The A-module structure on M ⊗[R] N

When M is both an R-module and an A-module, and Algebra R A, then many of the morphisms preserve the actions by A.

The Module instance itself is provided elsewhere as TensorProduct.leftModule. This file provides more general versions of the definitions already in LinearAlgebra/TensorProduct.

In this file, we use the convention that M, N, P, Q are all R-modules, but only M and P are simultaneously A-modules.

Main definitions

• TensorProduct.AlgebraTensorModule.curry
• TensorProduct.AlgebraTensorModule.uncurry
• TensorProduct.AlgebraTensorModule.lcurry
• TensorProduct.AlgebraTensorModule.lift
• TensorProduct.AlgebraTensorModule.lift.equiv
• TensorProduct.AlgebraTensorModule.mk
• TensorProduct.AlgebraTensorModule.map
• TensorProduct.AlgebraTensorModule.mapBilinear
• TensorProduct.AlgebraTensorModule.congr
• TensorProduct.AlgebraTensorModule.rid
• TensorProduct.AlgebraTensorModule.homTensorHomMap
• TensorProduct.AlgebraTensorModule.assoc
• TensorProduct.AlgebraTensorModule.leftComm
• TensorProduct.AlgebraTensorModule.rightComm
• TensorProduct.AlgebraTensorModule.tensorTensorTensorComm

Implementation notes

We could thus consider replacing the less general definitions with these ones. If we do this, we probably should still implement the less general ones as abbreviations to the more general ones with fewer type arguments.

theorem TensorProduct.AlgebraTensorModule.smul_eq_lsmul_rTensor {R : Type uR} {A : Type uA} {M : Type uM} {N : Type uN} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] (a : A) (x : TensorProduct R M N) : a • x = (LinearMap.rTensor N ((Algebra.lsmul R R M) a)) x

noncomputable def TensorProduct.AlgebraTensorModule.curry {R : Type uR} {A : Type uA} {M : Type uM} {N : Type uN} {P : Type uP} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module R P] [Module A P] [IsScalarTower R A P] (f : TensorProduct R M N →ₗ[A] P) : M →ₗ[A] N →ₗ[R] P

Heterobasic version of TensorProduct.curry:

Given a linear map M ⊗[R] N →[A] P, compose it with the canonical bilinear map M →[A] N →[R] M ⊗[R] N to form a bilinear map M →[A] N →[R] P.

Equations

• TensorProduct.AlgebraTensorModule.curry f = { toFun := ⇑(TensorProduct.curry (↑R f)), map_add' := ⋯, map_smul' := ⋯ }

@[simp]

theorem TensorProduct.AlgebraTensorModule.curry_apply {R : Type uR} {A : Type uA} {M : Type uM} {N : Type uN} {P : Type uP} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module R P] [Module A P] [IsScalarTower R A P] (f : TensorProduct R M N →ₗ[A] P) (a : M) : (curry f) a = (TensorProduct.curry (↑R f)) a

theorem TensorProduct.AlgebraTensorModule.restrictScalars_curry {R : Type uR} {A : Type uA} {M : Type uM} {N : Type uN} {P : Type uP} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module R P] [Module A P] [IsScalarTower R A P] (f : TensorProduct R M N →ₗ[A] P) : ↑R (curry f) = TensorProduct.curry (↑R f)

theorem TensorProduct.AlgebraTensorModule.curry_injective {R : Type uR} {A : Type uA} {M : Type uM} {N : Type uN} {P : Type uP} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module R P] [Module A P] [IsScalarTower R A P] : Function.Injective curry

Just as TensorProduct.ext is marked ext instead of TensorProduct.ext', this is a better ext lemma than TensorProduct.AlgebraTensorModule.ext below.

See note [partially-applied ext lemmas].

theorem TensorProduct.AlgebraTensorModule.curry_injective_iff {R : Type uR} {A : Type uA} {M : Type uM} {N : Type uN} {P : Type uP} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module R P] [Module A P] [IsScalarTower R A P] {a₁ a₂ : TensorProduct R M N →ₗ[A] P} : a₁ = a₂ ↔ curry a₁ = curry a₂

theorem TensorProduct.AlgebraTensorModule.ext {R : Type uR} {A : Type uA} {M : Type uM} {N : Type uN} {P : Type uP} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module R P] [Module A P] [IsScalarTower R A P] {g h : TensorProduct R M N →ₗ[A] P} (H : ∀ (x : M) (y : N), g (x ⊗ₜ[R] y) = h (x ⊗ₜ[R] y)) : g = h

noncomputable def TensorProduct.AlgebraTensorModule.lift {R : Type uR} {A : Type uA} {M : Type uM} {N : Type uN} {P : Type uP} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module R P] [Module A P] [IsScalarTower R A P] (f : M →ₗ[A] N →ₗ[R] P) : TensorProduct R M N →ₗ[A] P

Heterobasic version of TensorProduct.lift:

Constructing a linear map M ⊗[R] N →[A] P given a bilinear map M →[A] N →[R] P with the property that its composition with the canonical bilinear map M →[A] N →[R] M ⊗[R] N is the given bilinear map M →[A] N →[R] P.

Equations

• TensorProduct.AlgebraTensorModule.lift f = { toAddHom := (TensorProduct.lift (↑R f)).toAddHom, map_smul' := ⋯ }

@[simp]

theorem TensorProduct.AlgebraTensorModule.lift_apply {R : Type uR} {A : Type uA} {M : Type uM} {N : Type uN} {P : Type uP} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module R P] [Module A P] [IsScalarTower R A P] (f : M →ₗ[A] N →ₗ[R] P) (a : TensorProduct R M N) : (lift f) a = (TensorProduct.lift (↑R f)) a

@[simp]

theorem TensorProduct.AlgebraTensorModule.lift_tmul {R : Type uR} {A : Type uA} {M : Type uM} {N : Type uN} {P : Type uP} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module R P] [Module A P] [IsScalarTower R A P] (f : M →ₗ[A] N →ₗ[R] P) (x : M) (y : N) : (lift f) (x ⊗ₜ[R] y) = (f x) y

noncomputable def TensorProduct.AlgebraTensorModule.uncurry (R : Type uR) (A : Type uA) (B : Type uB) (M : Type uM) (N : Type uN) (P : Type uP) [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module R P] [Module A P] [IsScalarTower R A P] [Module B P] [IsScalarTower R B P] [SMulCommClass A B P] : (M →ₗ[A] N →ₗ[R] P) →ₗ[B] TensorProduct R M N →ₗ[A] P

Heterobasic version of TensorProduct.uncurry:

Linearly constructing a linear map M ⊗[R] N →[A] P given a bilinear map M →[A] N →[R] P with the property that its composition with the canonical bilinear map M →[A] N →[R] M ⊗[R] N is the given bilinear map M →[A] N →[R] P.

