (inferInstance : IsIso (DerivedCategory.Q.map ((firstComplexFunctor L R).map f))) |
|
|
| /-- Native derived inflation on the first block of the product algebra. -/ |
| def derivedFirst : DerivedCategory (ModuleCat.{u} L) ⥤ DerivedCategory (ModuleCat.{u} (L × R)) := |
| Localization.lift (firstComplexFunctor L R ⋙ DerivedCategory.Q) (first_inverts L R) DerivedCategory.Q |
|
|
| def derivedFirstComparison : DerivedCategory.Q ⋙ derivedFirst L R ≅ firstComplexFunctor L R ⋙ DerivedCategory.Q := |
| Localization.fac (firstComplexFunctor L R ⋙ DerivedCategory.Q) (first_inverts L R) DerivedCategory.Q |
| end LittleFinitistic.BlockModule |
|
|
| end |
|
|
| end OAI |
|
|
