File size: 2,528 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 | 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
|