openAI-math-dataset / lean /OAI /Algebra /AuslanderReiten /Complete /CompleteFunctorCokernelAction.lean
| import OAI.Algebra.AuslanderReiten.Complete.CompleteFunctorRawEndpoint | |
| import OAI.Algebra.AuslanderReiten.Complete.CompleteCokernelLifts | |
| import OAI.Algebra.AuslanderReiten.Homology.FullPositiveFunctorAction | |
| import OAI.Algebra.AuslanderReiten.Ext.ExtFunctorScalar | |
| namespace OAI | |
| noncomputable section | |
| open CategoryTheory CategoryTheory.Abelian CategoryTheory.Limits HomologicalComplex | |
| open CochainComplex.HomComplex | |
| open scoped ModuleCat.Algebra | |
| namespace ArExplicit.CompleteComparison | |
| universe u | |
| variable {k R : Type u} [Field k] [CharP k 2] [Ring R] [Algebra k R] | |
| variable [HasDerivedCategory (ModuleCat.{u} R)] | |
| variable (F : ModuleCat.{u} R ⥤ ModuleCat.{u} R) [F.Additive] | |
| [PreservesFiniteLimits F] [PreservesFiniteColimits F] [F.PreservesProjectiveObjects] | |
| variable (K : CochainComplex (ModuleCat.{u} R) ℤ) | |
| variable (hK : TotallyAcyclicComplex (chainOfCochain K)) | |
| variable (hFK : TotallyAcyclicComplex (chainOfCochain (functorComplex F K))) | |
| private theorem endpoint_before {C : Type*} [Category* C] | |
| {X X' Y Y' Z Z' : C} (d : X ≅ X') (e : Y ≅ Y') (c : Y ≅ Z) | |
| (f : Z ≅ Z') (E : X' ≅ Z') (a : X ⟶ Y) | |
| (ha : a ≫ e.hom = d.hom ≫ (E ≪≫ (e.symm ≪≫ c ≪≫ f).symm).hom) : | |
| a ≫ c.hom = (d ≪≫ E ≪≫ f.symm).hom := by | |
| have ht := congrArg (fun z => z ≫ e.inv ≫ c.hom) ha | |
| simpa only [Iso.trans_hom, Iso.trans_inv, Iso.symm_hom, Iso.symm_inv, | |
| Category.assoc, Iso.hom_inv_id_assoc, Iso.inv_hom_id, Category.comp_id] using ht | |
| private theorem endpoint_after {C : Type*} [Category* C] | |
| {X X' Y Y' Z Z' : C} (d : X ≅ X') (e : Y ≅ Y') (c : Y ≅ Z) | |
| (f : Z ≅ Z') (E : X' ≅ Z') (b : Y ⟶ X) | |
| (hb : b ≫ d.hom = e.hom ≫ (e.symm ≪≫ c ≪≫ f ≪≫ E.symm).hom) : | |
| c.inv ≫ b = (d ≪≫ E ≪≫ f.symm).inv := by | |
| apply (cancel_mono d.hom).1 | |
| have ht := congrArg (fun z => c.inv ≫ z) hb | |
| simpa only [Iso.trans_hom, Iso.trans_inv, Iso.symm_hom, Iso.symm_inv, | |
| Category.assoc, Iso.hom_inv_id_assoc, Iso.inv_hom_id_assoc, Iso.inv_hom_id, | |
| Category.comp_id] using ht | |
| omit [HasDerivedCategory (ModuleCat.{u} R)] in | |
| theorem cokernelMap_coordinates {L M : CochainComplex (ModuleCat.{u} R) ℤ} (f : L ⟶ M) : | |
| ModuleCat.ofHom (cokernelMap (extendMap f negIndexEmbedding) 0) ≫ | |
| (cochainCokernelLinearEquiv M).toModuleIso.hom = | |
| (cochainCokernelLinearEquiv L).toModuleIso.hom ≫ ModuleCat.ofHom (cochainCokernelMap f) := by | |
| apply ModuleCat.hom_ext | |
| exact cochainCokernelLinearEquiv_natural f | |
| omit [HasDerivedCategory (ModuleCat.{u} R)] in | |
| theorem rawFunctor_precomp_endpoint | |
| (a : K ⟶ functorComplex F K) | |
