Tensor product of algebras

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Algebraic structure → Ring theory Ring theory
Basic conceptsRings • Subrings • Ideal • Quotient ring • Fractional ideal • Total ring of fractions • Product of rings • Free product of associative algebras • Tensor product of algebras Ring homomorphisms • Kernel • Inner automorphism • Frobenius endomorphism Algebraic structures • Module • Associative algebra • Graded ring • Involutive ring • Category of rings • Initial ring Z {\displaystyle \mathbb {Z} } $\mathbb {Z}$ • Terminal ring 0 = Z / 1 Z {\displaystyle 0=\mathbb {Z} /1\mathbb {Z} } $0=\mathbb {Z} /1\mathbb {Z}$ Related structures • Field • Finite field • Non-associative ring • Lie ring • Jordan ring • Semiring • Semifield
Commutative algebra Commutative rings • Integral domain • Integrally closed domain • GCD domain • Unique factorization domain • Principal ideal domain • Euclidean domain • Field • Finite field • Composition ring • Polynomial ring • Formal power series ring Algebraic number theory • Algebraic number field • Ring of integers • Algebraic independence • Transcendental number theory • Transcendence degree p-adic number theory and decimals • Direct limit/Inverse limit • Zero ring Z / 1 Z {\displaystyle \mathbb {Z} /1\mathbb {Z} } $\mathbb {Z} /1\mathbb {Z}$ • Integers modulo pn Z / p n Z {\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} } $\mathbb {Z} /p^{n}\mathbb {Z}$ • Prüfer p-ring Z ( p ∞ ) {\displaystyle \mathbb {Z} (p^{\infty })} $\mathbb {Z} (p^{\infty })$ • Base-p circle ring T {\displaystyle \mathbb {T} } $\mathbb {T}$ • Base-p integers Z {\displaystyle \mathbb {Z} } $\mathbb {Z}$ • p-adic rationals Z {\displaystyle \mathbb {Z} } $\mathbb {Z}$ • Base-p real numbers R {\displaystyle \mathbb {R} } $\mathbb {R}$ • p-adic integers Z p {\displaystyle \mathbb {Z} _{p}} $\mathbb {Z} _{p}$ • p-adic numbers Q p {\displaystyle \mathbb {Q} _{p}} $\mathbb {Q} _{p}$ • p-adic solenoid T p {\displaystyle \mathbb {T} _{p}} $\mathbb {T} _{p}$ Algebraic geometry • Affine variety
Noncommutative algebra Noncommutative rings • Division ring • Semiprimitive ring • Simple ring • Commutator Noncommutative algebraic geometry Free algebra Clifford algebra • Geometric algebra Operator algebra
v t e

Basic conceptsRings • Subrings • Ideal • Quotient ring • Fractional ideal • Total ring of fractions • Product of rings • Free product of associative algebras • Tensor product of algebras

Ring homomorphisms

• Kernel • Inner automorphism • Frobenius endomorphism

Algebraic structures

• Module • Associative algebra • Graded ring • Involutive ring • Category of rings • Initial ring Z {\displaystyle \mathbb {Z} }

\mathbb {Z}

• Terminal ring 0 = Z / 1 Z {\displaystyle 0=\mathbb {Z} /1\mathbb {Z} }

0=\mathbb {Z} /1\mathbb {Z}

Related structures

• Field • Finite field • Non-associative ring • Lie ring • Jordan ring • Semiring • Semifield

Commutative algebra Commutative rings • Integral domain • Integrally closed domain • GCD domain • Unique factorization domain • Principal ideal domain • Euclidean domain • Field • Finite field • Composition ring • Polynomial ring • Formal power series ring

Algebraic number theory

• Algebraic number field • Ring of integers • Algebraic independence • Transcendental number theory • Transcendence degree

p-adic number theory and decimals

• Direct limit/Inverse limit • Zero ring Z / 1 Z {\displaystyle \mathbb {Z} /1\mathbb {Z} }

\mathbb {Z} /1\mathbb {Z}

• Integers modulo pn Z / p n Z {\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} }

\mathbb {Z} /p^{n}\mathbb {Z}

• Prüfer p-ring Z ( p ∞ ) {\displaystyle \mathbb {Z} (p^{\infty })}

\mathbb {Z} (p^{\infty })

