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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 | import OAI.Algebra.AuslanderReiten.Resolutions.FiniteCochainResolution
import OAI.Algebra.AuslanderReiten.Homology.ShortExactTriangle
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}
def resolutionCokernelMap (f : M.carrier →ₗ[R] N.carrier) :
CochainCokernel (M.completeCochain τ) →ₗ[R] CochainCokernel (N.completeCochain τ) :=
(N.completeCochainCokernelEquiv τ).symm.toLinearMap.comp
(f.comp (M.completeCochainCokernelEquiv τ).toLinearMap)
def resolutionMap (f : M.carrier →ₗ[R] N.carrier) : M.completeCochain τ ⟶ N.completeCochain τ :=
Classical.choose (cochainCokernelMap_surjective
(M.completeCochain_totallyAcyclic τ).to_chain
(N.completeCochain_totallyAcyclic τ).to_chain.toExactProjectiveComplex
(resolutionCokernelMap τ f))
@[simp]
theorem resolutionMap_cokernel (f : M.carrier →ₗ[R] N.carrier) :
cochainCokernelMap (resolutionMap τ f) = resolutionCokernelMap τ f :=
Classical.choose_spec (cochainCokernelMap_surjective
(M.completeCochain_totallyAcyclic τ).to_chain
(N.completeCochain_totallyAcyclic τ).to_chain.toExactProjectiveComplex
(resolutionCokernelMap τ f))
theorem resolutionCokernelMap_injective (f : M.carrier →ₗ[R] N.carrier)
(hf : Function.Injective f) : Function.Injective (resolutionCokernelMap τ f) :=
(N.completeCochainCokernelEquiv τ).symm.injective.comp
(hf.comp (M.completeCochainCokernelEquiv τ).injective)
theorem resolutionCokernelMap_surjective (f : M.carrier →ₗ[R] N.carrier)
(hf : Function.Surjective f) : Function.Surjective (resolutionCokernelMap τ f) :=
(N.completeCochainCokernelEquiv τ).symm.surjective.comp
(hf.comp (M.completeCochainCokernelEquiv τ).surjective)
theorem resolutionCokernelMap_exact
(f : M.carrier →ₗ[R] N.carrier) (g : N.carrier →ₗ[R] L.carrier)
(h : Function.Exact f g) : Function.Exact (resolutionCokernelMap τ f) (resolutionCokernelMap τ g) := by
intro x
constructor
· intro hx
have hx' : g ((N.completeCochainCokernelEquiv τ) x) = 0 := by
have hh := congrArg (L.completeCochainCokernelEquiv τ) hx
simpa [resolutionCokernelMap] using hh
obtain ⟨m, hm⟩ := (h _).mp hx'
refine ⟨(M.completeCochainCokernelEquiv τ).symm m, ?_⟩
simp [resolutionCokernelMap, hm]
· rintro ⟨m, rfl⟩
change (L.completeCochainCokernelEquiv τ).symm
(g ((N.completeCochainCokernelEquiv τ)
((N.completeCochainCokernelEquiv τ).symm
(f ((M.completeCochainCokernelEquiv τ) m))))) = 0
rw [LinearEquiv.apply_symm_apply, (h _).mpr ⟨_, rfl⟩, map_zero]
theorem exists_resolution_triangle
(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) :
∃ δ : (HomotopyCategory.quotient _ _).obj (L.completeCochain τ) ⟶
((HomotopyCategory.quotient _ _).obj (M.completeCochain τ))⟦(1 : ℤ)⟧,
Triangle.mk ((HomotopyCategory.quotient _ _).map (resolutionMap τ f))
((HomotopyCategory.quotient _ _).map (resolutionMap τ g)) δ ∈
distTriang (HomotopyCategory (ModuleCat.{u} R) (ComplexShape.up ℤ)) := by
let := τ.regular_injective
apply exists_shortExact_distinguished (resolutionMap τ f) (resolutionCokernelMap τ g)
(hF := M.completeCochain_totallyAcyclic τ)
(hG := N.completeCochain_totallyAcyclic τ)
(hH := L.completeCochain_totallyAcyclic τ)
(ψ := resolutionMap τ g)
· rw [resolutionMap_cokernel]
exact resolutionCokernelMap_exact τ f g hex
· rw [resolutionMap_cokernel]
exact resolutionCokernelMap_injective τ f hf
· exact resolutionCokernelMap_surjective τ g hg
· exact resolutionMap_cokernel τ g
end ArExplicit.FiniteLeftModule
end
end OAI
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