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Mathlib.LinearAlgebra.TensorProduct.Tower

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 #

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.

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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) :

(TensorProduct.AlgebraTensorModule.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 (TensorProduct.AlgebraTensorModule.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 TensorProduct.AlgebraTensorModule.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.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 : TensorProduct R M N →ₗ[A] P} {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.

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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) :

(TensorProduct.AlgebraTensorModule.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) :

(TensorProduct.AlgebraTensorModule.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] [Module B P] [IsScalarTower R A 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.

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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] [Module B P] [IsScalarTower R A P] [IsScalarTower R B P] [SMulCommClass A B P] (f : M →ₗ[A] N →ₗ[R] P) :

(TensorProduct.AlgebraTensorModule.uncurry R A B M N P) f = TensorProduct.AlgebraTensorModule.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] [Module B P] [IsScalarTower R A 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.

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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] [Module B P] [IsScalarTower R A P] [IsScalarTower R B P] [SMulCommClass A B P] (f : TensorProduct R M N →ₗ[A] P) :

(TensorProduct.AlgebraTensorModule.lcurry R A B M N P) f = TensorProduct.AlgebraTensorModule.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] [Module B P] [IsScalarTower R A 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.

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noncomputable def TensorProduct.AlgebraTensorModule.mk (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] :

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.

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theorem TensorProduct.AlgebraTensorModule.mk_apply (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] (m : M) :

(TensorProduct.AlgebraTensorModule.mk R A M N) m = { toAddHom := { toFun := fun (x : N) => m ⊗ₜ[R] x, map_add' := (_ : ∀ (n₁ n₂ : N), m ⊗ₜ[R] (n₁ + n₂) = m ⊗ₜ[R] n₁ + m ⊗ₜ[R] n₂) }, map_smul' := (_ : ∀ (c : R) (y : N), m ⊗ₜ[R] (c • y) = c • m ⊗ₜ[R] y) }

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

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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) :

(TensorProduct.AlgebraTensorModule.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] :

TensorProduct.AlgebraTensorModule.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) :

TensorProduct.AlgebraTensorModule.map (f₂ ∘ₗ f₁) (g₂ ∘ₗ g₁) = TensorProduct.AlgebraTensorModule.map f₂ g₂ ∘ₗ TensorProduct.AlgebraTensorModule.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] :

TensorProduct.AlgebraTensorModule.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 : M →ₗ[A] M) (f₂ : M →ₗ[A] M) (g₁ : N →ₗ[R] N) (g₂ : N →ₗ[R] N) :

TensorProduct.AlgebraTensorModule.map (f₁ * f₂) (g₁ * g₂) = TensorProduct.AlgebraTensorModule.map f₁ g₁ * TensorProduct.AlgebraTensorModule.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 : M →ₗ[A] P) (f₂ : M →ₗ[A] P) (g : N →ₗ[R] Q) :

TensorProduct.AlgebraTensorModule.map (f₁ + f₂) g = TensorProduct.AlgebraTensorModule.map f₁ g + TensorProduct.AlgebraTensorModule.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 : N →ₗ[R] Q) (g₂ : N →ₗ[R] Q) :

TensorProduct.AlgebraTensorModule.map f (g₁ + g₂) = TensorProduct.AlgebraTensorModule.map f g₁ + TensorProduct.AlgebraTensorModule.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) :

TensorProduct.AlgebraTensorModule.map f (r • g) = r • TensorProduct.AlgebraTensorModule.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] [Module B P] [IsScalarTower R A P] [IsScalarTower R B P] [SMulCommClass A B P] [AddCommMonoid Q] [Module R Q] (b : B) (f : M →ₗ[A] P) (g : N →ₗ[R] Q) :

TensorProduct.AlgebraTensorModule.map (b • f) g = b • TensorProduct.AlgebraTensorModule.map f g

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] [Module B P] [IsScalarTower R A P] [IsScalarTower R B P] [SMulCommClass A B P] [AddCommMonoid Q] [Module R Q] :

