openAI-math-dataset / lean /OAI /Algebra /Finitistic /Ordinary /OrdinaryFirstDerivedCompatibility.lean
| import OAI.Algebra.Finitistic.Ordinary.OrdinaryBlockDerived | |
| import Mathlib.Algebra.Homology.DerivedCategory.ExactFunctor | |
| namespace OAI | |
| noncomputable section | |
| open CategoryTheory CategoryTheory.Limits | |
| namespace LittleFinitistic.BlockModule | |
| universe u | |
| variable (L R : Type u) [Ring L] [Ring R] | |
| /-- First-block inflation is exact for arbitrary, possibly noncommutative rings. -/ | |
| instance firstInflation_preservesFiniteLimits : | |
| PreservesFiniteLimits (ProductModules.firstInflation.{u,u} L R) := by | |
| let := RestrictionHomology.preservesHomology (RingHom.fst L R) | |
| exact Functor.preservesFiniteLimits_of_preservesHomology _ | |
| instance firstInflation_preservesFiniteColimits : | |
| PreservesFiniteColimits (ProductModules.firstInflation.{u,u} L R) := by | |
| let := RestrictionHomology.preservesHomology (RingHom.fst L R) | |
| exact Functor.preservesFiniteColimits_of_preservesHomology _ | |
| variable [HasDerivedCategory.{u+1} (ModuleCat.{u} L)] | |
| variable [HasDerivedCategory.{u+1} (ModuleCat.{u} (L × R))] | |
| instance derivedFirst_lifting : Localization.Lifting DerivedCategory.Q | |
| (HomologicalComplex.quasiIso (ModuleCat.{u} L) (.up ℤ)) | |
| (firstComplexFunctor L R ⋙ DerivedCategory.Q) (derivedFirst L R) := | |
| ⟨derivedFirstComparison L R⟩ | |
| /-- The ordinary localization model is naturally the exact derived inflation. -/ | |
| def derivedFirstIso : derivedFirst L R ≅ (ProductModules.firstInflation L R).mapDerivedCategory := | |
| Localization.liftNatIso DerivedCategory.Q | |
| (HomologicalComplex.quasiIso (ModuleCat.{u} L) (.up ℤ)) | |
| (firstComplexFunctor L R ⋙ DerivedCategory.Q) | |
| (firstComplexFunctor L R ⋙ DerivedCategory.Q) | |
| (derivedFirst L R) (ProductModules.firstInflation L R).mapDerivedCategory (Iso.refl _) | |
| /-- The comparison agrees with both specified localization comparison maps. -/ | |
| @[reassoc (attr := simp)] | |
| theorem derivedFirstIso_hom_app_Q (Z : CochainComplex (ModuleCat.{u} L) ℤ) : | |
| (derivedFirstIso L R).hom.app (DerivedCategory.Q.obj Z) ≫ | |
| (ProductModules.firstInflation L R).mapDerivedCategoryFactors.hom.app Z = | |
| (derivedFirstComparison L R).hom.app Z := by | |
| simp only [derivedFirstIso, Localization.liftNatIso_hom, | |
| Localization.liftNatTrans_app, Iso.refl_hom, NatTrans.id_app, Category.id_comp] | |
| change ((derivedFirstComparison L R).hom.app Z ≫ | |
| (ProductModules.firstInflation L R).mapDerivedCategoryFactors.inv.app Z) ≫ | |
| (ProductModules.firstInflation L R).mapDerivedCategoryFactors.hom.app Z = _ | |
| simp | |
| /-- Transport the coherent integer shift structure of the exact derived functor. -/ | |
| instance derivedFirst_commShift : (derivedFirst L R).CommShift ℤ := | |
| Functor.CommShift.ofIso (derivedFirstIso L R).symm ℤ | |
| instance derivedFirstIso_inv_commShift : NatTrans.CommShift (derivedFirstIso L R).inv ℤ := | |
| Functor.CommShift.ofIso_compatibility (derivedFirstIso L R).symm ℤ | |
| instance derivedFirstIso_hom_commShift : NatTrans.CommShift (derivedFirstIso L R).hom ℤ := by | |
| let : NatTrans.CommShift (derivedFirstIso L R).symm.hom ℤ := | |
| derivedFirstIso_inv_commShift L R | |
| exact NatTrans.CommShift.of_iso_inv (derivedFirstIso L R).symm ℤ | |
| instance derivedFirst_isTriangulated : (derivedFirst L R).IsTriangulated := | |
| Functor.isTriangulated_of_iso (derivedFirstIso L R).symm | |
| end LittleFinitistic.BlockModule | |
| end | |
| end OAI | |