Documentation

Mathlib.RingTheory.TensorProduct.Basic

The tensor product of R-algebras #

This file provides results about the multiplicative structure on A ⊗[R] B when R is a commutative (semi)ring and A and B are both R-algebras. On these tensor products, multiplication is characterized by (a₁ ⊗ₜ b₁) * (a₂ ⊗ₜ b₂) = (a₁ * a₂) ⊗ₜ (b₁ * b₂).

Main declarations #

References #

The base-change of a linear map of R-modules to a linear map of A-modules #

noncomputable def LinearMap.baseChange {R : Type u_1} (A : Type u_2) {M : Type u_4} {N : Type u_5} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] (f : M →ₗ[R] N) :

baseChange A f for f : M →ₗ[R] N is the A-linear map A ⊗[R] M →ₗ[A] A ⊗[R] N.

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

Equations
Instances For
    @[simp]
    theorem LinearMap.baseChange_tmul {R : Type u_1} {A : Type u_2} {M : Type u_4} {N : Type u_5} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] (f : M →ₗ[R] N) (a : A) (x : M) :
    theorem LinearMap.baseChange_eq_ltensor {R : Type u_1} {A : Type u_2} {M : Type u_4} {N : Type u_5} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] (f : M →ₗ[R] N) :
    @[simp]
    theorem LinearMap.baseChange_add {R : Type u_1} {A : Type u_2} {M : Type u_4} {N : Type u_5} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] (f : M →ₗ[R] N) (g : M →ₗ[R] N) :
    @[simp]
    theorem LinearMap.baseChange_zero {R : Type u_1} {A : Type u_2} {M : Type u_4} {N : Type u_5} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] :
    @[simp]
    theorem LinearMap.baseChange_smul {R : Type u_1} {A : Type u_2} {M : Type u_4} {N : Type u_5} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] (r : R) (f : M →ₗ[R] N) :
    @[simp]
    theorem LinearMap.baseChange_id {R : Type u_1} {A : Type u_2} {M : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] :
    LinearMap.baseChange A LinearMap.id = LinearMap.id
    theorem LinearMap.baseChange_comp {R : Type u_1} {A : Type u_2} {M : Type u_4} {N : Type u_5} {P : Type u_6} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module R N] [Module R P] (f : M →ₗ[R] N) (g : N →ₗ[R] P) :
    @[simp]
    theorem LinearMap.baseChange_one (R : Type u_1) {A : Type u_2} (M : Type u_4) [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] :
    theorem LinearMap.baseChange_mul {R : Type u_1} {A : Type u_2} {M : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] (f : Module.End R M) (g : Module.End R M) :
    noncomputable def LinearMap.baseChangeHom (R : Type u_1) (A : Type u_2) (M : Type u_4) (N : Type u_5) [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] :

    baseChange as a linear map.

    When M = N, this is true more strongly as Module.End.baseChangeHom.

    Equations
    Instances For
      @[simp]
      theorem LinearMap.baseChangeHom_apply (R : Type u_1) (A : Type u_2) (M : Type u_4) (N : Type u_5) [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] (f : M →ₗ[R] N) :
      noncomputable def Module.End.baseChangeHom (R : Type u_1) (A : Type u_2) (M : Type u_4) [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] :

      baseChange as an AlgHom.

      Equations
      Instances For
        @[simp]
        theorem Module.End.baseChangeHom_apply_apply (R : Type u_1) (A : Type u_2) (M : Type u_4) [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] (a : Module.End R M) :
        ∀ (a_1 : TensorProduct R A M), ((Module.End.baseChangeHom R A M) a) a_1 = (TensorProduct.liftAux (R (LinearMap.comp { toAddHom := { toFun := fun (h : M →ₗ[R] TensorProduct R A M) => LinearMap.comp h a, map_add' := }, map_smul' := } (TensorProduct.AlgebraTensorModule.mk R A A M)))) a_1
        theorem LinearMap.baseChange_pow (R : Type u_1) (A : Type u_2) (M : Type u_4) [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] (f : Module.End R M) (n : ) :
        @[simp]
        theorem LinearMap.baseChange_sub {R : Type u_1} {A : Type u_2} {M : Type u_4} {N : Type u_5} [CommRing R] [Ring A] [Algebra R A] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (f : M →ₗ[R] N) (g : M →ₗ[R] N) :
        @[simp]
        theorem LinearMap.baseChange_neg {R : Type u_1} {A : Type u_2} {M : Type u_4} {N : Type u_5} [CommRing R] [Ring A] [Algebra R A] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (f : M →ₗ[R] N) :

        The R-algebra structure on A ⊗[R] B #

        Equations
        • Algebra.TensorProduct.instOneTensorProductToAddCommMonoidToAddCommMonoid = { one := 1 ⊗ₜ[R] 1 }
        Equations
        theorem Algebra.TensorProduct.natCast_def {R : Type uR} {A : Type uA} {B : Type uB} [CommSemiring R] [AddCommMonoidWithOne A] [Module R A] [AddCommMonoidWithOne B] [Module R B] (n : ) :
        n = n ⊗ₜ[R] 1
        theorem Algebra.TensorProduct.natCast_def' {R : Type uR} {A : Type uA} {B : Type uB} [CommSemiring R] [AddCommMonoidWithOne A] [Module R A] [AddCommMonoidWithOne B] [Module R B] (n : ) :
        n = 1 ⊗ₜ[R] n

        (Implementation detail) The multiplication map on A ⊗[R] B, as an R-bilinear map.

