openAI-math-dataset / lean /OAI /Algebra /AuslanderReiten /Tensor /BalancedTensorCompleteInverse.lean
| import OAI.Algebra.AuslanderReiten.Tensor.BalancedTensorCompleteAction | |
| namespace OAI | |
| noncomputable section | |
| open CategoryTheory CategoryTheory.Abelian CategoryTheory.Limits | |
| open scoped ModuleCat.Algebra | |
| namespace ArExplicit.BalancedTensor | |
| universe u | |
| variable {k R F : Type u} [Field k] [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] | |
| 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] | |
| 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) | |
| theorem completeActionIn_quotient : | |
| (HomotopyCategory.quotient _ _).map (completeActionIn τ M e) = | |
| (completeTensorIso τ M M e).inv := by | |
| apply (cancel_mono (completeTensorIso τ M M e).hom).1 | |
| have h := completeAction_inverse τ M e | |
| rw [CategoryTheory.Functor.map_comp, completeActionOut_quotient] at h | |
| exact h.trans (completeTensorIso τ M M e).inv_hom_id.symm | |
| end ArExplicit.BalancedTensor | |
| end | |
| end OAI | |