Equations

• TensorProduct.AlgebraTensorModule.uncurry R A B M N P = { toFun := TensorProduct.AlgebraTensorModule.lift, map_add' := ⋯, map_smul' := ⋯ }

@[simp]

theorem TensorProduct.AlgebraTensorModule.uncurry_apply (R : Type uR) (A : Type uA) (B : Type uB) (M : Type uM) (N : Type uN) (P : Type uP) [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module R P] [Module A P] [IsScalarTower R A P] [Module B P] [IsScalarTower R B P] [SMulCommClass A B P] (f : M →ₗ[A] N →ₗ[R] P) : (uncurry R A B M N P) f = lift f

noncomputable def TensorProduct.AlgebraTensorModule.lcurry (R : Type uR) (A : Type uA) (B : Type uB) (M : Type uM) (N : Type uN) (P : Type uP) [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module R P] [Module A P] [IsScalarTower R A P] [Module B P] [IsScalarTower R B P] [SMulCommClass A B P] : (TensorProduct R M N →ₗ[A] P) →ₗ[B] M →ₗ[A] N →ₗ[R] P

Heterobasic version of TensorProduct.lcurry:

Given a linear map M ⊗[R] N →[A] P, compose it with the canonical bilinear map M →[A] N →[R] M ⊗[R] N to form a bilinear map M →[A] N →[R] P.

Equations

• TensorProduct.AlgebraTensorModule.lcurry R A B M N P = { toFun := TensorProduct.AlgebraTensorModule.curry, map_add' := ⋯, map_smul' := ⋯ }

@[simp]

theorem TensorProduct.AlgebraTensorModule.lcurry_apply (R : Type uR) (A : Type uA) (B : Type uB) (M : Type uM) (N : Type uN) (P : Type uP) [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module R P] [Module A P] [IsScalarTower R A P] [Module B P] [IsScalarTower R B P] [SMulCommClass A B P] (f : TensorProduct R M N →ₗ[A] P) : (lcurry R A B M N P) f = curry f

noncomputable def TensorProduct.AlgebraTensorModule.lift.equiv (R : Type uR) (A : Type uA) (B : Type uB) (M : Type uM) (N : Type uN) (P : Type uP) [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module R P] [Module A P] [IsScalarTower R A P] [Module B P] [IsScalarTower R B P] [SMulCommClass A B P] : (M →ₗ[A] N →ₗ[R] P) ≃ₗ[B] TensorProduct R M N →ₗ[A] P

Heterobasic version of TensorProduct.lift.equiv:

A linear equivalence constructing a linear map M ⊗[R] N →[A] P given a bilinear map M →[A] N →[R] P with the property that its composition with the canonical bilinear map M →[A] N →[R] M ⊗[R] N is the given bilinear map M →[A] N →[R] P.

Equations

• One or more equations did not get rendered due to their size.

noncomputable def TensorProduct.AlgebraTensorModule.mk (R : Type uR) [CommSemiring R] (A : Type u_1) (M : Type u_2) (N : Type u_3) [Semiring A] [AddCommMonoid M] [Module R M] [Module A M] [SMulCommClass R A M] [AddCommMonoid N] [Module R N] : M →ₗ[A] N →ₗ[R] TensorProduct R M N

Heterobasic version of TensorProduct.mk:

The canonical bilinear map M →[A] N →[R] M ⊗[R] N.

Equations

• TensorProduct.AlgebraTensorModule.mk R A M N = { toAddHom := (TensorProduct.mk R M N).toAddHom, map_smul' := ⋯ }

@[simp]

theorem TensorProduct.AlgebraTensorModule.mk_apply (R : Type uR) [CommSemiring R] (A : Type u_1) (M : Type u_2) (N : Type u_3) [Semiring A] [AddCommMonoid M] [Module R M] [Module A M] [SMulCommClass R A M] [AddCommMonoid N] [Module R N] (m : M) : (mk R A M N) m = { toFun := fun (x2 : N) => m ⊗ₜ[R] x2, map_add' := ⋯, map_smul' := ⋯ }

noncomputable def TensorProduct.AlgebraTensorModule.map {R : Type uR} {A : Type uA} {M : Type uM} {N : Type uN} {P : Type uP} {Q : Type uQ} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module R P] [Module A P] [IsScalarTower R A P] [AddCommMonoid Q] [Module R Q] (f : M →ₗ[A] P) (g : N →ₗ[R] Q) : TensorProduct R M N →ₗ[A] TensorProduct R P Q

Heterobasic version of TensorProduct.map

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem TensorProduct.AlgebraTensorModule.map_tmul {R : Type uR} {A : Type uA} {M : Type uM} {N : Type uN} {P : Type uP} {Q : Type uQ} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module R P] [Module A P] [IsScalarTower R A P] [AddCommMonoid Q] [Module R Q] (f : M →ₗ[A] P) (g : N →ₗ[R] Q) (m : M) (n : N) : (map f g) (m ⊗ₜ[R] n) = f m ⊗ₜ[R] g n

