File size: 3,383 Bytes
d7f9a33 | 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 71 72 73 74 75 76 77 78 | import OAI.Algebra.AuslanderReiten.Modules.FiniteModuleNaturality
namespace OAI
noncomputable section
open CategoryTheory CategoryTheory.Pretriangulated
namespace ArExplicit
open CompleteComparison
universe u
variable {R : Type u} [Ring R]
theorem completeCochain_isZero_of_projectiveCokernel
(K : CochainComplex (ModuleCat.{u} R) ℤ) (hK : TotallyAcyclicCochain K)
[Module.Projective R (CochainCokernel K)] :
Limits.IsZero ((HomotopyCategory.quotient _ _).obj K) := by
let := hK.finite 0
let : Module.Finite R (CochainCokernel K) := inferInstance
rw [Limits.IsZero.iff_id_eq_zero]
apply (cochainHomotopyRawEquiv hK.to_chain hK.to_chain.toExactProjectiveComplex).injective
rw [map_zero]
rw [← (HomotopyCategory.quotient (ModuleCat.{u} R) (ComplexShape.up ℤ)).map_id,
cochainHomotopyRawEquiv_map]
exact StableHom.eq_zero_of_projective_source _
namespace FiniteLeftModule
variable {k : Type u} [Field k] [Algebra k R] [Module.Finite k R]
variable (τ : FrobeniusTrace k R)
variable (M : FiniteLeftModule k R)
theorem projective_completeCochain_isZero [Module.Projective R M.carrier] :
Limits.IsZero ((HomotopyCategory.quotient _ _).obj (M.completeCochain τ)) := by
let : Module.Projective R (CochainCokernel (M.completeCochain τ)) :=
Module.Projective.of_equiv (M.completeCochainCokernelEquiv τ).symm
exact completeCochain_isZero_of_projectiveCokernel _ (M.completeCochain_totallyAcyclic τ)
variable {M} {N L : FiniteLeftModule k R}
theorem nonempty_projectiveMiddleShiftIso [Module.Projective R N.carrier]
(f : M.carrier →ₗ[R] N.carrier) (g : N.carrier →ₗ[R] L.carrier)
(hf : Function.Injective f) (hg : Function.Surjective g) (hex : Function.Exact f g) :
Nonempty (((HomotopyCategory.quotient _ _).obj (L.completeCochain τ)) ≅
((HomotopyCategory.quotient _ _).obj (M.completeCochain τ))⟦(1 : ℤ)⟧) := by
obtain ⟨δ, hδ⟩ := exists_resolution_triangle τ f g hf hg hex
let T := Triangle.mk ((HomotopyCategory.quotient _ _).map (resolutionMap τ f))
((HomotopyCategory.quotient _ _).map (resolutionMap τ g)) δ
have hz : Limits.IsZero T.obj₂ := N.projective_completeCochain_isZero τ
have : IsIso δ := (T.isZero₂_iff_isIso₃ hδ).mp hz
exact ⟨asIso δ⟩
def projectiveMiddleShiftIso [Module.Projective R N.carrier]
(f : M.carrier →ₗ[R] N.carrier) (g : N.carrier →ₗ[R] L.carrier)
(hf : Function.Injective f) (hg : Function.Surjective g) (hex : Function.Exact f g) :
((HomotopyCategory.quotient _ _).obj (L.completeCochain τ)) ≅
((HomotopyCategory.quotient _ _).obj (M.completeCochain τ))⟦(1 : ℤ)⟧ := by
let T := resolutionTriangle τ f g hf hg hex
have hz : Limits.IsZero T.obj₂ := N.projective_completeCochain_isZero τ
haveI : IsIso (resolutionConnecting τ f g hf hg hex) :=
(T.isZero₂_iff_isIso₃ (resolutionTriangle_distinguished τ f g hf hg hex)).mp hz
exact asIso (resolutionConnecting τ f g hf hg hex)
@[simp]
theorem projectiveMiddleShiftIso_hom [Module.Projective R N.carrier]
(f : M.carrier →ₗ[R] N.carrier) (g : N.carrier →ₗ[R] L.carrier)
(hf : Function.Injective f) (hg : Function.Surjective g) (hex : Function.Exact f g) :
(projectiveMiddleShiftIso τ f g hf hg hex).hom = resolutionConnecting τ f g hf hg hex := rfl
end FiniteLeftModule
end ArExplicit
end
end OAI
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