File size: 3,294 Bytes
216fb31 | 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 68 69 70 | 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
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