• Base-p circle ring T {\displaystyle \mathbb {T} }

\mathbb {T}

• Base-p integers Z {\displaystyle \mathbb {Z} }

\mathbb {Z}

• p-adic rationals Z {\displaystyle \mathbb {Z} }

\mathbb {Z}

• Base-p real numbers R {\displaystyle \mathbb {R} }

\mathbb {R}

• p-adic integers Z p {\displaystyle \mathbb {Z} _{p}}

\mathbb {Z} _{p}

• p-adic numbers Q p {\displaystyle \mathbb {Q} _{p}}

\mathbb {Q} _{p}

• p-adic solenoid T p {\displaystyle \mathbb {T} _{p}}

\mathbb {T} _{p}

Algebraic geometry

• Affine variety

Noncommutative algebra Noncommutative rings • Division ring • Semiprimitive ring • Simple ring • Commutator

Noncommutative algebraic geometry

Free algebra

Clifford algebra

• Geometric algebra Operator algebra

In mathematics, the tensor product of two algebras over a commutative ring R is also an R-algebra. This gives the tensor product of algebras. When the ring is a field, the most common application of such products is to describe the product of algebra representations.

Definition

Let R be a commutative ring and let A and B be R-algebras. Since A and B may both be regarded as R-modules, their tensor product

A ⊗ R B {\displaystyle A\otimes _{R}B}

A\otimes _{R}B

is also an R-module. The tensor product can be given the structure of a ring by defining the product on elements of the form a ⊗ b by

( a 1 ⊗ b 1 ) ( a 2 ⊗ b 2 ) = a 1 a 2 ⊗ b 1 b 2 {\displaystyle (a_{1}\otimes b_{1})(a_{2}\otimes b_{2})=a_{1}a_{2}\otimes b_{1}b_{2}}

(a_{1}\otimes b_{1})(a_{2}\otimes b_{2})=a_{1}a_{2}\otimes b_{1}b_{2}

and then extending by linearity to all of A ⊗R B. This ring is an R-algebra, associative and unital with identity element given by 1A ⊗ 1B. where 1A and 1B are the identity elements of A and B. If A and B are commutative, then the tensor product is commutative as well.

The tensor product turns the category of R-algebras into a symmetric monoidal category.

Further properties

There are natural homomorphisms from A and B to A ⊗R B given by

a ↦ a ⊗ 1 B {\displaystyle a\mapsto a\otimes 1_{B}}

a\mapsto a\otimes 1_{B}

b ↦ 1 A ⊗ b {\displaystyle b\mapsto 1_{A}\otimes b}

b\mapsto 1_{A}\otimes b

These maps make the tensor product the coproduct in the category of commutative R-algebras. The tensor product is not the coproduct in the category of all R-algebras. There the coproduct is given by a more general free product of algebras. Nevertheless, the tensor product of non-commutative algebras can be described by a universal property similar to that of the coproduct:

Hom ( A ⊗ B , X ) ≅ { ( f , g ) ∈ Hom ( A , X ) × Hom ( B , X ) ∣ ∀ a ∈ A , b ∈ B : = 0 } , {\displaystyle {\text{Hom}}(A\otimes B,X)\cong \lbrace (f,g)\in {\text{Hom}}(A,X)\times {\text{Hom}}(B,X)\mid \forall a\in A,b\in B:=0\rbrace ,}

{\text{Hom}}(A\otimes B,X)\cong \lbrace (f,g)\in {\text{Hom}}(A,X)\times {\text{Hom}}(B,X)\mid \forall a\in A,b\in B:=0\rbrace ,

where denotes the commutator. The natural isomorphism is given by identifying a morphism ϕ : A ⊗ B → X {\displaystyle \phi :A\otimes B\to X} $\phi :A\otimes B\to X$ on the left hand side with the pair of morphisms ( f , g ) {\displaystyle (f,g)} $(f,g)$ on the right hand side where f ( a ) := ϕ ( a ⊗ 1 ) {\displaystyle f(a):=\phi (a\otimes 1)} $f(a):=\phi (a\otimes 1)$ and similarly g ( b ) := ϕ ( 1 ⊗ b ) {\displaystyle g(b):=\phi (1\otimes b)} $g(b):=\phi (1\otimes b)$ .

Applications

The tensor product of commutative algebras is of frequent use in algebraic geometry. For affine schemes X, Y, Z with morphisms from X and Z to Y, so X = Spec(A), Y = Spec(R), and Z = Spec(B) for some commutative rings A, R, B, the fiber product scheme is the affine scheme corresponding to the tensor product of algebras:

X × Y Z = Spec ⁡ ( A ⊗ R B ) . {\displaystyle X\times _{Y}Z=\operatorname {Spec} (A\otimes _{R}B).}

X\times _{Y}Z=\operatorname {Spec} (A\otimes _{R}B).

More generally, the fiber product of schemes is defined by gluing together affine fiber products of this form.

