| import OAI.Algebra.Finitistic.Bar.CountProjective |
| import OAI.Algebra.Finitistic.Bar.KProjectiveCoproduct |
| import OAI.Algebra.Finitistic.Bar.ExtensionColimits |
|
|
| namespace OAI |
| open CategoryTheory CategoryTheory.Limits HomologicalComplex TrivSqZeroExt |
| noncomputable section |
| namespace DerivedBar |
| universe u |
| variable (k D X N : Type u) [Field k] [Ring D] [Algebra k D] |
| [AddCommGroup X] [Module k X] [Module D X] [Module Dᵐᵒᵖ X] |
| [SMulCommClass D Dᵐᵒᵖ X] [IsScalarTower k D X] [IsScalarTower k Dᵐᵒᵖ X] |
| [AddCommGroup N] [Module k N] |
| [Module (TrivSqZeroExt D X) N] [IsScalarTower k (TrivSqZeroExt D X) N] |
| (hN : ∀ (x : X) (v : N), (inr x : TrivSqZeroExt D X) • v = 0) |
| local instance derivedCountCoproductHasDerivedCategory : |
| HasDerivedCategory.{u+1} (ModuleCat.{u} D) := |
| HasDerivedCategory.standard (ModuleCat.{u} D) |
| local notation "S" => SquareZero.twoSidedWeightComplex k D X N hN |
|
|
| /-- The original count direct sum, with its original inclusions, is an actual |
| coproduct after passing to the genuine integer derived category. -/ |
| def derivedCountCoproductIsColimit : |
| IsColimit ((integerQ D).mapCocone (directSumComplexCocone D S)) := by |
| letI : PreservesColimit (Discrete.functor S) (integerExtension D) := |
| extensionPreservesColimit ComplexShape.embeddingDownNat (Discrete.functor S) |
| let s := (integerExtension D).mapCocone (directSumComplexCocone D S) |
| let hs := isColimitOfPreserves (integerExtension D) (directSumComplexIsColimit D S) |
| letI : s.pt.IsKProjective := integerCountSum_isKProjective k D X N hN |
| letI (r : Discrete ℕ) : (((Discrete.functor S) ⋙ integerExtension D).obj r).IsKProjective := |
| integerCount_isKProjective k D X N hN r.as |
| exact kProjectiveCoproductIsColimit ((Discrete.functor S) ⋙ integerExtension D) s hs |
| end DerivedBar |
|
|
| end |
|
|
| end OAI |
|
|