        Equations
        Instances For
          @[simp]
          theorem Algebra.TensorProduct.mul_apply {R : Type uR} {A : Type uA} {B : Type uB} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] [NonUnitalNonAssocSemiring B] [Module R B] [SMulCommClass R B B] [IsScalarTower R B B] (a₁ : A) (a₂ : A) (b₁ : B) (b₂ : B) :
          (Algebra.TensorProduct.mul (a₁ ⊗ₜ[R] b₁)) (a₂ ⊗ₜ[R] b₂) = (a₁ * a₂) ⊗ₜ[R] (b₁ * b₂)
          noncomputable instance Algebra.TensorProduct.instMul {R : Type uR} {A : Type uA} {B : Type uB} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] [NonUnitalNonAssocSemiring B] [Module R B] [SMulCommClass R B B] [IsScalarTower R B B] :
          Equations
          • Algebra.TensorProduct.instMul = { mul := fun (a b : TensorProduct R A B) => (Algebra.TensorProduct.mul a) b }
          @[simp]
          theorem Algebra.TensorProduct.tmul_mul_tmul {R : Type uR} {A : Type uA} {B : Type uB} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] [NonUnitalNonAssocSemiring B] [Module R B] [SMulCommClass R B B] [IsScalarTower R B B] (a₁ : A) (a₂ : A) (b₁ : B) (b₂ : B) :
          a₁ ⊗ₜ[R] b₁ * a₂ ⊗ₜ[R] b₂ = (a₁ * a₂) ⊗ₜ[R] (b₁ * b₂)
          theorem SemiconjBy.tmul {R : Type uR} {A : Type uA} {B : Type uB} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] [NonUnitalNonAssocSemiring B] [Module R B] [SMulCommClass R B B] [IsScalarTower R B B] {a₁ : A} {a₂ : A} {a₃ : A} {b₁ : B} {b₂ : B} {b₃ : B} (ha : SemiconjBy a₁ a₂ a₃) (hb : SemiconjBy b₁ b₂ b₃) :
          SemiconjBy (a₁ ⊗ₜ[R] b₁) (a₂ ⊗ₜ[R] b₂) (a₃ ⊗ₜ[R] b₃)
          theorem Commute.tmul {R : Type uR} {A : Type uA} {B : Type uB} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] [NonUnitalNonAssocSemiring B] [Module R B] [SMulCommClass R B B] [IsScalarTower R B B] {a₁ : A} {a₂ : A} {b₁ : B} {b₂ : B} (ha : Commute a₁ a₂) (hb : Commute b₁ b₂) :
          Commute (a₁ ⊗ₜ[R] b₁) (a₂ ⊗ₜ[R] b₂)
          Equations
          noncomputable instance Algebra.TensorProduct.isScalarTower_right {R : Type uR} {S : Type uS} {A : Type uA} {B : Type uB} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] [NonUnitalNonAssocSemiring B] [Module R B] [SMulCommClass R B B] [IsScalarTower R B B] [Monoid S] [DistribMulAction S A] [IsScalarTower S A A] [SMulCommClass R S A] :
          Equations
          • =
          noncomputable instance Algebra.TensorProduct.sMulCommClass_right {R : Type uR} {S : Type uS} {A : Type uA} {B : Type uB} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] [NonUnitalNonAssocSemiring B] [Module R B] [SMulCommClass R B B] [IsScalarTower R B B] [Monoid S] [DistribMulAction S A] [SMulCommClass S A A] [SMulCommClass R S A] :
          Equations
          • =
          theorem Algebra.TensorProduct.one_mul {R : Type uR} {A : Type uA} {B : Type uB} [CommSemiring R] [NonAssocSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] [NonAssocSemiring B] [Module R B] [SMulCommClass R B B] [IsScalarTower R B B] (x : TensorProduct R A B) :
          (Algebra.TensorProduct.mul (1 ⊗ₜ[R] 1)) x = x
          theorem Algebra.TensorProduct.mul_one {R : Type uR} {A : Type uA} {B : Type uB} [CommSemiring R] [NonAssocSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] [NonAssocSemiring B] [Module R B] [SMulCommClass R B B] [IsScalarTower R B B] (x : TensorProduct R A B) :
          (Algebra.TensorProduct.mul x) (1 ⊗ₜ[R] 1) = x
          noncomputable instance Algebra.TensorProduct.instNonAssocSemiring {R : Type uR} {A : Type uA} {B : Type uB} [CommSemiring R] [NonAssocSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] [NonAssocSemiring B] [Module R B] [SMulCommClass R B B] [IsScalarTower R B B] :
          Equations
          • Algebra.TensorProduct.instNonAssocSemiring = let __spread.0 := Algebra.TensorProduct.instAddCommMonoidWithOne; NonAssocSemiring.mk
          theorem Algebra.TensorProduct.mul_assoc {R : Type uR} {A : Type uA} {B : Type uB} [CommSemiring R] [NonUnitalSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] [NonUnitalSemiring B] [Module R B] [SMulCommClass R B B] [IsScalarTower R B B] (x : TensorProduct R A B) (y : TensorProduct R A B) (z : TensorProduct R A B) :
          (Algebra.TensorProduct.mul ((Algebra.TensorProduct.mul x) y)) z = (Algebra.TensorProduct.mul x) ((Algebra.TensorProduct.mul y) z)
          Equations
          noncomputable instance Algebra.TensorProduct.instSemiring {R : Type uR} {A : Type uA} {B : Type uB} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] :
          Equations
          • Algebra.TensorProduct.instSemiring = Semiring.mk npowRec
          @[simp]
          theorem Algebra.TensorProduct.tmul_pow {R : Type uR} {A : Type uA} {B : Type uB} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] (a : A) (b : B) (k : ) :
          a ⊗ₜ[R] b ^ k = (a ^ k) ⊗ₜ[R] (b ^ k)
          noncomputable def Algebra.TensorProduct.includeLeftRingHom {R : Type uR} {A : Type uA} {B : Type uB} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] :

          The ring morphism A →+* A ⊗[R] B sending a to a ⊗ₜ 1.

          Equations
          • Algebra.TensorProduct.includeLeftRingHom = { toMonoidHom := { toOneHom := { toFun := fun (a : A) => a ⊗ₜ[R] 1, map_one' := }, map_mul' := }, map_zero' := , map_add' := }
          Instances For
            @[simp]
            theorem Algebra.TensorProduct.includeLeftRingHom_apply {R : Type uR} {A : Type uA} {B : Type uB} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] (a : A) :
            Algebra.TensorProduct.includeLeftRingHom a = a ⊗ₜ[R] 1
            noncomputable instance Algebra.TensorProduct.leftAlgebra {R : Type uR} {S : Type uS} {A : Type uA} {B : Type uB} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CommSemiring S] [Algebra S A] [SMulCommClass R S A] :
            Equations
            noncomputable instance Algebra.TensorProduct.instAlgebra {R : Type uR} {A : Type uA} {B : Type uB} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] :

            The tensor product of two R-algebras is an R-algebra.

            Equations
            • Algebra.TensorProduct.instAlgebra = inferInstance
            @[simp]
            theorem Algebra.TensorProduct.algebraMap_apply {R : Type uR} {S : Type uS} {A : Type uA} {B : Type uB} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CommSemiring S] [Algebra S A] [SMulCommClass R S A] (r : S) :
            (algebraMap S (TensorProduct R A B)) r = (algebraMap S A) r ⊗ₜ[R] 1
            theorem Algebra.TensorProduct.algebraMap_apply' {R : Type uR} {A : Type uA} {B : Type uB} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] (r : R) :
            (algebraMap R (TensorProduct R A B)) r = 1 ⊗ₜ[R] (algebraMap R B) r
            noncomputable def Algebra.TensorProduct.includeLeft {R : Type uR} {S : Type uS} {A : Type uA} {B : Type uB} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CommSemiring S] [Algebra S A] [SMulCommClass R S A] :

            The R-algebra morphism A →ₐ[R] A ⊗[R] B sending a to a ⊗ₜ 1.

            Equations
            • Algebra.TensorProduct.includeLeft = let __src := Algebra.TensorProduct.includeLeftRingHom; { toRingHom := __src, commutes' := }
            Instances For
              @[simp]
              theorem Algebra.TensorProduct.includeLeft_apply {R : Type uR} {S : Type uS} {A : Type uA} {B : Type uB} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CommSemiring S] [Algebra S A] [SMulCommClass R S A] (a : A) :
              Algebra.TensorProduct.includeLeft a = a ⊗ₜ[R] 1
              noncomputable def Algebra.TensorProduct.includeRight {R : Type uR} {A : Type uA} {B : Type uB} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] :

              The algebra morphism B →ₐ[R] A ⊗[R] B sending b to 1 ⊗ₜ b.

