File size: 3,316 Bytes
ae00908 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 | 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
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