(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

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@[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] [Module B P] [IsScalarTower R A P] [IsScalarTower R B P] [SMulCommClass A B P] [AddCommMonoid Q] [Module R Q] (f : M →ₗ[A] P) (g : N →ₗ[R] Q) :

((TensorProduct.AlgebraTensorModule.mapBilinear R A B M N P Q) f) g = TensorProduct.AlgebraTensorModule.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] [Module B P] [IsScalarTower R A P] [IsScalarTower R B P] [SMulCommClass A B P] [AddCommMonoid Q] [Module R Q] :

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)

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@[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] [Module B P] [IsScalarTower R A P] [IsScalarTower R B P] [SMulCommClass A B P] [AddCommMonoid Q] [Module R Q] (f : M →ₗ[A] P) (g : N →ₗ[R] Q) :

(TensorProduct.AlgebraTensorModule.homTensorHomMap R A B M N P Q) (f ⊗ₜ[R] g) = TensorProduct.AlgebraTensorModule.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

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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] :

TensorProduct.AlgebraTensorModule.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') :

TensorProduct.AlgebraTensorModule.congr (f₁ ≪≫ₗ f₂) (g₁ ≪≫ₗ g₂) = TensorProduct.AlgebraTensorModule.congr f₁ g₁ ≪≫ₗ TensorProduct.AlgebraTensorModule.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) :

TensorProduct.AlgebraTensorModule.congr (LinearEquiv.symm f) (LinearEquiv.symm g) = LinearEquiv.symm (TensorProduct.AlgebraTensorModule.congr f g)

@[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] :

TensorProduct.AlgebraTensorModule.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 : M ≃ₗ[A] M) (f₂ : M ≃ₗ[A] M) (g₁ : N ≃ₗ[R] N) (g₂ : N ≃ₗ[R] N) :

TensorProduct.AlgebraTensorModule.congr (f₁ * f₂) (g₁ * g₂) = TensorProduct.AlgebraTensorModule.congr f₁ g₁ * TensorProduct.AlgebraTensorModule.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) :

(TensorProduct.AlgebraTensorModule.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) :

(LinearEquiv.symm (TensorProduct.AlgebraTensorModule.congr f g)) (p ⊗ₜ[R] q) = (LinearEquiv.symm f) p ⊗ₜ[R] (LinearEquiv.symm g) 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.

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

TensorProduct.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) :

(TensorProduct.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) :

(LinearEquiv.symm (TensorProduct.AlgebraTensorModule.rid R A M)) 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 R P] [Module A P] [IsScalarTower R A P] [AddCommMonoid Q] [Module R Q] [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.

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@[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 R P] [Module A P] [IsScalarTower R A P] [AddCommMonoid Q] [Module R Q] [Algebra A B] [IsScalarTower A B M] (m : M) (p : P) (q : Q) :

(TensorProduct.AlgebraTensorModule.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 R P] [Module A P] [IsScalarTower R A P] [AddCommMonoid Q] [Module R Q] [Algebra A B] [IsScalarTower A B M] (m : M) (p : P) (q : Q) :

(LinearEquiv.symm (TensorProduct.AlgebraTensorModule.assoc R A B M P Q)) (m ⊗ₜ[A] p ⊗ₜ[R] q) = (m ⊗ₜ[A] p) ⊗ₜ[R] q

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.

