openAI-math-dataset / lean /OAI /Algebra /AuslanderReiten /Complete /ProjectiveCompleteResolution.lean
| 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 | |