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2c7d975 | 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 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 | import OAI.Algebra.AuslanderReiten.Modules.FiniteModuleTriangle
import OAI.Algebra.AuslanderReiten.Homology.ShortExactNaturality
namespace OAI
noncomputable section
open CategoryTheory CategoryTheory.Pretriangulated
namespace ArExplicit.FiniteLeftModule
open CompleteComparison
universe u
variable {k R : Type u} [Field k] [Ring R] [Algebra k R] [Module.Finite k R]
variable (τ : FrobeniusTrace k R)
variable {M N L : FiniteLeftModule k R}
theorem resolutionCokernelMap_comp (f : M.carrier →ₗ[R] N.carrier) (g : N.carrier →ₗ[R] L.carrier) :
resolutionCokernelMap τ (g.comp f) = (resolutionCokernelMap τ g).comp (resolutionCokernelMap τ f) := by
apply LinearMap.ext
intro x
simp [resolutionCokernelMap]
@[simp]
theorem resolutionMap_id :
(HomotopyCategory.quotient _ _).map (resolutionMap τ (LinearMap.id : M.carrier →ₗ[R] M.carrier)) = 𝟙 _ := by
apply (cochainHomotopyRawEquiv (M.completeCochain_totallyAcyclic τ).to_chain
(M.completeCochain_totallyAcyclic τ).to_chain.toExactProjectiveComplex).injective
rw [← (HomotopyCategory.quotient (ModuleCat.{u} R) (ComplexShape.up ℤ)).map_id,
cochainHomotopyRawEquiv_map, cochainHomotopyRawEquiv_map, resolutionMap_cokernel]
apply congrArg (StableHom.classOf R _ _)
apply LinearMap.ext
intro x
obtain ⟨x, rfl⟩ := (LinearMap.range ((M.completeCochain τ).d (-1) 0).hom).mkQ_surjective x
simp [resolutionCokernelMap, cochainCokernelMap_mk]
theorem resolutionMap_comp (f : M.carrier →ₗ[R] N.carrier) (g : N.carrier →ₗ[R] L.carrier) :
(HomotopyCategory.quotient _ _).map (resolutionMap τ (g.comp f)) =
(HomotopyCategory.quotient _ _).map (resolutionMap τ f) ≫
(HomotopyCategory.quotient _ _).map (resolutionMap τ g) := by
apply (cochainHomotopyRawEquiv (M.completeCochain_totallyAcyclic τ).to_chain
(L.completeCochain_totallyAcyclic τ).to_chain.toExactProjectiveComplex).injective
rw [← Functor.map_comp, cochainHomotopyRawEquiv_map, cochainHomotopyRawEquiv_map,
cochainCokernelMap_comp, resolutionMap_cokernel, resolutionMap_cokernel, resolutionMap_cokernel,
resolutionCokernelMap_comp]
theorem resolutionMap_surjective : Function.Surjective
(fun f : M.carrier →ₗ[R] N.carrier =>
(HomotopyCategory.quotient (ModuleCat.{u} R) (ComplexShape.up ℤ)).map (resolutionMap τ f)) := by
intro h
let E := cochainHomotopyRawEquiv (M.completeCochain_totallyAcyclic τ).to_chain
(N.completeCochain_totallyAcyclic τ).to_chain.toExactProjectiveComplex
obtain ⟨u, hu⟩ := StableHom.classOf_surjective (E h)
let f := (N.completeCochainCokernelEquiv τ).toLinearMap.comp
(u.comp (M.completeCochainCokernelEquiv τ).symm.toLinearMap)
refine ⟨f, E.injective ?_⟩
change cochainHomotopyRawEquiv _ _ _ = _
rw [cochainHomotopyRawEquiv_map, resolutionMap_cokernel]
have hf : resolutionCokernelMap τ f = u := by
apply LinearMap.ext
intro x