@[simp]

theorem TensorProduct.AlgebraTensorModule.map_id {R : Type uR} {A : Type uA} {M : Type uM} {N : Type uN} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] : map LinearMap.id LinearMap.id = LinearMap.id

theorem TensorProduct.AlgebraTensorModule.map_comp {R : Type uR} {A : Type uA} {M : Type uM} {N : Type uN} {P : Type uP} {Q : Type uQ} {P' : Type uP'} {Q' : Type uQ'} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module R P] [Module A P] [IsScalarTower R A P] [AddCommMonoid Q] [Module R Q] [AddCommMonoid P'] [Module R P'] [Module A P'] [IsScalarTower R A P'] [AddCommMonoid Q'] [Module R Q'] (f₂ : P →ₗ[A] P') (f₁ : M →ₗ[A] P) (g₂ : Q →ₗ[R] Q') (g₁ : N →ₗ[R] Q) : map (f₂ ∘ₗ f₁) (g₂ ∘ₗ g₁) = map f₂ g₂ ∘ₗ map f₁ g₁

@[simp]

theorem TensorProduct.AlgebraTensorModule.map_one {R : Type uR} {A : Type uA} {M : Type uM} {N : Type uN} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] : map 1 1 = 1

theorem TensorProduct.AlgebraTensorModule.map_mul {R : Type uR} {A : Type uA} {M : Type uM} {N : Type uN} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] (f₁ f₂ : M →ₗ[A] M) (g₁ g₂ : N →ₗ[R] N) : map (f₁ * f₂) (g₁ * g₂) = map f₁ g₁ * map f₂ g₂

theorem TensorProduct.AlgebraTensorModule.map_add_left {R : Type uR} {A : Type uA} {M : Type uM} {N : Type uN} {P : Type uP} {Q : Type uQ} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module R P] [Module A P] [IsScalarTower R A P] [AddCommMonoid Q] [Module R Q] (f₁ f₂ : M →ₗ[A] P) (g : N →ₗ[R] Q) : map (f₁ + f₂) g = map f₁ g + map f₂ g

theorem TensorProduct.AlgebraTensorModule.map_add_right {R : Type uR} {A : Type uA} {M : Type uM} {N : Type uN} {P : Type uP} {Q : Type uQ} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module R P] [Module A P] [IsScalarTower R A P] [AddCommMonoid Q] [Module R Q] (f : M →ₗ[A] P) (g₁ g₂ : N →ₗ[R] Q) : map f (g₁ + g₂) = map f g₁ + map f g₂

theorem TensorProduct.AlgebraTensorModule.map_smul_right {R : Type uR} {A : Type uA} {M : Type uM} {N : Type uN} {P : Type uP} {Q : Type uQ} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module R P] [Module A P] [IsScalarTower R A P] [AddCommMonoid Q] [Module R Q] (r : R) (f : M →ₗ[A] P) (g : N →ₗ[R] Q) : map f (r • g) = r • map f g

theorem TensorProduct.AlgebraTensorModule.map_smul_left {R : Type uR} {A : Type uA} {B : Type uB} {M : Type uM} {N : Type uN} {P : Type uP} {Q : Type uQ} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module R P] [Module A P] [IsScalarTower R A P] [AddCommMonoid Q] [Module R Q] [Module B P] [IsScalarTower R B P] [SMulCommClass A B P] (b : B) (f : M →ₗ[A] P) (g : N →ₗ[R] Q) : map (b • f) g = b • map f g

noncomputable def TensorProduct.AlgebraTensorModule.lTensor {R : Type uR} (A : Type uA) (M : Type uM) {N : Type uN} {Q : Type uQ} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] [AddCommMonoid Q] [Module R Q] : (N →ₗ[R] Q) →ₗ[R] TensorProduct R M N →ₗ[A] TensorProduct R M Q

Heterobasic version of LinearMap.lTensor

Equations

• TensorProduct.AlgebraTensorModule.lTensor A M = { toFun := fun (f : N →ₗ[R] Q) => TensorProduct.AlgebraTensorModule.map LinearMap.id f, map_add' := ⋯, map_smul' := ⋯ }

@[simp]

theorem TensorProduct.AlgebraTensorModule.coe_lTensor {R : Type uR} {A : Type uA} {M : Type uM} {N : Type uN} {Q : Type uQ} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] [AddCommMonoid Q] [Module R Q] (f : N →ₗ[R] Q) : ⇑((lTensor A M) f) = ⇑(LinearMap.lTensor M f)

@[simp]

theorem TensorProduct.AlgebraTensorModule.restrictScalars_lTensor {R : Type uR} {A : Type uA} {M : Type uM} {N : Type uN} {Q : Type uQ} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] [AddCommMonoid Q] [Module R Q] (f : N →ₗ[R] Q) : ↑R ((lTensor A M) f) = LinearMap.lTensor M f

@[simp]

theorem TensorProduct.AlgebraTensorModule.lTensor_tmul {R : Type uR} {A : Type uA} {M : Type uM} {N : Type uN} {Q : Type uQ} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] [AddCommMonoid Q] [Module R Q] (f : N →ₗ[R] Q) (m : M) (n : N) : ((lTensor A M) f) (m ⊗ₜ[R] n) = m ⊗ₜ[R] f n

@[simp]

theorem TensorProduct.AlgebraTensorModule.lTensor_id {R : Type uR} {A : Type uA} {M : Type uM} {N : Type uN} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] : (lTensor A M) LinearMap.id = LinearMap.id

theorem TensorProduct.AlgebraTensorModule.lTensor_comp {R : Type uR} {A : Type uA} {M : Type uM} {N : Type uN} {Q : Type uQ} {Q' : Type uQ'} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] [AddCommMonoid Q] [Module R Q] [AddCommMonoid Q'] [Module R Q'] (f₂ : Q →ₗ[R] Q') (f₁ : N →ₗ[R] Q) : (lTensor A M) (f₂ ∘ₗ f₁) = (lTensor A M) f₂ ∘ₗ (lTensor A M) f₁

