| import OAI.Algebra.AuslanderReiten.Tensor.BalancedTensorMap |
| import Mathlib.LinearAlgebra.TensorProduct.Associator |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace ArExplicit.BalancedTensor |
| open scoped TensorProduct |
| universe u |
| variable {k A B F G X Y Z : Type u} [CommRing k] [Ring A] [Ring B] |
| [AddCommGroup F] [AddCommGroup G] [AddCommGroup X] [AddCommGroup Y] [AddCommGroup Z] |
| [Module k F] [Module k G] [Module k X] [Module k Y] [Module k Z] |
| [Module Aᵐᵒᵖ F] [Module Bᵐᵒᵖ G] [Module A X] [Module B Y] |
|
|
| def doubleLift (b : F →ₗ[k] X →ₗ[k] G →ₗ[k] Y →ₗ[k] Z) |
| (hA : ∀ (f : F) (a : A) (x : X) (g : G) (y : Y), b (MulOpposite.op a • f) x g y = b f (a • x) g y) |
| (hB : ∀ (f : F) (x : X) (g : G) (a : B) (y : Y), b f x (MulOpposite.op a • g) y = b f x g (a • y)) : |
| BalancedTensor k A F X →ₗ[k] BalancedTensor k B G Y →ₗ[k] Z := |
| lift |
| { toFun f := |
| { toFun x := lift (b f x) (hB f x) |
| map_add' x x' := by |
| apply hom_ext |
| intro g y |
| simp only [lift_tmul, map_add, LinearMap.add_apply] |
| map_smul' a x := by |
| apply hom_ext |
| intro g y |
| simp only [lift_tmul, map_smul, LinearMap.smul_apply, RingHom.id_apply] } |
| map_add' f f' := by |
| apply LinearMap.ext |
| intro x |
| apply hom_ext |
| intro g y |
| simp only [LinearMap.coe_mk, AddHom.coe_mk, lift_tmul, map_add, LinearMap.add_apply] |
| map_smul' a f := by |
| apply LinearMap.ext |
| intro x |
| apply hom_ext |
| intro g y |
| simp only [LinearMap.coe_mk, AddHom.coe_mk, lift_tmul, map_smul, |
| LinearMap.smul_apply, RingHom.id_apply] } |
| (by |
| intro f a x |
| apply hom_ext |
| intro g y |
| exact hA f a x g y) |
|
|
| @[simp] theorem doubleLift_tmul (b : F →ₗ[k] X →ₗ[k] G →ₗ[k] Y →ₗ[k] Z) |
| (hA : ∀ (f : F) (a : A) (x : X) (g : G) (y : Y), b (MulOpposite.op a • f) x g y = b f (a • x) g y) |
| (hB : ∀ (f : F) (x : X) (g : G) (a : B) (y : Y), b f x (MulOpposite.op a • g) y = b f x g (a • y)) |
| (f : F) (x : X) (g : G) (y : Y) : |
| doubleLift b hA hB (tmul f x) (tmul g y) = b f x g y := rfl |
|
|
| def curryFour (f : ((F ⊗[k] X) ⊗[k] (G ⊗[k] Y)) →ₗ[k] Z) : |
| F →ₗ[k] X →ₗ[k] G →ₗ[k] Y →ₗ[k] Z := |
| TensorProduct.curry ((TensorProduct.lcurry (RingHom.id k) G Y Z).comp |
| (TensorProduct.curry f)) |
|
|
| @[simp] theorem curryFour_apply (h : ((F ⊗[k] X) ⊗[k] (G ⊗[k] Y)) →ₗ[k] Z) |
| (f : F) (x : X) (g : G) (y : Y) : |
| curryFour h f x g y = h ((f ⊗ₜ[k] x) ⊗ₜ[k] (g ⊗ₜ[k] y)) := rfl |
|
|
| end ArExplicit.BalancedTensor |
| |
| end |
|
|
| end OAI |
|
|
