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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
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