openAI-math-dataset / lean /OAI /Algebra /AuslanderReiten /Tensor /BalancedTensorPositiveAction.lean
| import OAI.Algebra.AuslanderReiten.Tensor.BalancedTensorCompleteAction | |
| namespace OAI | |
| noncomputable section | |
| open CategoryTheory CategoryTheory.Abelian CategoryTheory.Limits CochainComplex.HomComplex | |
| open scoped ModuleCat.Algebra | |
| namespace ArExplicit.BalancedTensor | |
| 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) | |
| omit [SMulCommClass k Rᵐᵒᵖ F] in | |
| theorem completeAction_scalar_positive (d : ℕ) (hd : 0 < d) (c : k) | |
| (hs : ∀ z : Ext M.carrier M.carrier d, | |
| selfExtIso (k := k) (nativeEvaluationIso M e) d | |
| (z.mapExactFunctor (functor (k := k) (R := R) (S := R) (F := F))) = c • z) | |
| (x : CohomologyClass (M.completeCochain τ) (M.completeCochain τ) (d : 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 | |
| cases d with | |
| | zero => exact False.elim (Nat.lt_irrefl 0 hd) | |
| | succ n => exact completeAction_scalar τ M e n c hs x | |
| end ArExplicit.BalancedTensor | |
| end | |
| end OAI | |