openAI-math-dataset / lean /OAI /Algebra /AuslanderReiten /Tensor /BalancedTensorCompleteAction.lean
| import OAI.Algebra.AuslanderReiten.Tensor.BalancedTensorResolution | |
| import OAI.Algebra.AuslanderReiten.Tensor.BalancedTensorCanonicalEndpoint | |
| import OAI.Algebra.AuslanderReiten.Complete.CompleteFunctorCokernelAction | |
| import OAI.Algebra.AuslanderReiten.Tensor.BalancedTensorNative | |
| namespace OAI | |
| noncomputable section | |
| open CategoryTheory CategoryTheory.Abelian CategoryTheory.Limits | |
| open CochainComplex.HomComplex | |
| open scoped ModuleCat.Algebra | |
| namespace ArExplicit.BalancedTensor | |
| open CompleteComparison | |
| universe u | |
| variable {k R F : Type u} [Field k] [CharP k 2] [Ring R] [Algebra k R] [Module.Finite k R] | |
| variable [AddCommGroup F] [Module k F] [Module Rᵐᵒᵖ F] [Module R F] | |
| variable [IsScalarTower k Rᵐᵒᵖ F] [SMulCommClass k Rᵐᵒᵖ F] | |
| variable [SMulCommClass k R F] [SMulCommClass R Rᵐᵒᵖ F] | |
| variable [Module.Finite Rᵐᵒᵖ F] [Module.Projective Rᵐᵒᵖ F] | |
| variable [Module.Finite R F] [Module.Projective R F] | |
| variable [HasDerivedCategory (ModuleCat.{u} R)] | |
| attribute [local instance] moduleCatBaseModule moduleCatBaseTower | |
| variable [PreservesFiniteLimits (functor (k := k) (R := R) (S := R) (F := F))] | |
| variable [PreservesFiniteColimits (functor (k := k) (R := R) (S := R) (F := F))] | |
| variable [(functor (k := k) (R := R) (S := R) (F := F)).PreservesProjectiveObjects] | |
| variable (τ : FrobeniusTrace k R) (M : FiniteLeftModule k R) | |
| variable (e : BalancedTensor k R F M.carrier ≃ₗ[R] M.carrier) | |
| def nativeEvaluationIso : | |
| (functor (k := k) (R := R) (S := R) (F := F)).obj M.carrier ≅ M.carrier := | |
| nativeObjectIso (k := k) (R := R) (S := R) (F := F) M.carrier ≪≫ e.toModuleIso | |
| def completeActionEndpoint : ModuleCat.of R (CochainCokernel (M.completeCochain τ)) ≅ | |
| (functor (k := k) (R := R) (S := R) (F := F)).obj | |
| (ModuleCat.of R (CochainCokernel (M.completeCochain τ))) := | |
| (M.completeCochainCokernelEquiv τ).toModuleIso ≪≫ (nativeEvaluationIso M e).symm ≪≫ | |
| (functor (k := k) (R := R) (S := R) (F := F)).mapIso | |
| (M.completeCochainCokernelEquiv τ).symm.toModuleIso | |
| def completeActionCokernelIso : | |
| ModuleCat.of R (CochainCokernel (completeTensor (F := F) τ M)) ≅ | |
| ModuleCat.of R (CochainCokernel (M.completeCochain τ)) := | |
| rawFunctorEndpointIso (functor (k := k) (R := R) (S := R) (F := F)) (M.completeCochain τ) | |
| (M.completeCochain_totallyAcyclic τ).to_chain | |
| (completeTensor_totallyAcyclic (F := F) τ M).to_chain ≪≫ | |
| (completeActionEndpoint τ M e).symm | |
| def completeActionIn : M.completeCochain τ ⟶ completeTensor (F := F) τ M := | |
| cokernelCochainLift (M.completeCochain_totallyAcyclic τ) | |
| (completeTensor_totallyAcyclic (F := F) τ M) | |
| (completeActionCokernelIso τ M e).inv.hom | |
| def completeActionOut : completeTensor (F := F) τ M ⟶ M.completeCochain τ := | |
| cokernelCochainLift (completeTensor_totallyAcyclic (F := F) τ M) | |
| (M.completeCochain_totallyAcyclic τ) (completeActionCokernelIso τ M e).hom.hom | |
