| import OAI.Algebra.AuslanderReiten.Tensor.BalancedTensorComplete | |
| import OAI.Algebra.AuslanderReiten.Tensor.BalancedTensorMapEquiv | |
| import OAI.Algebra.AuslanderReiten.Cones.CochainCokernelIso | |
| import OAI.Algebra.AuslanderReiten.Duality.FiniteStableResolutionIso | |
| namespace OAI | |
| noncomputable section | |
| namespace ArExplicit.BalancedTensor | |
| open CategoryTheory 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] [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] | |
| attribute [local instance] moduleCatBaseModule moduleCatBaseTower | |
| variable (τ : FrobeniusTrace k R) (M N : FiniteLeftModule k R) | |
| abbrev completeTensor := | |
| (((functor (k := k) (R := R) (S := R) (F := F)).mapHomologicalComplex _).obj | |
| (M.completeCochain τ)) | |
| def tensorCokernelEquiv : CochainCokernel (completeTensor (F := F) τ M) ≃ₗ[R] | |
| BalancedTensor k R F M.carrier := | |
| (cochainCokernelEquiv (k := k) (F := F) (S := R) (M.completeCochain τ)).trans | |
| (mapRightEquiv (k := k) (F := F) (S := R) (M.completeCochainCokernelEquiv τ)) | |
| omit [SMulCommClass k Rᵐᵒᵖ F] [Module.Finite R F] [Module.Projective R F] in | |
| @[simp] theorem tensorCokernelEquiv_mk (f : F) (x : (M.completeCochain τ).X 0) : | |
| tensorCokernelEquiv (F := F) τ M (Submodule.Quotient.mk (tmul f x)) = | |
| tmul f (M.completeCochainCokernelEquiv τ (Submodule.Quotient.mk x)) := by | |
| rw [tensorCokernelEquiv, LinearEquiv.trans_apply, | |
| cochainCokernelEquiv_mk_tmul, mapRightEquiv_tmul] | |
| omit [SMulCommClass k Rᵐᵒᵖ F] in | |
| theorem completeTensor_totallyAcyclic : TotallyAcyclicCochain (completeTensor (F := F) τ M) := by | |
| let := τ.regular_injective | |
| exact complex_totallyAcyclic (M.completeCochain τ) | |
| (M.completeCochain_totallyAcyclic τ).toExactProjectiveCochain | |
| def completeTensorIso (e : BalancedTensor k R F M.carrier ≃ₗ[R] N.carrier) : | |
| (HomotopyCategory.quotient _ _).obj (completeTensor (F := F) τ M) ≅ | |
| (HomotopyCategory.quotient _ _).obj (N.completeCochain τ) := | |
| rawCokernelHomotopyIso (completeTensor_totallyAcyclic (F := F) τ M) | |
| (N.completeCochain_totallyAcyclic τ) | |
| (((tensorCokernelEquiv (F := F) τ M).trans e).trans (N.completeCochainCokernelEquiv τ).symm) | |
| variable {G : Type u} [AddCommGroup G] [Module k G] [Module Rᵐᵒᵖ G] [Module R G] | |
| variable [IsScalarTower k Rᵐᵒᵖ G] [SMulCommClass 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] | |
| omit [SMulCommClass k Rᵐᵒᵖ F] [Module.Finite R F] [Module.Projective R F] in | |
| omit [SMulCommClass k Rᵐᵒᵖ G] [Module.Finite R G] [Module.Projective R G] in | |
| theorem tensorCokernelEquiv_natural (a : F →ₗ[Rᵐᵒᵖ] G) | |
| (ha : ∀ (r : R) f, a (r • f) = r • a f) | |
| (z : CochainCokernel (completeTensor (F := F) τ M)) : | |
| tensorCokernelEquiv (F := G) τ M | |
| (cochainCokernelMap (complexMap (k := k) a ha (M.completeCochain τ)) z) = | |
| mapLeft (k := k) (M := M.carrier) a (tensorCokernelEquiv (F := F) τ M z) := by | |
| obtain ⟨x,rfl⟩ := Submodule.mkQ_surjective | |
| (LinearMap.range ((completeTensor (F := F) τ M).d (-1) 0).hom) z | |
| induction x using induction_on with | |
| | zero => repeat' erw [map_zero] | |
| | add x y hx hy => | |
| repeat' erw [map_add] | |
| exact congrArg₂ (· + ·) hx hy | |
| | tmul f m => | |
| erw [cochainCokernelMap_mk] | |
| change tensorCokernelEquiv (F := G) τ M (Submodule.Quotient.mk (tmul (a f) m)) = _ | |
| erw [tensorCokernelEquiv_mk] | |
| variable (N' : FiniteLeftModule k R) | |
| omit [SMulCommClass k Rᵐᵒᵖ F] [SMulCommClass k Rᵐᵒᵖ G] in | |
| theorem completeTensorIso_natural | |
| (a : F →ₗ[Rᵐᵒᵖ] G) (ha : ∀ (r : R) f, a (r • f) = r • a f) | |
| (e : BalancedTensor k R F M.carrier ≃ₗ[R] N.carrier) | |
| (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)) ≫ | |
| (completeTensorIso τ M N' e').hom = | |
| (completeTensorIso τ M N e).hom ≫ | |
| (HomotopyCategory.quotient _ _).map (FiniteLeftModule.resolutionMap τ b) := by | |
| apply rawCokernelHomotopyIso_natural | |
| rw [FiniteLeftModule.resolutionMap_cokernel] | |
| apply LinearMap.ext | |
| intro z | |
| change (N'.completeCochainCokernelEquiv τ).symm | |
| (e' (tensorCokernelEquiv (F := G) τ M | |
| (cochainCokernelMap (complexMap (k := k) a ha (M.completeCochain τ)) z))) = | |
| (N'.completeCochainCokernelEquiv τ).symm | |
| (b ((N.completeCochainCokernelEquiv τ) | |
| ((N.completeCochainCokernelEquiv τ).symm (e (tensorCokernelEquiv (F := F) τ M z))))) | |
| rw [tensorCokernelEquiv_natural, LinearEquiv.apply_symm_apply, hab] | |
| end ArExplicit.BalancedTensor | |
| end | |
| end OAI | |