openAI-math-dataset / lean /OAI /Algebra /Finitistic /Ordinary /OrdinaryAssociativeTensorModules.lean
| import OAI.Algebra.Finitistic.Ordinary.OrdinaryBalancedActions | |
| namespace OAI | |
| noncomputable section | |
| namespace LittleFinitistic | |
| universe u | |
| /-- A tensor retaining the right action of the middle bimodule. -/ | |
| def RightTensor (k A : Type u) (_ : Type u) (M N : Type u) [Field k] [Ring A] [Algebra k A] | |
| [AddCommGroup M] [Module k M] [Module Aᵐᵒᵖ M] | |
| [AddCommGroup N] [Module k N] [Module A N] := BalancedTensor.Tensor k A M N | |
| /-- A tensor retaining the left action of the middle bimodule. -/ | |
| def LeftTensor (k B : Type u) (_ : Type u) (N Z : Type u) [Field k] [Ring B] [Algebra k B] | |
| [AddCommGroup N] [Module k N] [Module Bᵐᵒᵖ N] | |
| [AddCommGroup Z] [Module k Z] [Module B Z] := BalancedTensor.Tensor k B N Z | |
| variable (k A B M N Z : Type u) [Field k] [Ring A] [Ring B] [Algebra k A] [Algebra k B] | |
| variable [AddCommGroup M] [Module k M] [Module Aᵐᵒᵖ M] [IsScalarTower k Aᵐᵒᵖ M] | |
| variable [AddCommGroup N] [Module k N] [Module A N] [Module Bᵐᵒᵖ N] | |
| [IsScalarTower k A N] [IsScalarTower k Bᵐᵒᵖ N] [SMulCommClass A Bᵐᵒᵖ N] | |
| variable [AddCommGroup Z] [Module k Z] [Module B Z] [IsScalarTower k B Z] | |
| namespace RightTensor | |
| instance : AddCommGroup (RightTensor k A B M N) := inferInstanceAs (AddCommGroup (BalancedTensor.Tensor k A M N)) | |
| instance : Module k (RightTensor k A B M N) := inferInstanceAs (Module k (BalancedTensor.Tensor k A M N)) | |
| local instance : SMulCommClass Bᵐᵒᵖ A N := ⟨fun b a n => (smul_comm a b n).symm⟩ | |
| def action : Bᵐᵒᵖ →ₐ[k] Module.End k (RightTensor k A B M N) := | |
| BalancedTensor.rightAction | |
| instance : Module Bᵐᵒᵖ (RightTensor k A B M N) := Module.compHom _ (action k A B M N).toRingHom | |
| instance : IsScalarTower k Bᵐᵒᵖ (RightTensor k A B M N) where | |
| smul_assoc r b z := by | |
| change action k A B M N (r • b) z = r • action k A B M N b z | |
| rw [map_smul] | |
| rfl | |
| def mk : M →ₗ[k] N →ₗ[k] RightTensor k A B M N := BalancedTensor.mk k A M N | |
| omit [IsScalarTower k Aᵐᵒᵖ M] in | |
| @[simp] theorem smul_mk (b : Bᵐᵒᵖ) (m : M) (n : N) : | |
| b • mk k A B M N m n = mk k A B M N m (b • n) := rfl | |
| end RightTensor | |
| namespace LeftTensor | |
| instance : AddCommGroup (LeftTensor k B A N Z) := inferInstanceAs (AddCommGroup (BalancedTensor.Tensor k B N Z)) | |
| instance : Module k (LeftTensor k B A N Z) := inferInstanceAs (Module k (BalancedTensor.Tensor k B N Z)) | |
| def action : A →ₐ[k] Module.End k (LeftTensor k B A N Z) := BalancedTensor.leftAction | |
| instance : Module A (LeftTensor k B A N Z) := Module.compHom _ (action k A B N Z).toRingHom | |
| instance : IsScalarTower k A (LeftTensor k B A N Z) where | |
| smul_assoc r a z := by | |
| change action k A B N Z (r • a) z = r • action k A B N Z a z | |
| rw [map_smul] | |
| rfl | |
| def mk : N →ₗ[k] Z →ₗ[k] LeftTensor k B A N Z := BalancedTensor.mk k B N Z | |
| omit [IsScalarTower k B Z] in | |
| @[simp] theorem smul_mk (a : A) (n : N) (z : Z) : | |
| a • mk k A B N Z n z = mk k A B N Z (a • n) z := rfl | |
| end LeftTensor | |
| end LittleFinitistic | |
| end | |
| end OAI | |