              Equations
              • One or more equations did not get rendered due to their size.
              Instances For
                @[simp]
                theorem Algebra.TensorProduct.includeRight_apply {R : Type uR} {A : Type uA} {B : Type uB} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] (b : B) :
                Algebra.TensorProduct.includeRight b = 1 ⊗ₜ[R] b
                theorem Algebra.TensorProduct.includeLeftRingHom_comp_algebraMap {R : Type uR} {A : Type uA} {B : Type uB} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] :
                RingHom.comp Algebra.TensorProduct.includeLeftRingHom (algebraMap R A) = RingHom.comp Algebra.TensorProduct.includeRight.toRingHom (algebraMap R B)
                theorem Algebra.TensorProduct.ext {R : Type uR} {S : Type uS} {A : Type uA} {B : Type uB} {C : Type uC} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [Semiring C] [Algebra R C] [CommSemiring S] [Algebra S A] [Algebra R S] [Algebra S C] [IsScalarTower R S A] [IsScalarTower R S C] ⦃f : TensorProduct R A B →ₐ[S] C ⦃g : TensorProduct R A B →ₐ[S] C (ha : AlgHom.comp f Algebra.TensorProduct.includeLeft = AlgHom.comp g Algebra.TensorProduct.includeLeft) (hb : AlgHom.comp (AlgHom.restrictScalars R f) Algebra.TensorProduct.includeRight = AlgHom.comp (AlgHom.restrictScalars R g) Algebra.TensorProduct.includeRight) :
                f = g

                A version of TensorProduct.ext for AlgHom.

                Using this as the @[ext] lemma instead of Algebra.TensorProduct.ext' allows ext to apply lemmas specific to A →ₐ[S] _ and B →ₐ[R] _; notably this allows recursion into nested tensor products of algebras.

                See note [partially-applied ext lemmas].

                theorem Algebra.TensorProduct.ext' {R : Type uR} {S : Type uS} {A : Type uA} {B : Type uB} {C : Type uC} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [Semiring C] [Algebra R C] [CommSemiring S] [Algebra S A] [Algebra R S] [Algebra S C] [IsScalarTower R S A] [IsScalarTower R S C] {g : TensorProduct R A B →ₐ[S] C} {h : TensorProduct R A B →ₐ[S] C} (H : ∀ (a : A) (b : B), g (a ⊗ₜ[R] b) = h (a ⊗ₜ[R] b)) :
                g = h
                Equations
                • Algebra.TensorProduct.instAddCommGroupWithOne = let __spread.0 := Algebra.TensorProduct.instAddCommMonoidWithOne; AddCommGroupWithOne.mk
                theorem Algebra.TensorProduct.intCast_def {R : Type uR} {A : Type uA} {B : Type uB} [CommSemiring R] [AddCommGroupWithOne A] [Module R A] [AddCommGroupWithOne B] [Module R B] (z : ) :
                z = z ⊗ₜ[R] 1
                Equations
                • Algebra.TensorProduct.instNonUnitalNonAssocRing = let __spread.0 := Algebra.TensorProduct.instNonUnitalNonAssocSemiring; NonUnitalNonAssocRing.mk
                noncomputable instance Algebra.TensorProduct.instNonAssocRing {R : Type uR} {A : Type uA} {B : Type uB} [CommRing R] [NonAssocRing A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] [NonAssocRing B] [Module R B] [SMulCommClass R B B] [IsScalarTower R B B] :
                Equations
                • One or more equations did not get rendered due to their size.
                noncomputable instance Algebra.TensorProduct.instNonUnitalRing {R : Type uR} {A : Type uA} {B : Type uB} [CommRing R] [NonUnitalRing A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] [NonUnitalRing B] [Module R B] [SMulCommClass R B B] [IsScalarTower R B B] :
                Equations
                • Algebra.TensorProduct.instNonUnitalRing = let __spread.0 := Algebra.TensorProduct.instNonUnitalSemiring; NonUnitalRing.mk
                noncomputable instance Algebra.TensorProduct.instCommSemiring {R : Type uR} {A : Type uA} {B : Type uB} [CommSemiring R] [CommSemiring A] [Algebra R A] [CommSemiring B] [Algebra R B] :
                Equations
                noncomputable instance Algebra.TensorProduct.instRing {R : Type uR} {A : Type uA} {B : Type uB} [CommRing R] [Ring A] [Algebra R A] [Ring B] [Algebra R B] :
                Equations
                • Algebra.TensorProduct.instRing = let __spread.0 := TensorProduct.addCommGroup; let __spread.1 := Algebra.TensorProduct.instNonAssocRing; Ring.mk SubNegMonoid.zsmul
                theorem Algebra.TensorProduct.intCast_def' {R : Type uR} {A : Type uA} {B : Type uB} [CommRing R] [Ring A] [Algebra R A] [Ring B] [Algebra R B] (z : ) :
                z = 1 ⊗ₜ[R] z
                noncomputable instance Algebra.TensorProduct.instCommRing {R : Type uR} {A : Type uA} {B : Type uB} [CommRing R] [CommRing A] [Algebra R A] [CommRing B] [Algebra R B] :
                Equations
                @[reducible]
                noncomputable def Algebra.TensorProduct.rightAlgebra {R : Type uR} {A : Type uA} {B : Type uB} [CommRing R] [CommRing A] [Algebra R A] [CommRing B] [Algebra R B] :

                S ⊗[R] T has a T-algebra structure. This is not a global instance or else the action of S on S ⊗[R] S would be ambiguous.

                Equations
                • Algebra.TensorProduct.rightAlgebra = RingHom.toAlgebra Algebra.TensorProduct.includeRight.toRingHom
                Instances For
                  noncomputable instance Algebra.TensorProduct.right_isScalarTower {R : Type uR} {A : Type uA} {B : Type uB} [CommRing R] [CommRing A] [Algebra R A] [CommRing B] [Algebra R B] :
                  Equations
                  • =

                  We now build the structure maps for the symmetric monoidal category of R-algebras.

                  noncomputable def Algebra.TensorProduct.algHomOfLinearMapTensorProduct {R : Type uR} {S : Type uS} {A : Type uA} {B : Type uB} {C : Type uC} [CommSemiring R] [CommSemiring S] [Algebra R S] [Semiring A] [Algebra R A] [Algebra S A] [IsScalarTower R S A] [Semiring B] [Algebra R B] [Semiring C] [Algebra S C] (f : TensorProduct R A B →ₗ[S] C) (h_mul : ∀ (a₁ a₂ : A) (b₁ b₂ : B), f ((a₁ * a₂) ⊗ₜ[R] (b₁ * b₂)) = f (a₁ ⊗ₜ[R] b₁) * f (a₂ ⊗ₜ[R] b₂)) (h_one : f (1 ⊗ₜ[R] 1) = 1) :

                  Build an algebra morphism from a linear map out of a tensor product, and evidence that on pure tensors, it preserves multiplication and the identity.

                  Note that we state h_one using 1 ⊗ₜ[R] 1 instead of 1 so that lemmas about f applied to pure tensors can be directly applied by the caller (without needing TensorProduct.one_def).