simp [resolutionCokernelMap, f]
rw [hf]
exact hu
variable (f : M.carrier →ₗ[R] N.carrier) (g : N.carrier →ₗ[R] L.carrier)
variable (hf : Function.Injective f) (hg : Function.Surjective g) (hex : Function.Exact f g)
def resolutionTriangle : Triangle (HomotopyCategory (ModuleCat.{u} R) (ComplexShape.up ℤ)) := by
letI := τ.regular_injective
exact shortExactTriangle (resolutionMap τ f) (resolutionCokernelMap τ g)
(by rw [resolutionMap_cokernel]; exact resolutionCokernelMap_exact τ f g hex)
(M.completeCochain_totallyAcyclic τ) (N.completeCochain_totallyAcyclic τ)
(L.completeCochain_totallyAcyclic τ)
(by rw [resolutionMap_cokernel]; exact resolutionCokernelMap_injective τ f hf)
(resolutionCokernelMap_surjective τ g hg)
theorem resolutionTriangle_distinguished :
resolutionTriangle τ f g hf hg hex ∈
distTriang (HomotopyCategory (ModuleCat.{u} R) (ComplexShape.up ℤ)) := by
let := τ.regular_injective
apply shortExactTriangle_distinguished
@[simp]
theorem resolutionTriangle_first :
(resolutionTriangle τ f g hf hg hex).mor₁ =
(HomotopyCategory.quotient _ _).map (resolutionMap τ f) := rfl
theorem resolutionTriangle_second :
(resolutionTriangle τ f g hf hg hex).mor₂ =
(HomotopyCategory.quotient _ _).map (resolutionMap τ g) := by
let := τ.regular_injective
apply shortExactTriangle_second
exact resolutionMap_cokernel τ g
abbrev resolutionConnecting :
(HomotopyCategory.quotient _ _).obj (L.completeCochain τ) ⟶
((HomotopyCategory.quotient _ _).obj (M.completeCochain τ))⟦(1 : ℤ)⟧ :=
(resolutionTriangle τ f g hf hg hex).mor₃
variable {M₁ N₁ L₁ M₂ N₂ L₂ : FiniteLeftModule k R}
variable (f₁ : M₁.carrier →ₗ[R] N₁.carrier) (g₁ : N₁.carrier →ₗ[R] L₁.carrier)
(f₂ : M₂.carrier →ₗ[R] N₂.carrier) (g₂ : N₂.carrier →ₗ[R] L₂.carrier)
variable (hf₁ : Function.Injective f₁) (hg₁ : Function.Surjective g₁) (hex₁ : Function.Exact f₁ g₁)
(hf₂ : Function.Injective f₂) (hg₂ : Function.Surjective g₂) (hex₂ : Function.Exact f₂ g₂)
variable (a : M₁.carrier →ₗ[R] M₂.carrier) (b : N₁.carrier →ₗ[R] N₂.carrier)
(c : L₁.carrier →ₗ[R] L₂.carrier)
theorem resolutionConnecting_natural
(hab : b.comp f₁ = f₂.comp a) (hbc : g₂.comp b = c.comp g₁) :
(HomotopyCategory.quotient _ _).map (resolutionMap τ c) ≫
resolutionConnecting τ f₂ g₂ hf₂ hg₂ hex₂ =
resolutionConnecting τ f₁ g₁ hf₁ hg₁ hex₁ ≫
(CategoryTheory.shiftFunctor (HomotopyCategory (ModuleCat.{u} R) (ComplexShape.up ℤ)) 1).map
((HomotopyCategory.quotient _ _).map (resolutionMap τ a)) := by
let := τ.regular_injective
apply shortExactTriangle_connecting_natural
· rw [resolutionMap_cokernel, ← resolutionCokernelMap_comp,
resolutionMap_cokernel, ← resolutionCokernelMap_comp, hbc]
· rw [cochainCokernelMap_comp, cochainCokernelMap_comp,
resolutionMap_cokernel, resolutionMap_cokernel, resolutionMap_cokernel, resolutionMap_cokernel,
← resolutionCokernelMap_comp, ← resolutionCokernelMap_comp, hab]
end ArExplicit.FiniteLeftModule
end
end OAI
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