@[simp]

theorem TensorProduct.AlgebraTensorModule.lTensor_one {R : Type uR} {A : Type uA} {M : Type uM} {N : Type uN} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] : (lTensor A M) 1 = 1

theorem TensorProduct.AlgebraTensorModule.lTensor_mul {R : Type uR} {A : Type uA} {M : Type uM} {N : Type uN} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] (f₁ f₂ : N →ₗ[R] N) : (lTensor A M) (f₁ * f₂) = (lTensor A M) f₁ * (lTensor A M) f₂

noncomputable def TensorProduct.AlgebraTensorModule.rTensor (R : Type uR) {A : Type uA} {M : Type uM} (N : Type uN) {P : Type uP} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module R P] [Module A P] [IsScalarTower R A P] : (M →ₗ[A] P) →ₗ[R] TensorProduct R M N →ₗ[A] TensorProduct R P N

Heterobasic version of LinearMap.rTensor

Equations

• TensorProduct.AlgebraTensorModule.rTensor R N = { toFun := fun (f : M →ₗ[A] P) => TensorProduct.AlgebraTensorModule.map f LinearMap.id, map_add' := ⋯, map_smul' := ⋯ }

@[simp]

theorem TensorProduct.AlgebraTensorModule.coe_rTensor {R : Type uR} {A : Type uA} {M : Type uM} {N : Type uN} {P : Type uP} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module R P] [Module A P] [IsScalarTower R A P] (f : M →ₗ[A] P) : ⇑((rTensor R N) f) = ⇑(LinearMap.rTensor N (↑R f))

@[simp]

theorem TensorProduct.AlgebraTensorModule.restrictScalars_rTensor {R : Type uR} {A : Type uA} {M : Type uM} {N : Type uN} {P : Type uP} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module R P] [Module A P] [IsScalarTower R A P] (f : M →ₗ[A] P) : ↑R ((rTensor R N) f) = LinearMap.rTensor N (↑R f)

@[simp]

theorem TensorProduct.AlgebraTensorModule.rTensor_tmul {R : Type uR} {A : Type uA} {M : Type uM} {N : Type uN} {P : Type uP} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module R P] [Module A P] [IsScalarTower R A P] (f : M →ₗ[A] P) (m : M) (n : N) : ((rTensor R N) f) (m ⊗ₜ[R] n) = f m ⊗ₜ[R] n

@[simp]

theorem TensorProduct.AlgebraTensorModule.rTensor_id {R : Type uR} {A : Type uA} {M : Type uM} {N : Type uN} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] : (rTensor R N) LinearMap.id = LinearMap.id

theorem TensorProduct.AlgebraTensorModule.rTensor_comp {R : Type uR} {A : Type uA} {M : Type uM} {N : Type uN} {P : Type uP} {P' : Type uP'} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module R P] [Module A P] [IsScalarTower R A P] [AddCommMonoid P'] [Module R P'] [Module A P'] [IsScalarTower R A P'] (f₂ : P →ₗ[A] P') (f₁ : M →ₗ[A] P) : (rTensor R N) (f₂ ∘ₗ f₁) = (rTensor R N) f₂ ∘ₗ (rTensor R N) f₁

@[simp]

theorem TensorProduct.AlgebraTensorModule.rTensor_one {R : Type uR} {A : Type uA} {M : Type uM} {N : Type uN} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] : (rTensor R N) 1 = 1

theorem TensorProduct.AlgebraTensorModule.rTensor_mul {R : Type uR} {A : Type uA} {M : Type uM} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] (f₁ f₂ : M →ₗ[A] M) : (rTensor R M) (f₁ * f₂) = (rTensor R M) f₁ * (rTensor R M) f₂

noncomputable def TensorProduct.AlgebraTensorModule.mapBilinear (R : Type uR) (A : Type uA) (B : Type uB) (M : Type uM) (N : Type uN) (P : Type uP) (Q : Type uQ) [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module R P] [Module A P] [IsScalarTower R A P] [AddCommMonoid Q] [Module R Q] [Module B P] [IsScalarTower R B P] [SMulCommClass A B P] : (M →ₗ[A] P) →ₗ[B] (N →ₗ[R] Q) →ₗ[R] TensorProduct R M N →ₗ[A] TensorProduct R P Q

Heterobasic version of TensorProduct.map_bilinear

Equations

• TensorProduct.AlgebraTensorModule.mapBilinear R A B M N P Q = LinearMap.mk₂' B R TensorProduct.AlgebraTensorModule.map ⋯ ⋯ ⋯ ⋯

@[simp]

theorem TensorProduct.AlgebraTensorModule.mapBilinear_apply {R : Type uR} {A : Type uA} {B : Type uB} {M : Type uM} {N : Type uN} {P : Type uP} {Q : Type uQ} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module R P] [Module A P] [IsScalarTower R A P] [AddCommMonoid Q] [Module R Q] [Module B P] [IsScalarTower R B P] [SMulCommClass A B P] (f : M →ₗ[A] P) (g : N →ₗ[R] Q) : ((mapBilinear R A B M N P Q) f) g = map f g

noncomputable def TensorProduct.AlgebraTensorModule.homTensorHomMap (R : Type uR) (A : Type uA) (B : Type uB) (M : Type uM) (N : Type uN) (P : Type uP) (Q : Type uQ) [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module R P] [Module A P] [IsScalarTower R A P] [AddCommMonoid Q] [Module R Q] [Module B P] [IsScalarTower R B P] [SMulCommClass A B P] : TensorProduct R (M →ₗ[A] P) (N →ₗ[R] Q) →ₗ[B] TensorProduct R M N →ₗ[A] TensorProduct R P Q

Heterobasic version of TensorProduct.homTensorHomMap

Equations

• TensorProduct.AlgebraTensorModule.homTensorHomMap R A B M N P Q = TensorProduct.AlgebraTensorModule.lift (TensorProduct.AlgebraTensorModule.mapBilinear R A B M N P Q)