                  Equations
                  Instances For
                    @[simp]
                    theorem Algebra.TensorProduct.algHomOfLinearMapTensorProduct_apply {R : Type uR} {S : Type uS} {A : Type uA} {B : Type uB} {C : Type uC} [CommSemiring R] [CommSemiring S] [Algebra R S] [Semiring A] [Algebra R A] [Algebra S A] [IsScalarTower R S A] [Semiring B] [Algebra R B] [Semiring C] [Algebra S C] (f : TensorProduct R A B →ₗ[S] C) (h_mul : ∀ (a₁ a₂ : A) (b₁ b₂ : B), f ((a₁ * a₂) ⊗ₜ[R] (b₁ * b₂)) = f (a₁ ⊗ₜ[R] b₁) * f (a₂ ⊗ₜ[R] b₂)) (h_one : f (1 ⊗ₜ[R] 1) = 1) (x : TensorProduct R A B) :
                    noncomputable def Algebra.TensorProduct.algEquivOfLinearEquivTensorProduct {R : Type uR} {S : Type uS} {A : Type uA} {B : Type uB} {C : Type uC} [CommSemiring R] [CommSemiring S] [Algebra R S] [Semiring A] [Algebra R A] [Algebra S A] [IsScalarTower R S A] [Semiring B] [Algebra R B] [Semiring C] [Algebra S C] (f : TensorProduct R A B ≃ₗ[S] C) (h_mul : ∀ (a₁ a₂ : A) (b₁ b₂ : B), f ((a₁ * a₂) ⊗ₜ[R] (b₁ * b₂)) = f (a₁ ⊗ₜ[R] b₁) * f (a₂ ⊗ₜ[R] b₂)) (h_one : f (1 ⊗ₜ[R] 1) = 1) :

                    Build an algebra equivalence from a linear equivalence out of a tensor product, and evidence that on pure tensors, it preserves multiplication and the identity.

                    Note that we state h_one using 1 ⊗ₜ[R] 1 instead of 1 so that lemmas about f applied to pure tensors can be directly applied by the caller (without needing TensorProduct.one_def).

                    Equations
                    • One or more equations did not get rendered due to their size.
                    Instances For
                      @[simp]
                      theorem Algebra.TensorProduct.algEquivOfLinearEquivTensorProduct_apply {R : Type uR} {S : Type uS} {A : Type uA} {B : Type uB} {C : Type uC} [CommSemiring R] [CommSemiring S] [Algebra R S] [Semiring A] [Algebra R A] [Algebra S A] [IsScalarTower R S A] [Semiring B] [Algebra R B] [Semiring C] [Algebra S C] (f : TensorProduct R A B ≃ₗ[S] C) (h_mul : ∀ (a₁ a₂ : A) (b₁ b₂ : B), f ((a₁ * a₂) ⊗ₜ[R] (b₁ * b₂)) = f (a₁ ⊗ₜ[R] b₁) * f (a₂ ⊗ₜ[R] b₂)) (h_one : f (1 ⊗ₜ[R] 1) = 1) (x : TensorProduct R A B) :
                      noncomputable def Algebra.TensorProduct.algEquivOfLinearEquivTripleTensorProduct {R : Type uR} {A : Type uA} {B : Type uB} {C : Type uC} {D : Type uD} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [Semiring C] [Algebra R C] [Semiring D] [Algebra R D] (f : TensorProduct R (TensorProduct R A B) C ≃ₗ[R] D) (h_mul : ∀ (a₁ a₂ : A) (b₁ b₂ : B) (c₁ c₂ : C), f (((a₁ * a₂) ⊗ₜ[R] (b₁ * b₂)) ⊗ₜ[R] (c₁ * c₂)) = f ((a₁ ⊗ₜ[R] b₁) ⊗ₜ[R] c₁) * f ((a₂ ⊗ₜ[R] b₂) ⊗ₜ[R] c₂)) (h_one : f ((1 ⊗ₜ[R] 1) ⊗ₜ[R] 1) = 1) :

                      Build an algebra equivalence from a linear equivalence out of a triple tensor product, and evidence of multiplicativity on pure tensors.

                      Equations
                      Instances For
                        @[simp]
                        theorem Algebra.TensorProduct.algEquivOfLinearEquivTripleTensorProduct_apply {R : Type uR} {A : Type uA} {B : Type uB} {C : Type uC} {D : Type uD} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [Semiring C] [Algebra R C] [Semiring D] [Algebra R D] (f : TensorProduct R (TensorProduct R A B) C ≃ₗ[R] D) (h_mul : ∀ (a₁ a₂ : A) (b₁ b₂ : B) (c₁ c₂ : C), f (((a₁ * a₂) ⊗ₜ[R] (b₁ * b₂)) ⊗ₜ[R] (c₁ * c₂)) = f ((a₁ ⊗ₜ[R] b₁) ⊗ₜ[R] c₁) * f ((a₂ ⊗ₜ[R] b₂) ⊗ₜ[R] c₂)) (h_one : f ((1 ⊗ₜ[R] 1) ⊗ₜ[R] 1) = 1) (x : TensorProduct R (TensorProduct R A B) C) :
                        noncomputable def Algebra.TensorProduct.lift {R : Type uR} {S : Type uS} {A : Type uA} {B : Type uB} {C : Type uC} [CommSemiring R] [CommSemiring S] [Algebra R S] [Semiring A] [Algebra R A] [Algebra S A] [IsScalarTower R S A] [Semiring B] [Algebra R B] [Semiring C] [Algebra R C] [Algebra S C] [IsScalarTower R S C] (f : A →ₐ[S] C) (g : B →ₐ[R] C) (hfg : ∀ (x : A) (y : B), Commute (f x) (g y)) :

                        The forward direction of the universal property of tensor products of algebras; any algebra morphism from the tensor product can be factored as the product of two algebra morphisms that commute.

                        See Algebra.TensorProduct.liftEquiv for the fact that every morphism factors this way.

                        Equations
                        • One or more equations did not get rendered due to their size.
                        Instances For
                          @[simp]
                          theorem Algebra.TensorProduct.lift_tmul {R : Type uR} {S : Type uS} {A : Type uA} {B : Type uB} {C : Type uC} [CommSemiring R] [CommSemiring S] [Algebra R S] [Semiring A] [Algebra R A] [Algebra S A] [IsScalarTower R S A] [Semiring B] [Algebra R B] [Semiring C] [Algebra R C] [Algebra S C] [IsScalarTower R S C] (f : A →ₐ[S] C) (g : B →ₐ[R] C) (hfg : ∀ (x : A) (y : B), Commute (f x) (g y)) (a : A) (b : B) :
                          (Algebra.TensorProduct.lift f g hfg) (a ⊗ₜ[R] b) = f a * g b
                          @[simp]
                          theorem Algebra.TensorProduct.lift_includeLeft_includeRight {R : Type uR} {S : Type uS} {A : Type uA} {B : Type uB} [CommSemiring R] [CommSemiring S] [Algebra R S] [Semiring A] [Algebra R A] [Algebra S A] [IsScalarTower R S A] [Semiring B] [Algebra R B] :
                          Algebra.TensorProduct.lift Algebra.TensorProduct.includeLeft Algebra.TensorProduct.includeRight = AlgHom.id S (TensorProduct R A B)
                          @[simp]
                          theorem Algebra.TensorProduct.lift_comp_includeLeft {R : Type uR} {S : Type uS} {A : Type uA} {B : Type uB} {C : Type uC} [CommSemiring R] [CommSemiring S] [Algebra R S] [Semiring A] [Algebra R A] [Algebra S A] [IsScalarTower R S A] [Semiring B] [Algebra R B] [Semiring C] [Algebra R C] [Algebra S C] [IsScalarTower R S C] (f : A →ₐ[S] C) (g : B →ₐ[R] C) (hfg : ∀ (x : A) (y : B), Commute (f x) (g y)) :
                          AlgHom.comp (Algebra.TensorProduct.lift f g hfg) Algebra.TensorProduct.includeLeft = f
                          @[simp]
                          theorem Algebra.TensorProduct.lift_comp_includeRight {R : Type uR} {S : Type uS} {A : Type uA} {B : Type uB} {C : Type uC} [CommSemiring R] [CommSemiring S] [Algebra R S] [Semiring A] [Algebra R A] [Algebra S A] [IsScalarTower R S A] [Semiring B] [Algebra R B] [Semiring C] [Algebra R C] [Algebra S C] [IsScalarTower R S C] (f : A →ₐ[S] C) (g : B →ₐ[R] C) (hfg : ∀ (x : A) (y : B), Commute (f x) (g y)) :
                          AlgHom.comp (AlgHom.restrictScalars R (Algebra.TensorProduct.lift f g hfg)) Algebra.TensorProduct.includeRight = g
                          noncomputable def Algebra.TensorProduct.liftEquiv {R : Type uR} {S : Type uS} {A : Type uA} {B : Type uB} {C : Type uC} [CommSemiring R] [CommSemiring S] [Algebra R S] [Semiring A] [Algebra R A] [Algebra S A] [IsScalarTower R S A] [Semiring B] [Algebra R B] [Semiring C] [Algebra R C] [Algebra S C] [IsScalarTower R S C] :
                          { fg : (A →ₐ[S] C) × (B →ₐ[R] C) // ∀ (x : A) (y : B), Commute (fg.1 x) (fg.2 y) } (TensorProduct R A B →ₐ[S] C)