| omit [CharP k 2] [SMulCommClass k Rᵐᵒᵖ F] | |
| [HasDerivedCategory (ModuleCat.{u} R)] in | |
| theorem completeAction_inverse : | |
| (HomotopyCategory.quotient _ _).map | |
| (completeActionIn τ M e ≫ completeActionOut τ M e) = 𝟙 _ := by | |
| exact cokernelCochainLift_homotopy_inverse (M.completeCochain_totallyAcyclic τ) | |
| (completeTensor_totallyAcyclic (F := F) τ M) | |
| (completeActionCokernelIso τ M e).symm.toLinearEquiv | |
| omit [CharP k 2] [SMulCommClass k Rᵐᵒᵖ F] | |
| [HasDerivedCategory (ModuleCat.{u} R)] in | |
| theorem completeAction_inverse' : | |
| (HomotopyCategory.quotient _ _).map | |
| (completeActionOut τ M e ≫ completeActionIn τ M e) = 𝟙 _ := by | |
| exact cokernelCochainLift_homotopy_inverse (completeTensor_totallyAcyclic (F := F) τ M) | |
| (M.completeCochain_totallyAcyclic τ) (completeActionCokernelIso τ M e).toLinearEquiv | |
| omit [SMulCommClass k Rᵐᵒᵖ F] in | |
| theorem completeAction_scalar (n : ℕ) (c : k) | |
| (hs : ∀ z : Ext M.carrier M.carrier (n+1), | |
| selfExtIso (k := k) (nativeEvaluationIso M e) (n+1) | |
| (z.mapExactFunctor (functor (k := k) (R := R) (S := R) (F := F))) = c • z) | |
| (x : CohomologyClass (M.completeCochain τ) (M.completeCochain τ) ((n+1 : ℕ) : ℤ)) : | |
| fullHomPostcomp (completeActionOut τ M e) _ | |
| (fullHomPrecomp (completeActionIn τ M e) _ | |
| (mapFullHom (functor (k := k) (R := R) (S := R) (F := F)) _ _ _ x)) = c • x := by | |
| apply fullHom_rawFunctor_scalar (functor (k := k) (R := R) (S := R) (F := F)) | |
| (M.completeCochain τ) (M.completeCochain_totallyAcyclic τ).to_chain | |
| (completeTensor_totallyAcyclic (F := F) τ M).to_chain | |
| (completeActionIn τ M e) (completeActionOut τ M e) (completeActionEndpoint τ M e) | |
| · exact cokernelCochainLift_map _ _ _ | |
| · exact cokernelCochainLift_map _ _ _ | |
| · exact exactFunctor_scalar_transport (functor (k := k) (R := R) (S := R) (F := F)) | |
| (M.completeCochainCokernelEquiv τ).toModuleIso (nativeEvaluationIso M e).symm (n+1) c hs | |
| omit [CharP k 2] [SMulCommClass k Rᵐᵒᵖ F] | |
| [HasDerivedCategory (ModuleCat.{u} R)] in | |
| theorem completeActionCokernelIso_eq : completeActionCokernelIso τ M e = | |
| (((tensorCokernelEquiv (F := F) τ M).trans e).trans | |
| (M.completeCochainCokernelEquiv τ).symm).toModuleIso := by | |
| rw [completeActionCokernelIso, rawFunctorEndpointIso_eq] | |
| apply Iso.ext | |
| have ht := congrArg (fun f => | |
| (cochainCokernelEquiv (k := k) (F := F) (S := R) (M.completeCochain τ)).toModuleIso.hom ≫ | |
| f ≫ e.toModuleIso.hom ≫ (M.completeCochainCokernelEquiv τ).toModuleIso.inv) | |
| (nativeObjectIso_natural (k := k) (F := F) (S := R) | |
| (M.completeCochainCokernelEquiv τ).toLinearMap) | |
| exact ht | |
| omit [CharP k 2] [SMulCommClass k Rᵐᵒᵖ F] | |
| [HasDerivedCategory (ModuleCat.{u} R)] in | |
| theorem completeActionOut_quotient : | |
| (HomotopyCategory.quotient _ _).map (completeActionOut τ M e) = | |
| (completeTensorIso τ M M e).hom := by | |
| unfold completeActionOut | |
| rw [completeActionCokernelIso_eq] | |
| exact cokernelCochainLift_quotient _ _ _ | |
| end ArExplicit.BalancedTensor | |
| end | |
| end OAI | |