@[simp]

theorem TensorProduct.AlgebraTensorModule.homTensorHomMap_apply {R : Type uR} {A : Type uA} {B : Type uB} {M : Type uM} {N : Type uN} {P : Type uP} {Q : Type uQ} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module R P] [Module A P] [IsScalarTower R A P] [AddCommMonoid Q] [Module R Q] [Module B P] [IsScalarTower R B P] [SMulCommClass A B P] (f : M →ₗ[A] P) (g : N →ₗ[R] Q) : (homTensorHomMap R A B M N P Q) (f ⊗ₜ[R] g) = map f g

noncomputable def TensorProduct.AlgebraTensorModule.congr {R : Type uR} {A : Type uA} {M : Type uM} {N : Type uN} {P : Type uP} {Q : Type uQ} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module R P] [Module A P] [IsScalarTower R A P] [AddCommMonoid Q] [Module R Q] (f : M ≃ₗ[A] P) (g : N ≃ₗ[R] Q) : TensorProduct R M N ≃ₗ[A] TensorProduct R P Q

Heterobasic version of TensorProduct.congr

Equations

• TensorProduct.AlgebraTensorModule.congr f g = LinearEquiv.ofLinear (TensorProduct.AlgebraTensorModule.map ↑f ↑g) (TensorProduct.AlgebraTensorModule.map ↑f.symm ↑g.symm) ⋯ ⋯

@[simp]

theorem TensorProduct.AlgebraTensorModule.congr_refl {R : Type uR} {A : Type uA} {M : Type uM} {N : Type uN} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] : congr (LinearEquiv.refl A M) (LinearEquiv.refl R N) = LinearEquiv.refl A (TensorProduct R M N)

theorem TensorProduct.AlgebraTensorModule.congr_trans {R : Type uR} {A : Type uA} {M : Type uM} {N : Type uN} {P : Type uP} {Q : Type uQ} {P' : Type uP'} {Q' : Type uQ'} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module R P] [Module A P] [IsScalarTower R A P] [AddCommMonoid Q] [Module R Q] [AddCommMonoid P'] [Module R P'] [Module A P'] [IsScalarTower R A P'] [AddCommMonoid Q'] [Module R Q'] (f₁ : M ≃ₗ[A] P) (f₂ : P ≃ₗ[A] P') (g₁ : N ≃ₗ[R] Q) (g₂ : Q ≃ₗ[R] Q') : congr (f₁ ≪≫ₗ f₂) (g₁ ≪≫ₗ g₂) = congr f₁ g₁ ≪≫ₗ congr f₂ g₂

theorem TensorProduct.AlgebraTensorModule.congr_symm {R : Type uR} {A : Type uA} {M : Type uM} {N : Type uN} {P : Type uP} {Q : Type uQ} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module R P] [Module A P] [IsScalarTower R A P] [AddCommMonoid Q] [Module R Q] (f : M ≃ₗ[A] P) (g : N ≃ₗ[R] Q) : congr f.symm g.symm = (congr f g).symm

@[simp]

theorem TensorProduct.AlgebraTensorModule.congr_one {R : Type uR} {A : Type uA} {M : Type uM} {N : Type uN} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] : congr 1 1 = 1

theorem TensorProduct.AlgebraTensorModule.congr_mul {R : Type uR} {A : Type uA} {M : Type uM} {N : Type uN} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] (f₁ f₂ : M ≃ₗ[A] M) (g₁ g₂ : N ≃ₗ[R] N) : congr (f₁ * f₂) (g₁ * g₂) = congr f₁ g₁ * congr f₂ g₂

@[simp]

theorem TensorProduct.AlgebraTensorModule.congr_tmul {R : Type uR} {A : Type uA} {M : Type uM} {N : Type uN} {P : Type uP} {Q : Type uQ} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module R P] [Module A P] [IsScalarTower R A P] [AddCommMonoid Q] [Module R Q] (f : M ≃ₗ[A] P) (g : N ≃ₗ[R] Q) (m : M) (n : N) : (congr f g) (m ⊗ₜ[R] n) = f m ⊗ₜ[R] g n

@[simp]

theorem TensorProduct.AlgebraTensorModule.congr_symm_tmul {R : Type uR} {A : Type uA} {M : Type uM} {N : Type uN} {P : Type uP} {Q : Type uQ} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module R P] [Module A P] [IsScalarTower R A P] [AddCommMonoid Q] [Module R Q] (f : M ≃ₗ[A] P) (g : N ≃ₗ[R] Q) (p : P) (q : Q) : (congr f g).symm (p ⊗ₜ[R] q) = f.symm p ⊗ₜ[R] g.symm q

noncomputable def TensorProduct.AlgebraTensorModule.rid (R : Type uR) (A : Type uA) (M : Type uM) [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] : TensorProduct R M R ≃ₗ[A] M

Heterobasic version of TensorProduct.rid.

Equations

• One or more equations did not get rendered due to their size.

theorem TensorProduct.AlgebraTensorModule.rid_eq_rid (R : Type uR) (M : Type uM) [CommSemiring R] [AddCommMonoid M] [Module R M] : AlgebraTensorModule.rid R R M = TensorProduct.rid R M

@[simp]

theorem TensorProduct.AlgebraTensorModule.rid_tmul {R : Type uR} (A : Type uA) {M : Type uM} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] (r : R) (m : M) : (AlgebraTensorModule.rid R A M) (m ⊗ₜ[R] r) = r • m

@[simp]

theorem TensorProduct.AlgebraTensorModule.rid_symm_apply (R : Type uR) (A : Type uA) {M : Type uM} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] (m : M) : (AlgebraTensorModule.rid R A M).symm m = m ⊗ₜ[R] 1

noncomputable def TensorProduct.AlgebraTensorModule.assoc (R : Type uR) (A : Type uA) (B : Type uB) (M : Type uM) (P : Type uP) (Q : Type uQ) [CommSemiring R] [CommSemiring A] [Semiring B] [Algebra R A] [Algebra R B] [AddCommMonoid M] [Module R M] [Module A M] [Module B M] [IsScalarTower R A M] [IsScalarTower R B M] [SMulCommClass A B M] [AddCommMonoid P] [Module A P] [AddCommMonoid Q] [Module R Q] [Module R P] [IsScalarTower R A P] [Algebra A B] [IsScalarTower A B M] : TensorProduct R (TensorProduct A M P) Q ≃ₗ[B] TensorProduct A M (TensorProduct R P Q)

Heterobasic version of TensorProduct.assoc:

B-linear equivalence between (M ⊗[A] P) ⊗[R] Q and M ⊗[A] (P ⊗[R] Q).