                          The universal property of the tensor product of algebras.

                          Pairs of algebra morphisms that commute are equivalent to algebra morphisms from the tensor product.

                          This is Algebra.TensorProduct.lift as an equivalence.

                          See also GradedTensorProduct.liftEquiv for an alternative commutativity requirement for graded algebra.

                          Equations
                          • One or more equations did not get rendered due to their size.
                          Instances For
                            @[simp]
                            theorem Algebra.TensorProduct.liftEquiv_apply {R : Type uR} {S : Type uS} {A : Type uA} {B : Type uB} {C : Type uC} [CommSemiring R] [CommSemiring S] [Algebra R S] [Semiring A] [Algebra R A] [Algebra S A] [IsScalarTower R S A] [Semiring B] [Algebra R B] [Semiring C] [Algebra R C] [Algebra S C] [IsScalarTower R S C] (fg : { fg : (A →ₐ[S] C) × (B →ₐ[R] C) // ∀ (x : A) (y : B), Commute (fg.1 x) (fg.2 y) }) :
                            Algebra.TensorProduct.liftEquiv fg = Algebra.TensorProduct.lift (fg).1 (fg).2
                            @[simp]
                            theorem Algebra.TensorProduct.liftEquiv_symm_apply_coe {R : Type uR} {S : Type uS} {A : Type uA} {B : Type uB} {C : Type uC} [CommSemiring R] [CommSemiring S] [Algebra R S] [Semiring A] [Algebra R A] [Algebra S A] [IsScalarTower R S A] [Semiring B] [Algebra R B] [Semiring C] [Algebra R C] [Algebra S C] [IsScalarTower R S C] (f' : TensorProduct R A B →ₐ[S] C) :
                            (Algebra.TensorProduct.liftEquiv.symm f') = (AlgHom.comp f' Algebra.TensorProduct.includeLeft, AlgHom.comp (AlgHom.restrictScalars R f') Algebra.TensorProduct.includeRight)
                            noncomputable def Algebra.TensorProduct.lid (R : Type uR) (A : Type uA) [CommSemiring R] [Semiring A] [Algebra R A] :

                            The base ring is a left identity for the tensor product of algebra, up to algebra isomorphism.

                            Equations
                            Instances For
                              @[simp]
                              theorem Algebra.TensorProduct.lid_tmul {R : Type uR} {A : Type uA} [CommSemiring R] [Semiring A] [Algebra R A] (r : R) (a : A) :
                              noncomputable def Algebra.TensorProduct.rid (R : Type uR) (S : Type uS) (A : Type uA) [CommSemiring R] [CommSemiring S] [Algebra R S] [Semiring A] [Algebra R A] [Algebra S A] [IsScalarTower R S A] :

                              The base ring is a right identity for the tensor product of algebra, up to algebra isomorphism.

                              Note that if A is commutative this can be instantiated with S = A.

                              Equations
                              Instances For
                                @[simp]
                                theorem Algebra.TensorProduct.rid_tmul {R : Type uR} (S : Type uS) {A : Type uA} [CommSemiring R] [CommSemiring S] [Algebra R S] [Semiring A] [Algebra R A] [Algebra S A] [IsScalarTower R S A] (r : R) (a : A) :
                                @[simp]
                                theorem Algebra.TensorProduct.rid_symm_apply (R : Type uR) (S : Type uS) {A : Type uA} [CommSemiring R] [CommSemiring S] [Algebra R S] [Semiring A] [Algebra R A] [Algebra S A] [IsScalarTower R S A] (a : A) :
                                noncomputable def Algebra.TensorProduct.comm (R : Type uR) (A : Type uA) (B : Type uB) [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] :

                                The tensor product of R-algebras is commutative, up to algebra isomorphism.

                                Equations
                                Instances For
                                  @[simp]
                                  theorem Algebra.TensorProduct.comm_tmul (R : Type uR) {A : Type uA} {B : Type uB} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] (a : A) (b : B) :
                                  @[simp]
                                  theorem Algebra.TensorProduct.comm_symm_tmul (R : Type uR) {A : Type uA} {B : Type uB} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] (a : A) (b : B) :
                                  theorem Algebra.TensorProduct.adjoin_tmul_eq_top (R : Type uR) (A : Type uA) (B : Type uB) [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] :
                                  Algebra.adjoin R {t : TensorProduct R A B | ∃ (a : A) (b : B), a ⊗ₜ[R] b = t} =
                                  theorem Algebra.TensorProduct.assoc_aux_1 {R : Type uR} {A : Type uA} {B : Type uB} {C : Type uC} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [Semiring C] [Algebra R C] (a₁ : A) (a₂ : A) (b₁ : B) (b₂ : B) (c₁ : C) (c₂ : C) :
                                  (TensorProduct.assoc R A B C) (((a₁ * a₂) ⊗ₜ[R] (b₁ * b₂)) ⊗ₜ[R] (c₁ * c₂)) = (TensorProduct.assoc R A B C) ((a₁ ⊗ₜ[R] b₁) ⊗ₜ[R] c₁) * (TensorProduct.assoc R A B C) ((a₂ ⊗ₜ[R] b₂) ⊗ₜ[R] c₂)
                                  theorem Algebra.TensorProduct.assoc_aux_2 {R : Type uR} {A : Type uA} {B : Type uB} {C : Type uC} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [Semiring C] [Algebra R C] :
                                  (TensorProduct.assoc R A B C) ((1 ⊗ₜ[R] 1) ⊗ₜ[R] 1) = 1
                                  noncomputable def Algebra.TensorProduct.assoc (R : Type uR) (A : Type uA) (B : Type uB) (C : Type uC) [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [Semiring C] [Algebra R C] :

                                  The associator for tensor product of R-algebras, as an algebra isomorphism.