| (E : ModuleCat.of R (CochainCokernel K) ≅ F.obj (ModuleCat.of R (CochainCokernel K))) | |
| (ha : cochainCokernelMap a = | |
| (E ≪≫ (rawFunctorEndpointIso F K hK hFK).symm).hom.hom) : | |
| ModuleCat.ofHom (cokernelMap (extendMap a negIndexEmbedding) 0) ≫ | |
| (completeFunctorEndpointIso F K hK hFK).hom = | |
| ((cochainCokernelLinearEquiv K).toModuleIso ≪≫ E ≪≫ | |
| F.mapIso (cochainCokernelLinearEquiv K).symm.toModuleIso).hom := by | |
| have hh := cokernelMap_coordinates a | |
| rw [ha] at hh | |
| exact endpoint_before (cochainCokernelLinearEquiv K).toModuleIso | |
| (cochainCokernelLinearEquiv (functorComplex F K)).toModuleIso | |
| (completeFunctorEndpointIso F K hK hFK) | |
| (F.mapIso (cochainCokernelLinearEquiv K).toModuleIso) E _ hh | |
| omit [HasDerivedCategory (ModuleCat.{u} R)] in | |
| theorem rawFunctor_postcomp_endpoint | |
| (b : functorComplex F K ⟶ K) | |
| (E : ModuleCat.of R (CochainCokernel K) ≅ F.obj (ModuleCat.of R (CochainCokernel K))) | |
| (hb : cochainCokernelMap b = | |
| (rawFunctorEndpointIso F K hK hFK ≪≫ E.symm).hom.hom) : | |
| (completeFunctorEndpointIso F K hK hFK).inv ≫ | |
| ModuleCat.ofHom (cokernelMap (extendMap b negIndexEmbedding) 0) = | |
| ((cochainCokernelLinearEquiv K).toModuleIso ≪≫ E ≪≫ | |
| F.mapIso (cochainCokernelLinearEquiv K).symm.toModuleIso).inv := by | |
| have hh := cokernelMap_coordinates b | |
| rw [hb] at hh | |
| exact endpoint_after (cochainCokernelLinearEquiv K).toModuleIso | |
| (cochainCokernelLinearEquiv (functorComplex F K)).toModuleIso | |
| (completeFunctorEndpointIso F K hK hFK) | |
| (F.mapIso (cochainCokernelLinearEquiv K).toModuleIso) E _ hh | |
| theorem fullHom_rawFunctor_scalar | |
| (a : K ⟶ functorComplex F K) (b : functorComplex F K ⟶ K) | |
| (E : ModuleCat.of R (CochainCokernel K) ≅ F.obj (ModuleCat.of R (CochainCokernel K))) | |
| (ha : cochainCokernelMap a = | |
| (E ≪≫ (rawFunctorEndpointIso F K hK hFK).symm).hom.hom) | |
| (hb : cochainCokernelMap b = | |
| (rawFunctorEndpointIso F K hK hFK ≪≫ E.symm).hom.hom) | |
| (n : ℕ) (c : k) | |
| (hs : ∀ z : Ext (ModuleCat.of R (CochainCokernel K)) | |
| (ModuleCat.of R (CochainCokernel K)) (n+1), | |
| selfExtIso (k := k) E.symm (n+1) (z.mapExactFunctor F) = c • z) | |
| (x : CohomologyClass K K ((n+1 : ℕ) : ℤ)) : | |
| fullHomPostcomp b _ (fullHomPrecomp a _ (mapFullHom F K K _ x)) = c • x := by | |
| apply fullHom_conjugatedFunctor_scalar F K hK hFK a b | |
| ((cochainCokernelLinearEquiv K).toModuleIso ≪≫ E ≪≫ | |
| F.mapIso (cochainCokernelLinearEquiv K).symm.toModuleIso) | |
| (rawFunctor_precomp_endpoint F K hK hFK a E ha) | |
| (rawFunctor_postcomp_endpoint F K hK hFK b E hb) n c | |
| exact exactFunctor_scalar_transport F (cochainCokernelLinearEquiv K).toModuleIso E (n+1) c hs | |
| end ArExplicit.CompleteComparison | |
| end | |
| end OAI | |