Note this is especially useful with A = R (where it is a "more linear" version of TensorProduct.assoc), or with B = A.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem TensorProduct.AlgebraTensorModule.assoc_tmul (R : Type uR) (A : Type uA) (B : Type uB) {M : Type uM} {P : Type uP} {Q : Type uQ} [CommSemiring R] [CommSemiring A] [Semiring B] [Algebra R A] [Algebra R B] [AddCommMonoid M] [Module R M] [Module A M] [Module B M] [IsScalarTower R A M] [IsScalarTower R B M] [SMulCommClass A B M] [AddCommMonoid P] [Module A P] [AddCommMonoid Q] [Module R Q] [Module R P] [IsScalarTower R A P] [Algebra A B] [IsScalarTower A B M] (m : M) (p : P) (q : Q) : (assoc R A B M P Q) ((m ⊗ₜ[A] p) ⊗ₜ[R] q) = m ⊗ₜ[A] p ⊗ₜ[R] q

@[simp]

theorem TensorProduct.AlgebraTensorModule.assoc_symm_tmul (R : Type uR) (A : Type uA) (B : Type uB) {M : Type uM} {P : Type uP} {Q : Type uQ} [CommSemiring R] [CommSemiring A] [Semiring B] [Algebra R A] [Algebra R B] [AddCommMonoid M] [Module R M] [Module A M] [Module B M] [IsScalarTower R A M] [IsScalarTower R B M] [SMulCommClass A B M] [AddCommMonoid P] [Module A P] [AddCommMonoid Q] [Module R Q] [Module R P] [IsScalarTower R A P] [Algebra A B] [IsScalarTower A B M] (m : M) (p : P) (q : Q) : (assoc R A B M P Q).symm (m ⊗ₜ[A] p ⊗ₜ[R] q) = (m ⊗ₜ[A] p) ⊗ₜ[R] q

theorem TensorProduct.AlgebraTensorModule.rTensor_tensor (R : Type uR) (A : Type uA) {M : Type uM} {N : Type uN} {P : Type uP} (P' : Type uP') [CommSemiring R] [CommSemiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module A P] [AddCommMonoid P'] [Module A P'] [Module R P] [IsScalarTower R A P] [Module R P'] [IsScalarTower R A P'] (g : P →ₗ[A] P') : LinearMap.rTensor (TensorProduct R M N) g = ↑(assoc R A A P' M N) ∘ₗ map (LinearMap.rTensor M g) LinearMap.id ∘ₗ ↑(assoc R A A P M N).symm

noncomputable def TensorProduct.AlgebraTensorModule.cancelBaseChange (R : Type uR) (A : Type uA) (B : Type uB) (M : Type uM) (N : Type uN) [CommSemiring R] [CommSemiring A] [Semiring B] [Algebra R A] [Algebra R B] [AddCommMonoid M] [Module R M] [Module A M] [Module B M] [IsScalarTower R A M] [IsScalarTower R B M] [SMulCommClass A B M] [AddCommMonoid N] [Module R N] [Algebra A B] [IsScalarTower A B M] : TensorProduct A M (TensorProduct R A N) ≃ₗ[B] TensorProduct R M N

B-linear equivalence between M ⊗[A] (A ⊗[R] N) and M ⊗[R] N. In particular useful with B = A.

Equations

• One or more equations did not get rendered due to their size.

noncomputable def TensorProduct.AlgebraTensorModule.distribBaseChange (R : Type uR) (A : Type uA) (M : Type uM) (N : Type uN) [CommSemiring R] [CommSemiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] : TensorProduct R A (TensorProduct R M N) ≃ₗ[A] TensorProduct A (TensorProduct R A M) (TensorProduct R A N)

Base change distributes over tensor product.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem TensorProduct.AlgebraTensorModule.cancelBaseChange_tmul (R : Type uR) (A : Type uA) (B : Type uB) {M : Type uM} {N : Type uN} [CommSemiring R] [CommSemiring A] [Semiring B] [Algebra R A] [Algebra R B] [AddCommMonoid M] [Module R M] [Module A M] [Module B M] [IsScalarTower R A M] [IsScalarTower R B M] [SMulCommClass A B M] [AddCommMonoid N] [Module R N] [Algebra A B] [IsScalarTower A B M] (m : M) (n : N) (a : A) : (cancelBaseChange R A B M N) (m ⊗ₜ[A] a ⊗ₜ[R] n) = (a • m) ⊗ₜ[R] n

@[simp]

theorem TensorProduct.AlgebraTensorModule.cancelBaseChange_symm_tmul (R : Type uR) (A : Type uA) (B : Type uB) {M : Type uM} {N : Type uN} [CommSemiring R] [CommSemiring A] [Semiring B] [Algebra R A] [Algebra R B] [AddCommMonoid M] [Module R M] [Module A M] [Module B M] [IsScalarTower R A M] [IsScalarTower R B M] [SMulCommClass A B M] [AddCommMonoid N] [Module R N] [Algebra A B] [IsScalarTower A B M] (m : M) (n : N) : (cancelBaseChange R A B M N).symm (m ⊗ₜ[R] n) = m ⊗ₜ[A] 1 ⊗ₜ[R] n

theorem TensorProduct.AlgebraTensorModule.lTensor_comp_cancelBaseChange (R : Type uR) (A : Type uA) (B : Type uB) {M : Type uM} {N : Type uN} {Q : Type uQ} [CommSemiring R] [CommSemiring A] [Semiring B] [Algebra R A] [Algebra R B] [AddCommMonoid M] [Module R M] [Module A M] [Module B M] [IsScalarTower R A M] [IsScalarTower R B M] [SMulCommClass A B M] [AddCommMonoid N] [Module R N] [AddCommMonoid Q] [Module R Q] [Algebra A B] [IsScalarTower A B M] (f : N →ₗ[R] Q) : (lTensor B M) f ∘ₗ ↑(cancelBaseChange R A B M N) = ↑(cancelBaseChange R A B M Q) ∘ₗ (lTensor B M) ((lTensor A A) f)

noncomputable def TensorProduct.AlgebraTensorModule.leftComm (R : Type uR) (A : Type uA) (M : Type uM) (P : Type uP) (Q : Type uQ) [CommSemiring R] [CommSemiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid P] [Module A P] [AddCommMonoid Q] [Module R Q] [Module R P] [IsScalarTower R A P] : TensorProduct A M (TensorProduct R P Q) ≃ₗ[A] TensorProduct A P (TensorProduct R M Q)