                                  Equations
                                  Instances For
                                    @[simp]
                                    theorem Algebra.TensorProduct.assoc_tmul (R : Type uR) {A : Type uA} {B : Type uB} {C : Type uC} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [Semiring C] [Algebra R C] (a : A) (b : B) (c : C) :
                                    @[simp]
                                    theorem Algebra.TensorProduct.assoc_symm_tmul (R : Type uR) {A : Type uA} {B : Type uB} {C : Type uC} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [Semiring C] [Algebra R C] (a : A) (b : B) (c : C) :
                                    noncomputable def Algebra.TensorProduct.map {R : Type uR} {S : Type uS} {A : Type uA} {B : Type uB} {C : Type uC} {D : Type uD} [CommSemiring R] [CommSemiring S] [Algebra R S] [Semiring A] [Algebra R A] [Algebra S A] [IsScalarTower R S A] [Semiring B] [Algebra R B] [Algebra S B] [IsScalarTower R S B] [Semiring C] [Algebra R C] [Semiring D] [Algebra R D] (f : A →ₐ[S] B) (g : C →ₐ[R] D) :

                                    The tensor product of a pair of algebra morphisms.

                                    Equations
                                    Instances For
                                      @[simp]
                                      theorem Algebra.TensorProduct.map_tmul {R : Type uR} {S : Type uS} {A : Type uA} {B : Type uB} {C : Type uC} {D : Type uD} [CommSemiring R] [CommSemiring S] [Algebra R S] [Semiring A] [Algebra R A] [Algebra S A] [IsScalarTower R S A] [Semiring B] [Algebra R B] [Algebra S B] [IsScalarTower R S B] [Semiring C] [Algebra R C] [Semiring D] [Algebra R D] (f : A →ₐ[S] B) (g : C →ₐ[R] D) (a : A) (c : C) :
                                      @[simp]
                                      theorem Algebra.TensorProduct.map_id {R : Type uR} {S : Type uS} {A : Type uA} {C : Type uC} [CommSemiring R] [CommSemiring S] [Algebra R S] [Semiring A] [Algebra R A] [Algebra S A] [IsScalarTower R S A] [Semiring C] [Algebra R C] :
                                      theorem Algebra.TensorProduct.map_comp {R : Type uR} {S : Type uS} {A : Type uA} {B : Type uB} {C : Type uC} {D : Type uD} {E : Type uE} {F : Type uF} [CommSemiring R] [CommSemiring S] [Algebra R S] [Semiring A] [Algebra R A] [Algebra S A] [IsScalarTower R S A] [Semiring B] [Algebra R B] [Algebra S B] [IsScalarTower R S B] [Semiring C] [Algebra R C] [Algebra S C] [IsScalarTower R S C] [Semiring D] [Algebra R D] [Semiring E] [Algebra R E] [Semiring F] [Algebra R F] (f₂ : B →ₐ[S] C) (f₁ : A →ₐ[S] B) (g₂ : E →ₐ[R] F) (g₁ : D →ₐ[R] E) :
                                      @[simp]
                                      theorem Algebra.TensorProduct.map_comp_includeLeft {R : Type uR} {S : Type uS} {A : Type uA} {B : Type uB} {C : Type uC} {D : Type uD} [CommSemiring R] [CommSemiring S] [Algebra R S] [Semiring A] [Algebra R A] [Algebra S A] [IsScalarTower R S A] [Semiring B] [Algebra R B] [Algebra S B] [IsScalarTower R S B] [Semiring C] [Algebra R C] [Semiring D] [Algebra R D] (f : A →ₐ[S] B) (g : C →ₐ[R] D) :
                                      AlgHom.comp (Algebra.TensorProduct.map f g) Algebra.TensorProduct.includeLeft = AlgHom.comp Algebra.TensorProduct.includeLeft f
                                      @[simp]
                                      theorem Algebra.TensorProduct.map_restrictScalars_comp_includeRight {R : Type uR} {S : Type uS} {A : Type uA} {B : Type uB} {C : Type uC} {D : Type uD} [CommSemiring R] [CommSemiring S] [Algebra R S] [Semiring A] [Algebra R A] [Algebra S A] [IsScalarTower R S A] [Semiring B] [Algebra R B] [Algebra S B] [IsScalarTower R S B] [Semiring C] [Algebra R C] [Semiring D] [Algebra R D] (f : A →ₐ[S] B) (g : C →ₐ[R] D) :
                                      AlgHom.comp (AlgHom.restrictScalars R (Algebra.TensorProduct.map f g)) Algebra.TensorProduct.includeRight = AlgHom.comp Algebra.TensorProduct.includeRight g
                                      @[simp]
                                      theorem Algebra.TensorProduct.map_comp_includeRight {R : Type uR} {A : Type uA} {B : Type uB} {C : Type uC} {D : Type uD} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [Semiring C] [Algebra R C] [Semiring D] [Algebra R D] (f : A →ₐ[R] B) (g : C →ₐ[R] D) :
                                      AlgHom.comp (Algebra.TensorProduct.map f g) Algebra.TensorProduct.includeRight = AlgHom.comp Algebra.TensorProduct.includeRight g
                                      theorem Algebra.TensorProduct.map_range {R : Type uR} {A : Type uA} {B : Type uB} {C : Type uC} {D : Type uD} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [Semiring C] [Algebra R C] [Semiring D] [Algebra R D] (f : A →ₐ[R] B) (g : C →ₐ[R] D) :
                                      AlgHom.range (Algebra.TensorProduct.map f g) = AlgHom.range (AlgHom.comp Algebra.TensorProduct.includeLeft f) AlgHom.range (AlgHom.comp Algebra.TensorProduct.includeRight g)
                                      noncomputable def Algebra.TensorProduct.congr {R : Type uR} {S : Type uS} {A : Type uA} {B : Type uB} {C : Type uC} {D : Type uD} [CommSemiring R] [CommSemiring S] [Algebra R S] [Semiring A] [Algebra R A] [Algebra S A] [IsScalarTower R S A] [Semiring B] [Algebra R B] [Algebra S B] [IsScalarTower R S B] [Semiring C] [Algebra R C] [Semiring D] [Algebra R D] (f : A ≃ₐ[S] B) (g : C ≃ₐ[R] D) :

                                      Construct an isomorphism between tensor products of an S-algebra with an R-algebra from S- and R- isomorphisms between the tensor factors.

