| import OAI.Algebra.Finitistic.Tensor.BalancedTensorMaps |
| import Mathlib.RingTheory.TensorProduct.Maps |
|
|
| namespace OAI |
| noncomputable section |
| open scoped TensorProduct |
| namespace LittleFinitistic.BalancedTensor |
| universe u |
| variable {k A M N S T : Type u} [Field k] [Ring A] [Algebra k A] |
| variable [AddCommGroup M] [Module k M] [Module Aᵐᵒᵖ M] [IsScalarTower k Aᵐᵒᵖ M] |
| variable [AddCommGroup N] [Module k N] [Module A N] [IsScalarTower k A N] |
|
|
| /-- Tensor an actual left-module homomorphism with the identity. -/ |
| def mapRight (f : N →ₗ[A] N) : Tensor k A M N →ₗ[k] Tensor k A M N := |
| lift ((mk k A M N).compl₁₂ LinearMap.id (f.restrictScalars k)) (by |
| intro a m n |
| change mk k A M N (MulOpposite.op a • m) (f n) = mk k A M N m (f (a • n)) |
| rw [f.map_smul, balance]) |
|
|
| omit [IsScalarTower k Aᵐᵒᵖ M] in |
| @[simp] theorem mapRight_mk (f : N →ₗ[A] N) (m : M) (n : N) : |
| mapRight (k := k) (M := M) f (mk k A M N m n) = mk k A M N m (f n) := rfl |
|
|
| variable [Ring S] [Algebra k S] [Module S M] [IsScalarTower k S M] |
| [SMulCommClass S Aᵐᵒᵖ M] |
|
|
| /-- The left coefficient action is induced from the first balanced-tensor factor. -/ |
| def leftAction : S →ₐ[k] Module.End k (Tensor k A M N) where |
| toFun s := mapLeft N (Algebra.lsmul k Aᵐᵒᵖ M s) |
| map_one' := by apply ext; intro m n; simp |
| map_mul' s t := by apply ext; intro m n; simp |
| map_zero' := by apply ext; intro m n; simp |
| map_add' s t := by apply ext; intro m n; simp |
| commutes' r := by apply ext; intro m n; simp |
|
|
| omit [IsScalarTower k A N] in |
| @[simp] theorem leftAction_mk (s : S) (m : M) (n : N) : |
| leftAction (k := k) (A := A) (N := N) s (mk k A M N m n) = mk k A M N (s • m) n := rfl |
|
|
| variable [Ring T] [Algebra k T] [Module T N] [IsScalarTower k T N] |
| [SMulCommClass T A N] |
|
|
| /-- The right coefficient action is induced from the second balanced-tensor factor. -/ |
| def rightAction : T →ₐ[k] Module.End k (Tensor k A M N) where |
| toFun t := mapRight (Algebra.lsmul k A N t) |
| map_one' := by apply ext; intro m n; simp |
| map_mul' s t := by apply ext; intro m n; simp |
| map_zero' := by apply ext; intro m n; simp |
| map_add' s t := by apply ext; intro m n; simp |
| commutes' r := by apply ext; intro m n; simp |
|
|
| omit [IsScalarTower k Aᵐᵒᵖ M] in |
| @[simp] theorem rightAction_mk (t : T) (m : M) (n : N) : |
| rightAction (k := k) (A := A) (M := M) t (mk k A M N m n) = mk k A M N m (t • n) := rfl |
|
|
| /-- Both outside actions on a balanced tensor, combined without forgetting either factor. -/ |
| def bimoduleAction : (S ⊗[k] T) →ₐ[k] Module.End k (Tensor k A M N) := |
| Algebra.TensorProduct.lift (leftAction (k := k) (A := A) (M := M) (N := N)) |
| (rightAction (k := k) (A := A) (M := M) (N := N)) (by |
| intro s t |
| apply ext |
| intro m n |
| rfl) |
|
|
| @[simp] theorem bimoduleAction_tmul_mk (s : S) (t : T) (m : M) (n : N) : |
| bimoduleAction (k := k) (A := A) (M := M) (N := N) (s ⊗ₜ[k] t) (mk k A M N m n) = |
| mk k A M N (s • m) (t • n) := rfl |
| end LittleFinitistic.BalancedTensor |
|
|
| end |
|
|
| end OAI |
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|
