| 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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|
