| import OAI.Algebra.AuslanderReiten.Triangular.Triangular |
| import OAI.Algebra.AuslanderReiten.Tensor.BalancedTensorUnit |
|
|
| namespace OAI |
|
|
| namespace ArExplicit |
|
|
| variable (k R F M N : Type*) [CommRing k] [Ring R] |
| variable [AddCommGroup F] [Module k F] [Module R F] [Module Rᵐᵒᵖ F] |
| variable [SMulCommClass k R F] [SMulCommClass R Rᵐᵒᵖ F] |
| variable [AddCommGroup M] [Module k M] [Module R M] |
| variable [AddCommGroup N] [Module R N] |
|
|
| @[ext] structure TriangularModule (φ : BalancedTensor k R F M →ₗ[R] N) where |
| top : M |
| bottom : N |
|
|
| namespace TriangularModule |
|
|
| variable {k R F M N} |
| variable (φ : BalancedTensor k R F M →ₗ[R] N) |
|
|
| def coordinates : TriangularModule k R F M N φ ≃ M × N where |
| toFun x := (x.top, x.bottom) |
| invFun x := ⟨x.1, x.2⟩ |
| left_inv _ := rfl |
| right_inv _ := rfl |
| |
| instance : AddCommGroup (TriangularModule k R F M N φ) := (coordinates φ).addCommGroup |
| |
| def coordinateAddEquiv : TriangularModule k R F M N φ ≃+ M × N where |
| __ := coordinates φ |
| map_add' _ _ := rfl |
|
|
| @[simp] theorem top_add (x y : TriangularModule k R F M N φ) : (x + y).top = x.top + y.top := rfl |
| @[simp] theorem bottom_add (x y : TriangularModule k R F M N φ) : |
| (x + y).bottom = x.bottom + y.bottom := rfl |
| @[simp] theorem top_zero : (0 : TriangularModule k R F M N φ).top = 0 := rfl |
| @[simp] theorem bottom_zero : (0 : TriangularModule k R F M N φ).bottom = 0 := rfl |
| |
| instance : SMul (Triangular R F) (TriangularModule k R F M N φ) := |
| ⟨fun a x => ⟨a.top • x.top, φ (BalancedTensor.tmul a.off x.top) + a.bottom • x.bottom⟩⟩ |
| |
| @[simp] theorem top_smul (a : Triangular R F) (x : TriangularModule k R F M N φ) : |
| (a • x).top = a.top • x.top := rfl |
| @[simp] theorem bottom_smul (a : Triangular R F) (x : TriangularModule k R F M N φ) : |
| (a • x).bottom = φ (BalancedTensor.tmul a.off x.top) + a.bottom • x.bottom := rfl |
| |
| instance : Module (Triangular R F) (TriangularModule k R F M N φ) where |
| one_smul x := by ext <;> simp |
| mul_smul a b x := by |
| ext |
| · exact mul_smul a.top b.top x.top |
| · simp only [bottom_smul, top_smul, Triangular.off_mul, Triangular.bottom_mul, |
| BalancedTensor.add_tmul, BalancedTensor.balance, map_add, mul_smul, smul_add] |
| rw [← BalancedTensor.smul_tmul, φ.map_smul] |
| abel |
| smul_add a x y := by |
| ext <;> simp [BalancedTensor.tmul_add, smul_add, add_add_add_comm] |
| smul_zero a := by ext <;> simp |
| add_smul a b x := by |
| ext <;> simp [BalancedTensor.add_tmul, add_smul, add_add_add_comm] |
| zero_smul x := by ext <;> simp |
|
|
| section Scalars |
|
|
| variable [Module k N] |
| |
| instance : Module k (TriangularModule k R F M N φ) := (coordinateAddEquiv φ).module k |
| |
| @[simp] theorem top_scalar (c : k) (x : TriangularModule k R F M N φ) : |
| (c • x).top = c • x.top := rfl |
| @[simp] theorem bottom_scalar (c : k) (x : TriangularModule k R F M N φ) : |
| (c • x).bottom = c • x.bottom := rfl |
| |
| variable [Algebra k R] [IsScalarTower k R F] [IsScalarTower k Rᵐᵒᵖ F] |
| variable [SMulCommClass k Rᵐᵒᵖ F] |
| variable [IsScalarTower k R M] [IsScalarTower k R N] |
| |
| instance : IsScalarTower k (Triangular R F) (TriangularModule k R F M N φ) where |
| smul_assoc c a x := by |
| ext |
| · exact smul_assoc c a.top x.top |
| · simp only [bottom_smul, Triangular.off_smul, Triangular.bottom_smul, |
| bottom_scalar, BalancedTensor.scalar_tmul, LinearMap.map_smul_of_tower, smul_assoc, smul_add] |
| |
| instance [SMulCommClass k R M] [SMulCommClass k R N] : |
| SMulCommClass k (Triangular R F) (TriangularModule k R F M N φ) where |
| smul_comm c a x := by |
| ext |
| · exact smul_comm c a.top x.top |
| · simp only [bottom_scalar, bottom_smul, top_scalar, BalancedTensor.tmul_scalar, |
| LinearMap.map_smul_of_tower, smul_add, smul_comm c a.bottom] |
|
|
| def coordinateLinearEquiv : TriangularModule k R F M N φ ≃ₗ[k] M × N := |
| (coordinateAddEquiv φ).linearEquiv k |
| |
| instance [Module.Finite k M] [Module.Finite k N] : |
| Module.Finite k (TriangularModule k R F M N φ) := |
| Module.Finite.equiv (coordinateLinearEquiv φ).symm |
| |
| end Scalars |
| |
| end TriangularModule |
| end ArExplicit |
| |
| end OAI |
| |