                                      Equations
                                      Instances For
                                        @[simp]
                                        theorem Algebra.TensorProduct.congr_apply {R : Type uR} {S : Type uS} {A : Type uA} {B : Type uB} {C : Type uC} {D : Type uD} [CommSemiring R] [CommSemiring S] [Algebra R S] [Semiring A] [Algebra R A] [Algebra S A] [IsScalarTower R S A] [Semiring B] [Algebra R B] [Algebra S B] [IsScalarTower R S B] [Semiring C] [Algebra R C] [Semiring D] [Algebra R D] (f : A ≃ₐ[S] B) (g : C ≃ₐ[R] D) (x : TensorProduct R A C) :
                                        @[simp]
                                        theorem Algebra.TensorProduct.congr_symm_apply {R : Type uR} {S : Type uS} {A : Type uA} {B : Type uB} {C : Type uC} {D : Type uD} [CommSemiring R] [CommSemiring S] [Algebra R S] [Semiring A] [Algebra R A] [Algebra S A] [IsScalarTower R S A] [Semiring B] [Algebra R B] [Algebra S B] [IsScalarTower R S B] [Semiring C] [Algebra R C] [Semiring D] [Algebra R D] (f : A ≃ₐ[S] B) (g : C ≃ₐ[R] D) (x : TensorProduct R B D) :
                                        @[simp]
                                        theorem Algebra.TensorProduct.congr_refl {R : Type uR} {S : Type uS} {A : Type uA} {C : Type uC} [CommSemiring R] [CommSemiring S] [Algebra R S] [Semiring A] [Algebra R A] [Algebra S A] [IsScalarTower R S A] [Semiring C] [Algebra R C] :
                                        Algebra.TensorProduct.congr AlgEquiv.refl AlgEquiv.refl = AlgEquiv.refl
                                        theorem Algebra.TensorProduct.congr_trans {R : Type uR} {S : Type uS} {A : Type uA} {B : Type uB} {C : Type uC} {D : Type uD} {E : Type uE} {F : Type uF} [CommSemiring R] [CommSemiring S] [Algebra R S] [Semiring A] [Algebra R A] [Algebra S A] [IsScalarTower R S A] [Semiring B] [Algebra R B] [Algebra S B] [IsScalarTower R S B] [Semiring C] [Algebra R C] [Algebra S C] [IsScalarTower R S C] [Semiring D] [Algebra R D] [Semiring E] [Algebra R E] [Semiring F] [Algebra R F] (f₁ : A ≃ₐ[S] B) (f₂ : B ≃ₐ[S] C) (g₁ : D ≃ₐ[R] E) (g₂ : E ≃ₐ[R] F) :
                                        theorem Algebra.TensorProduct.congr_symm {R : Type uR} {S : Type uS} {A : Type uA} {B : Type uB} {C : Type uC} {D : Type uD} [CommSemiring R] [CommSemiring S] [Algebra R S] [Semiring A] [Algebra R A] [Algebra S A] [IsScalarTower R S A] [Semiring B] [Algebra R B] [Algebra S B] [IsScalarTower R S B] [Semiring C] [Algebra R C] [Semiring D] [Algebra R D] (f : A ≃ₐ[S] B) (g : C ≃ₐ[R] D) :
                                        @[inline, reducible]
                                        noncomputable abbrev Algebra.TensorProduct.productLeftAlgHom {R : Type uR} {S : Type uS} {A : Type uA} {B : Type uB} {C : Type uC} [CommSemiring R] [CommSemiring S] [Algebra R S] [Semiring A] [Algebra R A] [Algebra S A] [IsScalarTower R S A] [Semiring B] [Algebra R B] [CommSemiring C] [Algebra R C] [Algebra S C] [IsScalarTower R S C] (f : A →ₐ[S] C) (g : B →ₐ[R] C) :

                                        If A, B, C are R-algebras, A and C are also S-algebras (forming a tower as ·/S/R), then the product map of f : A →ₐ[S] C and g : B →ₐ[R] C is an S-algebra homomorphism.

                                        This is just a special case of Algebra.TensorProduct.lift for when C is commutative.

                                        Equations
                                        Instances For
                                          @[simp]
                                          theorem Algebra.TensorProduct.lmul'_apply_tmul {R : Type uR} {S : Type uS} [CommSemiring R] [CommSemiring S] [Algebra R S] (a : S) (b : S) :
                                          @[simp]
                                          theorem Algebra.TensorProduct.lmul'_comp_includeLeft {R : Type uR} {S : Type uS} [CommSemiring R] [CommSemiring S] [Algebra R S] :
                                          AlgHom.comp (Algebra.TensorProduct.lmul' R) Algebra.TensorProduct.includeLeft = AlgHom.id R S
                                          @[simp]
                                          theorem Algebra.TensorProduct.lmul'_comp_includeRight {R : Type uR} {S : Type uS} [CommSemiring R] [CommSemiring S] [Algebra R S] :
                                          AlgHom.comp (Algebra.TensorProduct.lmul' R) Algebra.TensorProduct.includeRight = AlgHom.id R S
                                          noncomputable def Algebra.TensorProduct.productMap {R : Type uR} {S : Type uS} {A : Type uA} {B : Type uB} [CommSemiring R] [Semiring A] [Semiring B] [CommSemiring S] [Algebra R A] [Algebra R B] [Algebra R S] (f : A →ₐ[R] S) (g : B →ₐ[R] S) :

                                          If S is commutative, for a pair of morphisms f : A →ₐ[R] S, g : B →ₐ[R] S, We obtain a map A ⊗[R] B →ₐ[R] S that commutes with f, g via a ⊗ b ↦ f(a) * g(b).

                                          This is a special case of Algebra.TensorProduct.productLeftAlgHom for when the two base rings are the same.

                                          Equations
                                          Instances For
                                            @[simp]
                                            theorem Algebra.TensorProduct.productMap_apply_tmul {R : Type uR} {S : Type uS} {A : Type uA} {B : Type uB} [CommSemiring R] [Semiring A] [Semiring B] [CommSemiring S] [Algebra R A] [Algebra R B] [Algebra R S] (f : A →ₐ[R] S) (g : B →ₐ[R] S) (a : A) (b : B) :
                                            theorem Algebra.TensorProduct.productMap_left_apply {R : Type uR} {S : Type uS} {A : Type uA} {B : Type uB} [CommSemiring R] [Semiring A] [Semiring B] [CommSemiring S] [Algebra R A] [Algebra R B] [Algebra R S] (f : A →ₐ[R] S) (g : B →ₐ[R] S) (a : A) :
                                            @[simp]
                                            theorem Algebra.TensorProduct.productMap_left {R : Type uR} {S : Type uS} {A : Type uA} {B : Type uB} [CommSemiring R] [Semiring A] [Semiring B] [CommSemiring S] [Algebra R A] [Algebra R B] [Algebra R S] (f : A →ₐ[R] S) (g : B →ₐ[R] S) :
                                            AlgHom.comp (Algebra.TensorProduct.productMap f g) Algebra.TensorProduct.includeLeft = f
                                            theorem Algebra.TensorProduct.productMap_right_apply {R : Type uR} {S : Type uS} {A : Type uA} {B : Type uB} [CommSemiring R] [Semiring A] [Semiring B] [CommSemiring S] [Algebra R A] [Algebra R B] [Algebra R S] (f : A →ₐ[R] S) (g : B →ₐ[R] S) (b : B) :
                                            @[simp]
                                            theorem Algebra.TensorProduct.productMap_right {R : Type uR} {S : Type uS} {A : Type uA} {B : Type uB} [CommSemiring R] [Semiring A] [Semiring B] [CommSemiring S] [Algebra R A] [Algebra R B] [Algebra R S] (f : A →ₐ[R] S) (g : B →ₐ[R] S) :
                                            AlgHom.comp (Algebra.TensorProduct.productMap f g) Algebra.TensorProduct.includeRight = g
                                            noncomputable def Algebra.TensorProduct.basisAux {R : Type uR} (A : Type uA) {M : Type uM} {ι : Type uι} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] (b : Basis ι R M) :

                                            Given an R-algebra A and an R-basis of M, this is an R-linear isomorphism A ⊗[R] M ≃ (ι →₀ A) (which is in fact A-linear).

