| import OAI.Algebra.AuslanderReiten.Tensor.BalancedTensorCategory | |
| import OAI.Algebra.AuslanderReiten.Cones.CochainCokernel | |
| namespace OAI | |
| noncomputable section | |
| namespace ArExplicit.BalancedTensor | |
| open CategoryTheory CompleteComparison | |
| universe u | |
| variable {k R S F : Type u} [Field k] [Ring R] [Ring S] [Algebra k R] | |
| variable [AddCommGroup F] [Module k F] [Module Rᵐᵒᵖ F] [Module S F] | |
| variable [IsScalarTower k Rᵐᵒᵖ F] | |
| variable [SMulCommClass k S F] [SMulCommClass S Rᵐᵒᵖ F] | |
| variable [Module.Finite Rᵐᵒᵖ F] [Module.Projective Rᵐᵒᵖ F] | |
| variable {L M : Type u} [AddCommGroup L] [AddCommGroup M] | |
| variable [Module k L] [Module k M] [Module R L] [Module R M] | |
| variable [IsScalarTower k R L] [IsScalarTower k R M] | |
| variable [SMulCommClass k R M] | |
| def cokernelEquiv (d : L →ₗ[R] M) : | |
| (BalancedTensor k R F M ⧸ LinearMap.range (mapRight (k := k) (F := F) (S := S) d)) ≃ₗ[S] | |
| BalancedTensor k R F (M ⧸ LinearMap.range d) := | |
| (mapRight_exact (k := k) (F := F) (S := S) d (LinearMap.range d).mkQ | |
| (LinearMap.exact_map_mkQ_range d)).linearEquivOfSurjective | |
| (mapRight_surjective (k := k) (F := F) (S := S) _ (Submodule.mkQ_surjective _)) | |
| theorem cokernelEquiv_mk (d : L →ₗ[R] M) (x : BalancedTensor k R F M) : | |
| cokernelEquiv (k := k) (F := F) (S := S) d (Submodule.Quotient.mk x) = | |
| mapRight (k := k) (F := F) (S := S) (LinearMap.range d).mkQ x := by | |
| apply (cokernelEquiv (k := k) (F := F) (S := S) d).symm.injective | |
| rw [LinearEquiv.symm_apply_apply] | |
| exact (Function.Exact.linearEquivOfSurjective_symm_apply _ _ x).symm | |
| @[simp] theorem cokernelEquiv_mk_tmul (d : L →ₗ[R] M) (f : F) (m : M) : | |
| cokernelEquiv (k := k) (F := F) (S := S) d (Submodule.Quotient.mk (tmul f m)) = | |
| tmul f (Submodule.Quotient.mk m) := by | |
| rw [cokernelEquiv_mk, mapRight_tmul] | |
| rfl | |
| attribute [local instance] moduleCatBaseModule moduleCatBaseTower | |
| def cochainCokernelEquiv (P : CochainComplex (ModuleCat.{u} R) ℤ) : | |
| CochainCokernel (((functor (k := k) (R := R) (S := S) (F := F)).mapHomologicalComplex _).obj P) ≃ₗ[S] | |
| BalancedTensor k R F (CochainCokernel P) := | |
| cokernelEquiv (k := k) (F := F) (S := S) (P.d (-1) 0).hom | |
| @[simp] theorem cochainCokernelEquiv_mk_tmul | |
| (P : CochainComplex (ModuleCat.{u} R) ℤ) (f : F) (x : P.X 0) : | |
| cochainCokernelEquiv (k := k) (F := F) (S := S) P | |
| (Submodule.Quotient.mk (tmul f x)) = tmul f (Submodule.Quotient.mk x) := | |
| cokernelEquiv_mk_tmul (k := k) (F := F) (S := S) (P.d (-1) 0).hom f x | |
| variable {G : Type u} [AddCommGroup G] [Module k G] [Module Rᵐᵒᵖ G] [Module S G] | |
| variable [IsScalarTower k Rᵐᵒᵖ G] | |
| variable [SMulCommClass k S G] [SMulCommClass S Rᵐᵒᵖ G] | |
| variable [Module.Finite Rᵐᵒᵖ G] [Module.Projective Rᵐᵒᵖ G] | |
| theorem cochainCokernelEquiv_natural | |
| (a : F →ₗ[Rᵐᵒᵖ] G) (ha : ∀ (s : S) f, a (s • f) = s • a f) | |
| (P : CochainComplex (ModuleCat.{u} R) ℤ) | |
| (z : CochainCokernel (((functor (k := k) (R := R) (S := S) (F := F)).mapHomologicalComplex _).obj P)) : | |
| cochainCokernelEquiv (k := k) (F := G) (S := S) P | |
| (cochainCokernelMap (complexMap (k := k) a ha P) z) = | |
| mapLeft (k := k) (M := CochainCokernel P) a | |
| (cochainCokernelEquiv (k := k) (F := F) (S := S) P z) := by | |
| obtain ⟨x,rfl⟩ := Submodule.mkQ_surjective | |
| (LinearMap.range (((((functor (k := k) (R := R) (S := S) (F := F)).mapHomologicalComplex _).obj P).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 cochainCokernelEquiv (k := k) (F := G) (S := S) P | |
| (Submodule.Quotient.mk (tmul (a f) m)) = _ | |
| erw [cochainCokernelEquiv_mk_tmul] | |
| end ArExplicit.BalancedTensor | |
| end | |
| end OAI | |