| import OAI.Algebra.Finitistic.Tensor.BalancedTensor |
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| namespace OAI |
| noncomputable section |
| namespace LittleFinitistic.BalancedTensor |
| universe u v w w' w'' z |
| variable {k : Type u} [Field k] {A : Type v} [Ring A] [Algebra k A] |
| variable {M : Type w} [AddCommGroup M] [Module k M] [Module Aᵐᵒᵖ M] |
| variable {M' : Type w'} [AddCommGroup M'] [Module k M'] [Module Aᵐᵒᵖ M'] |
| variable {M'' : Type w''} [AddCommGroup M''] [Module k M''] [Module Aᵐᵒᵖ M''] |
| variable (N : Type z) [AddCommGroup N] [Module k N] [Module A N] |
| variable [IsScalarTower k Aᵐᵒᵖ M] [IsScalarTower k Aᵐᵒᵖ M'] |
| [IsScalarTower k Aᵐᵒᵖ M''] |
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|
| /-- Tensor an actual right-module homomorphism with the identity. -/ |
| def mapLeft (f : M →ₗ[Aᵐᵒᵖ] M') : Tensor k A M N →ₗ[k] Tensor k A M' N := |
| lift ((mk k A M' N).compl₁₂ (f.restrictScalars k) LinearMap.id) (by |
| intro a m n |
| change mk k A M' N (f (MulOpposite.op a • m)) n = mk k A M' N (f m) (a • n) |
| rw [map_smul] |
| exact balance a (f m) n) |
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| @[simp] theorem mapLeft_mk (f : M →ₗ[Aᵐᵒᵖ] M') (m : M) (n : N) : |
| mapLeft (k := k) N f (mk k A M N m n) = mk k A M' N (f m) n := rfl |
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| @[simp] theorem mapLeft_id : mapLeft (k := k) N (LinearMap.id : M →ₗ[Aᵐᵒᵖ] M) = LinearMap.id := by |
| apply ext |
| intro m n |
| rfl |
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| @[simp] theorem mapLeft_comp (f : M →ₗ[Aᵐᵒᵖ] M') (g : M' →ₗ[Aᵐᵒᵖ] M'') : |
| mapLeft (k := k) N (g.comp f) = (mapLeft (k := k) N g).comp (mapLeft (k := k) N f) := by |
| apply ext |
| intro m n |
| rfl |
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| @[simp] theorem mapLeft_add (f g : M →ₗ[Aᵐᵒᵖ] M') : |
| mapLeft (k := k) N (f+g) = mapLeft (k := k) N f + mapLeft (k := k) N g := by |
| apply ext |
| intro m n |
| simp |
| end LittleFinitistic.BalancedTensor |
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| end |
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| end OAI |
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