                                            Equations
                                            • One or more equations did not get rendered due to their size.
                                            Instances For
                                              theorem Algebra.TensorProduct.basisAux_tmul {R : Type uR} {A : Type uA} {M : Type uM} {ι : Type uι} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] (b : Basis ι R M) (a : A) (m : M) :
                                              theorem Algebra.TensorProduct.basisAux_map_smul {R : Type uR} {A : Type uA} {M : Type uM} {ι : Type uι} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] (b : Basis ι R M) (a : A) (x : TensorProduct R A M) :
                                              noncomputable def Algebra.TensorProduct.basis {R : Type uR} (A : Type uA) {M : Type uM} {ι : Type uι} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] (b : Basis ι R M) :
                                              Basis ι A (TensorProduct R A M)

                                              Given a R-algebra A, this is the A-basis of A ⊗[R] M induced by a R-basis of M.

                                              Equations
                                              • One or more equations did not get rendered due to their size.
                                              Instances For
                                                @[simp]
                                                theorem Algebra.TensorProduct.basis_repr_tmul {R : Type uR} {A : Type uA} {M : Type uM} {ι : Type uι} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] (b : Basis ι R M) (a : A) (m : M) :
                                                (Algebra.TensorProduct.basis A b).repr (a ⊗ₜ[R] m) = a Finsupp.mapRange (algebraMap R A) (b.repr m)
                                                theorem Algebra.TensorProduct.basis_repr_symm_apply {R : Type uR} {A : Type uA} {M : Type uM} {ι : Type uι} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] (b : Basis ι R M) (a : A) (i : ι) :
                                                @[simp]
                                                theorem Algebra.TensorProduct.basis_apply {R : Type uR} {A : Type uA} {M : Type uM} {ι : Type uι} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] (b : Basis ι R M) (i : ι) :
                                                theorem Algebra.TensorProduct.basis_repr_symm_apply' {R : Type uR} {A : Type uA} {M : Type uM} {ι : Type uι} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] (b : Basis ι R M) (a : A) (i : ι) :
                                                @[simp]
                                                theorem LinearMap.toMatrix_baseChange {R : Type u_1} {M₁ : Type u_2} {M₂ : Type u_3} {ι : Type u_4} {ι₂ : Type u_5} (A : Type u_6) [Fintype ι] [Fintype ι₂] [DecidableEq ι] [CommSemiring R] [CommSemiring A] [Algebra R A] [AddCommMonoid M₁] [Module R M₁] [AddCommMonoid M₂] [Module R M₂] (f : M₁ →ₗ[R] M₂) (b₁ : Basis ι R M₁) (b₂ : Basis ι₂ R M₂) :
                                                noncomputable def Module.endTensorEndAlgHom {R : Type u_1} {S : Type u_2} {A : Type u_3} {M : Type u_4} {N : Type u_5} [CommSemiring R] [CommSemiring S] [Semiring A] [AddCommMonoid M] [AddCommMonoid N] [Algebra R S] [Algebra S A] [Algebra R A] [Module R M] [Module S M] [Module A M] [Module R N] [IsScalarTower R A M] [IsScalarTower S A M] [IsScalarTower R S M] :

                                                The algebra homomorphism from End M ⊗ End N to End (M ⊗ N) sending f ⊗ₜ g to the TensorProduct.map f g, the tensor product of the two maps.

                                                This is an AlgHom version of TensorProduct.AlgebraTensorModule.homTensorHomMap. Like that definition, this is generalized across many different rings; namely a tower of algebras A/S/R.

                                                Equations
                                                Instances For
                                                  theorem Module.endTensorEndAlgHom_apply {R : Type u_1} {S : Type u_2} {A : Type u_3} {M : Type u_4} {N : Type u_5} [CommSemiring R] [CommSemiring S] [Semiring A] [AddCommMonoid M] [AddCommMonoid N] [Algebra R S] [Algebra S A] [Algebra R A] [Module R M] [Module S M] [Module A M] [Module R N] [IsScalarTower R A M] [IsScalarTower S A M] [IsScalarTower R S M] (f : Module.End A M) (g : Module.End R N) :
                                                  Module.endTensorEndAlgHom (f ⊗ₜ[R] g) = TensorProduct.AlgebraTensorModule.map f g
                                                  theorem Subalgebra.finiteDimensional_sup {K : Type u_1} {L : Type u_2} [Field K] [CommRing L] [Algebra K L] (E1 : Subalgebra K L) (E2 : Subalgebra K L) [FiniteDimensional K E1] [FiniteDimensional K E2] :
                                                  FiniteDimensional K (E1 E2)
                                                  noncomputable def TensorProduct.Algebra.moduleAux {R : Type u_1} {A : Type u_2} {B : Type u_3} {M : Type u_4} [CommSemiring R] [AddCommMonoid M] [Module R M] [Semiring A] [Semiring B] [Module A M] [Module B M] [Algebra R A] [Algebra R B] [IsScalarTower R A M] [IsScalarTower R B M] :

                                                  An auxiliary definition, used for constructing the Module (A ⊗[R] B) M in TensorProduct.Algebra.module below.

                                                  Equations
                                                  Instances For
                                                    theorem TensorProduct.Algebra.moduleAux_apply {R : Type u_1} {A : Type u_2} {B : Type u_3} {M : Type u_4} [CommSemiring R] [AddCommMonoid M] [Module R M] [Semiring A] [Semiring B] [Module A M] [Module B M] [Algebra R A] [Algebra R B] [IsScalarTower R A M] [IsScalarTower R B M] (a : A) (b : B) (m : M) :
                                                    (TensorProduct.Algebra.moduleAux (a ⊗ₜ[R] b)) m = a b m
                                                    noncomputable def TensorProduct.Algebra.module {R : Type u_1} {A : Type u_2} {B : Type u_3} {M : Type u_4} [CommSemiring R] [AddCommMonoid M] [Module R M] [Semiring A] [Semiring B] [Module A M] [Module B M] [Algebra R A] [Algebra R B] [IsScalarTower R A M] [IsScalarTower R B M] [SMulCommClass A B M] :

                                                    If M is a representation of two different R-algebras A and B whose actions commute, then it is a representation the R-algebra A ⊗[R] B.

                                                    An important example arises from a semiring S; allowing S to act on itself via left and right multiplication, the roles of R, A, B, M are played by , S, Sᵐᵒᵖ, S. This example is important because a submodule of S as a Module over S ⊗[ℕ] Sᵐᵒᵖ is a two-sided ideal.

                                                    NB: This is not an instance because in the case B = A and M = A ⊗[R] A we would have a diamond of smul actions. Furthermore, this would not be a mere definitional diamond but a true mathematical diamond in which A ⊗[R] A had two distinct scalar actions on itself: one from its multiplication, and one from this would-be instance. Arguably we could live with this but in any case the real fix is to address the ambiguity in notation, probably along the lines outlined here: https://leanprover.zulipchat.com/#narrow/stream/144837-PR-reviews/topic/.234773.20base.20change/near/240929258

                                                    Equations
                                                    Instances For
                                                      theorem TensorProduct.Algebra.smul_def {R : Type u_1} {A : Type u_2} {B : Type u_3} {M : Type u_4} [CommSemiring R] [AddCommMonoid M] [Module R M] [Semiring A] [Semiring B] [Module A M] [Module B M] [Algebra R A] [Algebra R B] [IsScalarTower R A M] [IsScalarTower R B M] [SMulCommClass A B M] (a : A) (b : B) (m : M) :
                                                      a ⊗ₜ[R] b m = a b m