| import OAI.Algebra.Finitistic.Bar.DerivedCountShift |
| import OAI.Algebra.Finitistic.Bar.DirectSumCoproduct |
| import Mathlib.LinearAlgebra.DirectSum.Basis |
| import Mathlib.Algebra.Homology.HomotopyCategory.KProjective |
|
|
| namespace OAI |
| open CategoryTheory CategoryTheory.Limits TensorProduct TrivSqZeroExt |
| open scoped TensorProduct DirectSum |
| 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] |
| open Multibar.Indexed |
| local instance countProjectiveShapeDecidableEq (n r : ℕ) : DecidableEq (Shape n r) := |
| Classical.decEq _ |
| |
| instance outerTensor_free {n r : ℕ} (p : Shape n r) : |
| Module.Free D (outerTensor k D X N p) := |
| Module.Free.of_basis (Algebra.TensorProduct.basis D |
| (Module.Free.chooseBasis k (tensor k D X N p))) |
| |
| instance countWeight_free (n r : ℕ) : |
| Module.Free D (SquareZero.twoSidedWeight k D X N n r) := |
| Module.Free.of_equiv (decomposition k D X N n r) |
| |
| variable [Module (TrivSqZeroExt D X) N] [IsScalarTower k (TrivSqZeroExt D X) N] |
| (hN : ∀ (x : X) (v : N), (inr x : TrivSqZeroExt D X) • v = 0) |
| |
| instance countComplex_projective (r n : ℕ) : |
| Projective ((SquareZero.twoSidedWeightComplex k D X N hN r).X n) := by |
| change Projective (ModuleCat.of D (SquareZero.twoSidedWeight k D X N n r)) |
| infer_instance |
| |
| instance countSum_projective (n : ℕ) : |
| Projective ((directSumComplex (SquareZero.twoSidedWeightComplex k D X N hN)).X n) := by |
| classical |
| let (r : ℕ) : Module.Free D ((SquareZero.twoSidedWeightComplex k D X N hN r).X n) := |
| countWeight_free k D X N n r |
| change Projective (ModuleCat.of D (⨁ r : ℕ, (SquareZero.twoSidedWeightComplex k D X N hN r).X n)) |
| infer_instance |
| |
| instance integerCount_isKProjective (r : ℕ) : |
| ((integerExtension D).obj (SquareZero.twoSidedWeightComplex k D X N hN r)).IsKProjective := by |
| change CochainComplex.IsKProjective ((SquareZero.twoSidedWeightComplex k D X N hN r).extend |
| ComplexShape.embeddingDownNat) |
| infer_instance |
| |
| instance integerCountSum_isKProjective : |
| ((integerExtension D).obj (directSumComplex |
| (SquareZero.twoSidedWeightComplex k D X N hN))).IsKProjective := by |
| change CochainComplex.IsKProjective ((directSumComplex (SquareZero.twoSidedWeightComplex k D X N hN)).extend |
| ComplexShape.embeddingDownNat) |
| infer_instance |
| end DerivedBar |
| |
| end |
| |
| end OAI |
| |
