openAI-math-dataset / lean /OAI /Algebra /AuslanderReiten /Tensor /BalancedTensorComparisonMaps.lean
| import OAI.Algebra.AuslanderReiten.Tensor.BalancedTensorResolution | |
| import OAI.Algebra.AuslanderReiten.Complete.CompleteCokernelLifts | |
| namespace OAI | |
| noncomputable section | |
| open CategoryTheory CochainComplex.HomComplex | |
| open scoped ModuleCat.Algebra | |
| namespace ArExplicit.BalancedTensor | |
| open CompleteComparison | |
| 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 (τ : FrobeniusTrace k R) (M N : FiniteLeftModule k R) | |
| variable (e : BalancedTensor k R F M.carrier ≃ₗ[R] N.carrier) | |
| def comparisonOut : completeTensor (F := F) τ M ⟶ N.completeCochain τ := | |
| cokernelCochainLift (completeTensor_totallyAcyclic (F := F) τ M) | |
| (N.completeCochain_totallyAcyclic τ) | |
| (((tensorCokernelEquiv (F := F) τ M).trans e).trans | |
| (N.completeCochainCokernelEquiv τ).symm).toLinearMap | |
| def comparisonIn : N.completeCochain τ ⟶ completeTensor (F := F) τ M := | |
| cokernelCochainLift (N.completeCochain_totallyAcyclic τ) | |
| (completeTensor_totallyAcyclic (F := F) τ M) | |
| (((tensorCokernelEquiv (F := F) τ M).trans e).trans | |
| (N.completeCochainCokernelEquiv τ).symm).symm.toLinearMap | |
| theorem comparisonOut_quotient : | |
| (HomotopyCategory.quotient _ _).map (comparisonOut τ M N e) = | |
| (completeTensorIso τ M N e).hom := | |
| cokernelCochainLift_quotient _ _ _ | |
| theorem comparisonIn_quotient : | |
| (HomotopyCategory.quotient _ _).map (comparisonIn τ M N e) = | |
| (completeTensorIso τ M N e).inv := | |
| cokernelCochainLift_quotient_inv _ _ _ | |
| theorem comparisonOut_bijective (K : CochainComplex (ModuleCat.{u} R) ℤ) (n : ℤ) : | |
| Function.Bijective (fullHomPostcomp (K := K) (comparisonOut τ M N e) n) := by | |
| apply fullHomPostcomp_bijective k n (comparisonOut τ M N e) (comparisonIn τ M N e) | |
| · rw [CategoryTheory.Functor.map_comp, comparisonOut_quotient, comparisonIn_quotient, | |
| Iso.hom_inv_id] | |
| · rw [CategoryTheory.Functor.map_comp, comparisonOut_quotient, comparisonIn_quotient, | |
| Iso.inv_hom_id] | |
| variable {G : Type u} [AddCommGroup G] [Module k G] [Module Rᵐᵒᵖ G] [Module R G] | |
| variable [IsScalarTower k Rᵐᵒᵖ G] | |
| variable [SMulCommClass k R G] [SMulCommClass R Rᵐᵒᵖ G] | |
| variable [Module.Finite Rᵐᵒᵖ G] [Module.Projective Rᵐᵒᵖ G] | |
| variable [Module.Finite R G] [Module.Projective R G] | |
| variable (N' : FiniteLeftModule k R) | |
| theorem comparisonOut_natural | |
| (a : F →ₗ[Rᵐᵒᵖ] G) (ha : ∀ (r : R) f, a (r • f) = r • a f) | |
| (e' : BalancedTensor k R G M.carrier ≃ₗ[R] N'.carrier) | |
| (b : N.carrier →ₗ[R] N'.carrier) | |
| (hab : ∀ x, e' (mapLeft (k := k) (M := M.carrier) a x) = b (e x)) : | |
| (HomotopyCategory.quotient _ _).map | |
| (complexMap (k := k) a ha (M.completeCochain τ) ≫ comparisonOut τ M N' e') = | |
| (HomotopyCategory.quotient _ _).map | |
| (comparisonOut τ M N e ≫ FiniteLeftModule.resolutionMap τ b) := by | |
| rw [CategoryTheory.Functor.map_comp, CategoryTheory.Functor.map_comp, | |
| comparisonOut_quotient, comparisonOut_quotient] | |
| exact completeTensorIso_natural τ M N N' a ha e e' b hab | |
| theorem comparisonOut_fullHom_natural | |
| (a : F →ₗ[Rᵐᵒᵖ] G) (ha : ∀ (r : R) f, a (r • f) = r • a f) | |
| (e' : BalancedTensor k R G M.carrier ≃ₗ[R] N'.carrier) | |
| (b : N.carrier →ₗ[R] N'.carrier) | |
| (hab : ∀ x, e' (mapLeft (k := k) (M := M.carrier) a x) = b (e x)) | |
| (K : CochainComplex (ModuleCat.{u} R) ℤ) (n : ℤ) | |
| (x : CohomologyClass K (completeTensor (F := F) τ M) n) : | |
| fullHomPostcomp (comparisonOut τ M N' e') n | |
| (fullHomPostcomp (complexMap (k := k) a ha (M.completeCochain τ)) n x) = | |
| fullHomPostcomp (FiniteLeftModule.resolutionMap τ b) n | |
| (fullHomPostcomp (comparisonOut τ M N e) n x) := by | |
| rw [fullHomPostcomp_comp, fullHomPostcomp_comp] | |
| exact fullHomPostcomp_congr k n (comparisonOut_natural τ M N e N' a ha e' b hab) x | |
| end ArExplicit.BalancedTensor | |
| end | |
| end OAI | |