Examples

The tensor product can be used as a means of taking intersections of two subschemes in a scheme: consider the C {\displaystyle \mathbb {C} } $\mathbb {C}$ -algebras C / f {\displaystyle \mathbb {C} /f} $\mathbb {C} /f$ , C / g {\displaystyle \mathbb {C} /g} $\mathbb {C} /g$ , then their tensor product is C / ( f ) ⊗ C C / ( g ) ≅ C / ( f , g ) {\displaystyle \mathbb {C} /(f)\otimes _{\mathbb {C} }\mathbb {C} /(g)\cong \mathbb {C} /(f,g)} $\mathbb {C} /(f)\otimes _{\mathbb {C} }\mathbb {C} /(g)\cong \mathbb {C} /(f,g)$ , which describes the intersection of the algebraic curves f = 0 and g = 0 in the affine plane over C.
More generally, if A {\displaystyle A} $A$ is a commutative ring and I , J ⊆ A {\displaystyle I,J\subseteq A} $I,J\subseteq A$ are ideals, then A I ⊗ A A J ≅ A I + J {\displaystyle {\frac {A}{I}}\otimes _{A}{\frac {A}{J}}\cong {\frac {A}{I+J}}} ${\frac {A}{I}}\otimes _{A}{\frac {A}{J}}\cong {\frac {A}{I+J}}$ , with a unique isomorphism sending ( a + I ) ⊗ ( b + J ) {\displaystyle (a+I)\otimes (b+J)} $(a+I)\otimes (b+J)$ to ( a b + I + J ) {\displaystyle (ab+I+J)} $(ab+I+J)$ .
Tensor products can be used as a means of changing coefficients. For example, Z / ( x 3 + 5 x 2 + x − 1 ) ⊗ Z Z / 5 ≅ Z / 5 / ( x 3 + x − 1 ) {\displaystyle \mathbb {Z} /(x^{3}+5x^{2}+x-1)\otimes _{\mathbb {Z} }\mathbb {Z} /5\cong \mathbb {Z} /5/(x^{3}+x-1)} $\mathbb {Z} /(x^{3}+5x^{2}+x-1)\otimes _{\mathbb {Z} }\mathbb {Z} /5\cong \mathbb {Z} /5/(x^{3}+x-1)$ and Z / ( f ) ⊗ Z C ≅ C / ( f ) {\displaystyle \mathbb {Z} /(f)\otimes _{\mathbb {Z} }\mathbb {C} \cong \mathbb {C} /(f)} $\mathbb {Z} /(f)\otimes _{\mathbb {Z} }\mathbb {C} \cong \mathbb {C} /(f)$ .
Tensor products also can be used for taking products of affine schemes over a field. For example, C / ( f ( x ) ) ⊗ C C / ( g ( y ) ) {\displaystyle \mathbb {C} /(f(x))\otimes _{\mathbb {C} }\mathbb {C} /(g(y))} $\mathbb {C} /(f(x))\otimes _{\mathbb {C} }\mathbb {C} /(g(y))$ is isomorphic to the algebra C / ( f ( x ) , g ( y ) ) {\displaystyle \mathbb {C} /(f(x),g(y))} $\mathbb {C} /(f(x),g(y))$ which corresponds to an affine surface in A C 4 {\displaystyle \mathbb {A} _{\mathbb {C} }^{4}} $\mathbb {A} _{\mathbb {C} }^{4}$ if f and g are not zero.
Given R {\displaystyle R} $R$ -algebras A {\displaystyle A} $A$ and B {\displaystyle B} $B$ whose underlying rings are graded-commutative rings, the tensor product A ⊗ R B {\displaystyle A\otimes _{R}B} $A\otimes _{R}B$ becomes a graded commutative ring by defining ( a ⊗ b ) ( a ′ ⊗ b ′ ) = ( − 1 ) | b | | a ′ | a a ′ ⊗ b b ′ {\displaystyle (a\otimes b)(a'\otimes b')=(-1)^{|b||a'|}aa'\otimes bb'} $(a\otimes b)(a'\otimes b')=(-1)^{|b||a'|}aa'\otimes bb'$ for homogeneous a {\displaystyle a} $a$ , a ′ {\displaystyle a'} $a'$ , b {\displaystyle b} $b$ , and b ′ {\displaystyle b'} $b'$ .

Notes

^ Kassel (1995), p. 32.
^ Lang 2002, pp. 629–630.
^ Kassel (1995), p. 32.
^ Kassel (1995), p. 32.

References

Kassel, Christian (1995), Quantum groups, Graduate texts in mathematics, vol. 155, Springer, ISBN 978-0-387-94370-1.
Lang, Serge (2002) . Algebra. Graduate Texts in Mathematics. Vol. 21. Springer. ISBN 0-387-95385-X.