| import OAI.Algebra.AuslanderReiten.Tensor.EnvelopingBimodule |
| import OAI.Algebra.AuslanderReiten.Tensor.BalancedTensorMap |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open CategoryTheory |
| open scoped TensorProduct ModuleCat.Algebra |
|
|
| namespace ArExplicit.EnvelopingBimodule |
|
|
| universe u |
| variable {k R : Type u} [Field k] [Ring R] [Algebra k R] |
| variable {M N P : ModuleCat.{u} (R ⊗[k] Rᵐᵒᵖ)} |
| variable [Module k M] [IsScalarTower k (R ⊗[k] Rᵐᵒᵖ) M] |
| variable [Module k N] [IsScalarTower k (R ⊗[k] Rᵐᵒᵖ) N] |
| variable [Module k P] [IsScalarTower k (R ⊗[k] Rᵐᵒᵖ) P] |
|
|
| def map (f : M ⟶ N) : EnvelopingBimodule k R M →ₗ[k] EnvelopingBimodule k R N where |
| toFun x := ⟨f.hom x.value⟩ |
| map_add' x y := by |
| ext |
| exact f.hom.map_add x.value y.value |
| map_smul' c x := by |
| ext |
| exact (f.hom.restrictScalars k).map_smul c x.value |
|
|
| @[simp] theorem map_value (f : M ⟶ N) (x : EnvelopingBimodule k R M) : |
| (map f x).value = f.hom x.value := rfl |
|
|
| @[simp] theorem map_left_smul (f : M ⟶ N) (a : R) |
| (x : EnvelopingBimodule k R M) : map f (a • x) = a • map f x := by |
| ext |
| exact f.hom.map_smul (a ⊗ₜ[k] (1 : Rᵐᵒᵖ)) x.value |
|
|
| @[simp] theorem map_right_smul (f : M ⟶ N) (b : Rᵐᵒᵖ) |
| (x : EnvelopingBimodule k R M) : map f (b • x) = b • map f x := by |
| ext |
| exact f.hom.map_smul ((1 : R) ⊗ₜ[k] b) x.value |
|
|
| def mapLeft (f : M ⟶ N) : EnvelopingBimodule k R M →ₗ[R] EnvelopingBimodule k R N where |
| toFun := map f |
| map_add' := (map f).map_add |
| map_smul' := map_left_smul f |
|
|
| @[simp] theorem map_id : map (𝟙 M) = LinearMap.id := by |
| ext x |
| rfl |
|
|
| @[simp] theorem map_comp (f : M ⟶ N) (g : N ⟶ P) : |
| map (f ≫ g) = (map g).comp (map f) := by |
| ext x |
| rfl |
|
|
| @[simp] theorem map_zero : map (0 : M ⟶ N) = 0 := by |
| ext x |
| rfl |
|
|
| @[simp] theorem map_add (f g : M ⟶ N) : map (f + g) = map f + map g := by |
| ext x |
| rfl |
|
|
| variable (X : ModuleCat.{u} R) [Module k X] [IsScalarTower k R X] |
|
|
| def tensorMap (f : M ⟶ N) : |
| BalancedTensor k R (EnvelopingBimodule k R M) X →ₗ[R] |
| BalancedTensor k R (EnvelopingBimodule k R N) X where |
| __ := (BalancedTensor.map (map f) (LinearMap.id : X →ₗ[k] X) |
| (map_right_smul f) (fun _ _ => rfl)).toAddHom |
| map_smul' a z := by |
| change BalancedTensor.map (map f) (LinearMap.id : X →ₗ[k] X) |
| (map_right_smul f) (fun _ _ => rfl) (a • z) = |
| a • BalancedTensor.map (map f) (LinearMap.id : X →ₗ[k] X) |
| (map_right_smul f) (fun _ _ => rfl) z |
| induction z using BalancedTensor.induction_on with |
| | zero => simp |
| | tmul m x => simp only [BalancedTensor.smul_tmul, BalancedTensor.map_tmul, |
| LinearMap.id_apply, map_left_smul] |
| | add z w hz hw => simp_all |
|
|
| omit [IsScalarTower k R X] |
|
|
| @[simp] theorem tensorMap_tmul (f : M ⟶ N) |
| (m : EnvelopingBimodule k R M) (x : X) : |
| tensorMap X f (BalancedTensor.tmul m x) = BalancedTensor.tmul (map f m) x := rfl |
|
|
| @[simp] theorem tensorMap_id : tensorMap X (𝟙 M) = LinearMap.id := by |
| apply LinearMap.ext |
| intro z |
| induction z using BalancedTensor.induction_on with |
| | zero => simp |
| | tmul m x => rfl |
| | add z w hz hw => simp_all |
|
|
| @[simp] theorem tensorMap_comp (f : M ⟶ N) (g : N ⟶ P) : |
| tensorMap X (f ≫ g) = (tensorMap X g).comp (tensorMap X f) := by |
| apply LinearMap.ext |
| intro z |
| induction z using BalancedTensor.induction_on with |
| | zero => simp |
| | tmul m x => rfl |
| | add z w hz hw => simp_all |
|
|
| @[simp] theorem tensorMap_zero : tensorMap X (0 : M ⟶ N) = 0 := by |
| apply LinearMap.ext |
| intro z |
| induction z using BalancedTensor.induction_on with |
| | zero => simp |
| | tmul m x => simp |
| | add z w hz hw => simp_all |
|
|
| @[simp] theorem tensorMap_add (f g : M ⟶ N) : |
| tensorMap X (f + g) = tensorMap X f + tensorMap X g := by |
| apply LinearMap.ext |
| intro z |
| induction z using BalancedTensor.induction_on with |
| | zero => simp |
| | tmul m x => simp [BalancedTensor.add_tmul] |
| | add z w hz hw => |
| simp only [_root_.LinearMap.map_add, hz, hw] |
|
|
| end ArExplicit.EnvelopingBimodule |
|
|
| end |
|
|
| end OAI |
|
|