Heterobasic version of TensorProduct.leftComm

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem TensorProduct.AlgebraTensorModule.leftComm_tmul (R : Type uR) (A : Type uA) {M : Type uM} {P : Type uP} {Q : Type uQ} [CommSemiring R] [CommSemiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid P] [Module A P] [AddCommMonoid Q] [Module R Q] [Module R P] [IsScalarTower R A P] (m : M) (p : P) (q : Q) : (leftComm R A M P Q) (m ⊗ₜ[A] p ⊗ₜ[R] q) = p ⊗ₜ[A] m ⊗ₜ[R] q

@[simp]

theorem TensorProduct.AlgebraTensorModule.leftComm_symm_tmul (R : Type uR) (A : Type uA) {M : Type uM} {P : Type uP} {Q : Type uQ} [CommSemiring R] [CommSemiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid P] [Module A P] [AddCommMonoid Q] [Module R Q] [Module R P] [IsScalarTower R A P] (m : M) (p : P) (q : Q) : (leftComm R A M P Q).symm (p ⊗ₜ[A] m ⊗ₜ[R] q) = m ⊗ₜ[A] p ⊗ₜ[R] q

noncomputable def TensorProduct.AlgebraTensorModule.rightComm (R : Type uR) (S : Type uS) (B : Type uB) (M : Type uM) (P : Type uP) (Q : Type uQ) [CommSemiring R] [Semiring B] [Algebra R B] [AddCommMonoid M] [Module R M] [Module B M] [IsScalarTower R B M] [AddCommMonoid P] [AddCommMonoid Q] [Module R Q] [CommSemiring S] [Module S M] [Module S P] [Algebra S B] [IsScalarTower S B M] [SMulCommClass R S M] [SMulCommClass S R M] : TensorProduct R (TensorProduct S M P) Q ≃ₗ[B] TensorProduct S (TensorProduct R M Q) P

A tensor product analogue of mul_right_comm.

Suppose we have a diagram of algebras R → B ← S, and a B-module M, S-module P, R-module Q, then

(M ⊗ˢ P)      ⎛ M ⎞ ⊗ˢ P
 ⊗ᴿ       ≅ᴮ  ⎜ ⊗ᴿ⎟
 Q            ⎝ Q ⎠

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem TensorProduct.AlgebraTensorModule.rightComm_tmul (R : Type uR) {S : Type uS} (B : Type uB) {M : Type uM} {P : Type uP} {Q : Type uQ} [CommSemiring R] [Semiring B] [Algebra R B] [AddCommMonoid M] [Module R M] [Module B M] [IsScalarTower R B M] [AddCommMonoid P] [AddCommMonoid Q] [Module R Q] [CommSemiring S] [Module S M] [Module S P] [Algebra S B] [IsScalarTower S B M] [SMulCommClass R S M] [SMulCommClass S R M] (m : M) (p : P) (q : Q) : (rightComm R S B M P Q) ((m ⊗ₜ[S] p) ⊗ₜ[R] q) = (m ⊗ₜ[R] q) ⊗ₜ[S] p

@[simp]

theorem TensorProduct.AlgebraTensorModule.rightComm_symm (R : Type uR) {S : Type uS} (B : Type uB) {M : Type uM} {P : Type uP} {Q : Type uQ} [CommSemiring R] [Semiring B] [Algebra R B] [AddCommMonoid M] [Module R M] [Module B M] [IsScalarTower R B M] [AddCommMonoid P] [AddCommMonoid Q] [Module R Q] [CommSemiring S] [Module S M] [Module S P] [Algebra S B] [IsScalarTower S B M] [SMulCommClass R S M] [SMulCommClass S R M] : (rightComm R S B M P Q).symm = rightComm S R B M Q P

theorem TensorProduct.AlgebraTensorModule.rightComm_symm_tmul (R : Type uR) {S : Type uS} (B : Type uB) {M : Type uM} {P : Type uP} {Q : Type uQ} [CommSemiring R] [Semiring B] [Algebra R B] [AddCommMonoid M] [Module R M] [Module B M] [IsScalarTower R B M] [AddCommMonoid P] [AddCommMonoid Q] [Module R Q] [CommSemiring S] [Module S M] [Module S P] [Algebra S B] [IsScalarTower S B M] [SMulCommClass R S M] [SMulCommClass S R M] (m : M) (p : P) (q : Q) : (rightComm R S B M P Q).symm ((m ⊗ₜ[R] q) ⊗ₜ[S] p) = (m ⊗ₜ[S] p) ⊗ₜ[R] q

noncomputable def TensorProduct.AlgebraTensorModule.tensorTensorTensorComm (R : Type uR) (S : Type uS) (A : Type uA) (B : Type uB) (M : Type uM) (N : Type uN) (P : Type uP) (Q : Type uQ) [CommSemiring R] [CommSemiring A] [Semiring B] [Algebra R A] [Algebra R B] [AddCommMonoid M] [Module R M] [Module A M] [Module B M] [IsScalarTower R A M] [IsScalarTower R B M] [SMulCommClass A B M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module A P] [AddCommMonoid Q] [Module R Q] [Module R P] [IsScalarTower R A P] [Algebra A B] [IsScalarTower A B M] [CommSemiring S] [Algebra R S] [Algebra S B] [Module S M] [Module S N] [IsScalarTower R S M] [SMulCommClass A S M] [SMulCommClass S A M] [IsScalarTower S B M] [IsScalarTower R S N] : TensorProduct A (TensorProduct S M N) (TensorProduct R P Q) ≃ₗ[B] TensorProduct S (TensorProduct A M P) (TensorProduct R N Q)

Heterobasic version of tensorTensorTensorComm.

