| 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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|