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@[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) :

(TensorProduct.AlgebraTensorModule.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) :

(LinearEquiv.symm (TensorProduct.AlgebraTensorModule.cancelBaseChange R A B M N)) (m ⊗ₜ[R] n) = m ⊗ₜ[A] 1 ⊗ₜ[R] n

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 R P] [Module A P] [IsScalarTower R A P] [AddCommMonoid Q] [Module R Q] :

TensorProduct A M (TensorProduct R P Q) ≃ₗ[A] TensorProduct A P (TensorProduct R M Q)

Heterobasic version of TensorProduct.leftComm

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@[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 R P] [Module A P] [IsScalarTower R A P] [AddCommMonoid Q] [Module R Q] (m : M) (p : P) (q : Q) :

(TensorProduct.AlgebraTensorModule.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 R P] [Module A P] [IsScalarTower R A P] [AddCommMonoid Q] [Module R Q] (m : M) (p : P) (q : Q) :

(LinearEquiv.symm (TensorProduct.AlgebraTensorModule.leftComm R A M P Q)) (p ⊗ₜ[A] m ⊗ₜ[R] q) = m ⊗ₜ[A] p ⊗ₜ[R] q

noncomputable def TensorProduct.AlgebraTensorModule.rightComm (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] :

TensorProduct R (TensorProduct A M P) Q ≃ₗ[A] TensorProduct A (TensorProduct R M Q) P

A tensor product analogue of mul_right_comm.

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@[simp]

theorem TensorProduct.AlgebraTensorModule.rightComm_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] (m : M) (p : P) (q : Q) :

(TensorProduct.AlgebraTensorModule.rightComm R A M P Q) ((m ⊗ₜ[A] p) ⊗ₜ[R] q) = (m ⊗ₜ[R] q) ⊗ₜ[A] p

@[simp]

theorem TensorProduct.AlgebraTensorModule.rightComm_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] (m : M) (p : P) (q : Q) :

(LinearEquiv.symm (TensorProduct.AlgebraTensorModule.rightComm R A M P Q)) ((m ⊗ₜ[R] q) ⊗ₜ[A] p) = (m ⊗ₜ[A] p) ⊗ₜ[R] q

noncomputable def TensorProduct.AlgebraTensorModule.tensorTensorTensorComm (R : Type uR) (A : Type uA) (M : Type uM) (N : Type uN) (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 N] [Module R N] [AddCommMonoid P] [Module R P] [Module A P] [IsScalarTower R A P] [AddCommMonoid Q] [Module R Q] :

TensorProduct A (TensorProduct R M N) (TensorProduct R P Q) ≃ₗ[A] TensorProduct R (TensorProduct A M P) (TensorProduct R N Q)

Heterobasic version of tensorTensorTensorComm.

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@[simp]

theorem TensorProduct.AlgebraTensorModule.tensorTensorTensorComm_tmul (R : Type uR) (A : Type uA) {M : Type uM} {N : Type uN} {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 N] [Module R N] [AddCommMonoid P] [Module R P] [Module A P] [IsScalarTower R A P] [AddCommMonoid Q] [Module R Q] (m : M) (n : N) (p : P) (q : Q) :

(TensorProduct.AlgebraTensorModule.tensorTensorTensorComm R A M N P Q) ((m ⊗ₜ[R] n) ⊗ₜ[A] p ⊗ₜ[R] q) = (m ⊗ₜ[A] p) ⊗ₜ[R] n ⊗ₜ[R] q

@[simp]

theorem TensorProduct.AlgebraTensorModule.tensorTensorTensorComm_symm_tmul (R : Type uR) (A : Type uA) {M : Type uM} {N : Type uN} {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 N] [Module R N] [AddCommMonoid P] [Module R P] [Module A P] [IsScalarTower R A P] [AddCommMonoid Q] [Module R Q] (m : M) (n : N) (p : P) (q : Q) :

(LinearEquiv.symm (TensorProduct.AlgebraTensorModule.tensorTensorTensorComm R A M N P Q)) ((m ⊗ₜ[A] p) ⊗ₜ[R] n ⊗ₜ[R] q) = (m ⊗ₜ[R] n) ⊗ₜ[A] p ⊗ₜ[R] q

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)

Instances For

@[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] :

Submodule.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] :

Submodule.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 ∈ Submodule.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) :

Submodule.baseChange A (Submodule.span R s) = Submodule.span A (⇑((TensorProduct.mk R A M) 1) '' s)