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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 | 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
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