Suppose we have towers of algebras R → S → B and R → A → B, and a B-module M, S-module N, A-module P, R-module Q, then

(M ⊗ˢ N)      ⎛ M ⎞ ⊗ˢ ⎛ N ⎞
 ⊗ᴬ       ≅ᴮ  ⎜ ⊗ᴬ⎟    ⎜ ⊗ᴿ⎟
(P ⊗ᴿ Q)      ⎝ P ⎠    ⎝ Q ⎠

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem TensorProduct.AlgebraTensorModule.tensorTensorTensorComm_tmul (R : Type uR) (S : Type uS) (A : Type uA) (B : Type uB) {M : Type uM} {N : Type uN} {P : Type uP} {Q : Type uQ} [CommSemiring R] [CommSemiring A] [Semiring B] [Algebra R A] [Algebra R B] [AddCommMonoid M] [Module R M] [Module A M] [Module B M] [IsScalarTower R A M] [IsScalarTower R B M] [SMulCommClass A B M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module A P] [AddCommMonoid Q] [Module R Q] [Module R P] [IsScalarTower R A P] [Algebra A B] [IsScalarTower A B M] [CommSemiring S] [Algebra R S] [Algebra S B] [Module S M] [Module S N] [IsScalarTower R S M] [SMulCommClass A S M] [SMulCommClass S A M] [IsScalarTower S B M] [IsScalarTower R S N] (m : M) (n : N) (p : P) (q : Q) : (tensorTensorTensorComm R S A B M N P Q) ((m ⊗ₜ[S] n) ⊗ₜ[A] p ⊗ₜ[R] q) = (m ⊗ₜ[A] p) ⊗ₜ[S] n ⊗ₜ[R] q

@[simp]

theorem TensorProduct.AlgebraTensorModule.tensorTensorTensorComm_symm (R : Type uR) (S : Type uS) (A : Type uA) (B : Type uB) {M : Type uM} {N : Type uN} {P : Type uP} {Q : Type uQ} [CommSemiring R] [CommSemiring A] [Semiring B] [Algebra R A] [Algebra R B] [AddCommMonoid M] [Module R M] [Module A M] [Module B M] [IsScalarTower R A M] [IsScalarTower R B M] [SMulCommClass A B M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module A P] [AddCommMonoid Q] [Module R Q] [Module R P] [IsScalarTower R A P] [Algebra A B] [IsScalarTower A B M] [CommSemiring S] [Algebra R S] [Algebra S B] [Module S M] [Module S N] [IsScalarTower R S M] [SMulCommClass A S M] [SMulCommClass S A M] [IsScalarTower S B M] [IsScalarTower R S N] : (tensorTensorTensorComm R S A B M N P Q).symm = tensorTensorTensorComm R A S B M P N Q

theorem TensorProduct.AlgebraTensorModule.tensorTensorTensorComm_symm_tmul (R : Type uR) (S : Type uS) (A : Type uA) (B : Type uB) {M : Type uM} {N : Type uN} {P : Type uP} {Q : Type uQ} [CommSemiring R] [CommSemiring A] [Semiring B] [Algebra R A] [Algebra R B] [AddCommMonoid M] [Module R M] [Module A M] [Module B M] [IsScalarTower R A M] [IsScalarTower R B M] [SMulCommClass A B M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module A P] [AddCommMonoid Q] [Module R Q] [Module R P] [IsScalarTower R A P] [Algebra A B] [IsScalarTower A B M] [CommSemiring S] [Algebra R S] [Algebra S B] [Module S M] [Module S N] [IsScalarTower R S M] [SMulCommClass A S M] [SMulCommClass S A M] [IsScalarTower S B M] [IsScalarTower R S N] (m : M) (n : N) (p : P) (q : Q) : (tensorTensorTensorComm R S A B M N P Q).symm ((m ⊗ₜ[A] p) ⊗ₜ[S] n ⊗ₜ[R] q) = (m ⊗ₜ[S] n) ⊗ₜ[A] p ⊗ₜ[R] q

theorem TensorProduct.AlgebraTensorModule.tensorTensorTensorComm_eq (R : Type uR) {M : Type uM} {N : Type uN} {P : Type uP} {Q : Type uQ} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [AddCommMonoid Q] [Module R Q] [Module R P] : tensorTensorTensorComm R R R R M N P Q = TensorProduct.tensorTensorTensorComm R M N P Q

The heterobasic version of tensorTensorTensorComm coincides with the regular version.

noncomputable def Submodule.baseChange {R : Type u_1} {M : Type u_2} (A : Type u_3) [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] (p : Submodule R M) : Submodule A (TensorProduct R A M)

If A is an R-algebra, any R-submodule p of an R-module M may be pushed forward to an A-submodule of A ⊗ M.

This "base change" operation is also known as "extension of scalars".

Equations

• Submodule.baseChange A p = Submodule.span A ↑(Submodule.map ((TensorProduct.mk R A M) 1) p)

@[simp]

theorem Submodule.baseChange_bot {R : Type u_1} {M : Type u_2} (A : Type u_3) [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] : baseChange A ⊥ = ⊥

@[simp]

theorem Submodule.baseChange_top {R : Type u_1} {M : Type u_2} (A : Type u_3) [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] : baseChange A ⊤ = ⊤

theorem Submodule.tmul_mem_baseChange_of_mem {R : Type u_1} {M : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] {p : Submodule R M} (a : A) {m : M} (hm : m ∈ p) : a ⊗ₜ[R] m ∈ baseChange A p

@[simp]

theorem Submodule.baseChange_span {R : Type u_1} {M : Type u_2} (A : Type u_3) [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] (s : Set M) : baseChange A (span R s) = span A (⇑((TensorProduct.mk R A M) 1) '' s)