Tensor product of algebras

<h2 id="definition">Definition</h2>
Let R be a commutative ring and let A and B be <a href="/facts/Associative_algebra/DRfoF6KV">R-algebras</a>. Since A and B may both be regarded as <a href="/facts/Module_(mathematics)/jP0LIhFL">R-modules</a>, their <a href="/facts/Tensor_product_of_modules/K3oaFIS4">tensor product</a>

A
        
          ⊗
          
            R
          
        
        B
      
    
    {\displaystyle 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 class="footnote-ref" id="fnref:1" href="#fn:1">1</a><a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>

(
        
          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}}

and then extending by linearity to all of A ⊗R B. This ring is an R-algebra, associative and unital with the identity element given by 1A ⊗ 1B.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> 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 <a href="/facts/Category_(mathematics)/7WzxUThf">category</a> of R-algebras into a <a href="/facts/Symmetric_monoidal_category/AxhumNny">symmetric monoidal category</a>.

<h2 id="further-properties">Further properties</h2>
There are natural homomorphisms from A and B to A ⊗R B given by<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>

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

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

These maps make the tensor product the <a href="/facts/Coproduct/0bJ8SCdu">coproduct</a> in the <a href="/facts/Category_of_commutative_algebras/pIwY32bc">category of commutative R-algebras</a>. The tensor product is not the coproduct in the category of all R-algebras. There the coproduct is given by a more general <a href="/facts/Free_product_of_algebras/2YLxKbJj">free product of algebras</a>. Nevertheless, the tensor product of non-commutative algebras can be described by a <a href="/facts/Universal_property/5XJBYMNz">universal property</a> similar to that of the coproduct:

Hom
        
        (
        A
        ⊗
        B
        ,
        X
        )
        ≅
        {
        (
        f
        ,
        g
        )
        ∈
        
          Hom
        
        (
        A
        ,
        X
        )
        ×
        
          Hom
        
        (
        B
        ,
        X
        )
        ∣
        ∀
        a
        ∈
        A
        ,
        b
        ∈
        B
        :
        [
        f
        (
        a
        )
        ,
        g
        (
        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:[f(a),g(b)]=0\rbrace ,}

where [-, -] denotes the <a href="/facts/Commutator/eYxzFmFY">commutator</a>.
The <a href="/facts/Natural_isomorphism/xwPumTHt">natural isomorphism</a> is given by identifying a morphism 
 
 
 
 ϕ
 :
 A
 ⊗
 B
 →
 X
 
 
 {\displaystyle \phi :A\otimes B\to X}
 
 on the left hand side with the pair of morphisms 
 
 
 
 (
 f
 ,
 g
 )
 
 
 {\displaystyle (f,g)}
 
 on the right hand side where 
 
 
 
 f
 (
 a
 )
 :=
 ϕ
 (
 a
 ⊗
 1
 )
 
 
 {\displaystyle f(a):=\phi (a\otimes 1)}
 
 and similarly 
 
 
 
 g
 (
 b
 )
 :=
 ϕ
 (
 1
 ⊗
 b
 )
 
 
 {\displaystyle g(b):=\phi (1\otimes b)}
 
.

<h2 id="applications">Applications</h2>
The tensor product of commutative algebras is of frequent use in <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic geometry</a>. For <a href="/facts/Affine_scheme/2k2nDE5w">affine schemes</a> 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 <a href="/facts/Fiber_product_of_schemes/gSFuGC7l">fiber product scheme</a> 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).}

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

<h2 id="examples">Examples</h2>
See also: <a href="/facts/Tensor_product_of_modules/K3oaFIS4">tensor product of modules § Examples</a>
<ul><li>The tensor product can be used as a means of taking <a href="/facts/Scheme-theoretic_intersection/ajm2778v">intersections</a> of two subschemes in a <a href="/facts/Scheme_(mathematics)/c7uSqGqL">scheme</a>: consider the 
 
 
 
 
 C
 
 [
 x
 ,
 y
 ]
 
 
 {\displaystyle \mathbb {C} [x,y]}
 
-algebras 
 
 
 
 
 C
 
 [
 x
 ,
 y
 ]
 
 /
 
 f
 
 
 {\displaystyle \mathbb {C} [x,y]/f}
 
, 
 
 
 
 
 C
 
 [
 x
 ,
 y
 ]
 
 /
 
 g
 
 
 {\displaystyle \mathbb {C} [x,y]/g}
 
, then their tensor product is 
 
 
 
 
 C
 
 [
 x
 ,
 y
 ]
 
 /
 
 (
 f
 )
 
 ⊗
 
 
 C
 
 [
 x
 ,
 y
 ]
 
 
 
 C
 
 [
 x
 ,
 y
 ]
 
 /
 
 (
 g
 )
 ≅
 
 C
 
 [
 x
 ,
 y
 ]
 
 /
 
 (
 f
 ,
 g
 )
 
 
 {\displaystyle \mathbb {C} [x,y]/(f)\otimes _{\mathbb {C} [x,y]}\mathbb {C} [x,y]/(g)\cong \mathbb {C} [x,y]/(f,g)}
 
, which describes the intersection of the <a href="/facts/Algebraic_curve/LnWsPmDP">algebraic curves</a> f = 0 and g = 0 in the affine plane over C.</li>
<li>More generally, if 
 
 
 
 A
 
 
 {\displaystyle A}
 
 is a commutative ring and 
 
 
 
 I
 ,
 J
 ⊆
 A
 
 
 {\displaystyle 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}}}
 
, with a unique isomorphism sending 
 
 
 
 (
 a
 +
 I
 )
 ⊗
 (
 b
 +
 J
 )
 
 
 {\displaystyle (a+I)\otimes (b+J)}
 
 to 
 
 
 
 (
 a
 b
 +
 I
 +
 J
 )
 
 
 {\displaystyle (ab+I+J)}
 
.</li>
<li>Tensor products can be used as a means of changing coefficients. For example, 
 
 
 
 
 Z
 
 [
 x
 ,
 y
 ]
 
 /
 
 (
 
 x
 
 3
 
 
 +
 5
 
 x
 
 2
 
 
 +
 x
 −
 1
 )
 
 ⊗
 
 
 Z
 
 
 
 
 Z
 
 
 /
 
 5
 ≅
 
 Z
 
 
 /
 
 5
 [
 x
 ,
 y
 ]
 
 /
 
 (
 
 x
 
 3
 
 
 +
 x
 −
 1
 )
 
 
 {\displaystyle \mathbb {Z} [x,y]/(x^{3}+5x^{2}+x-1)\otimes _{\mathbb {Z} }\mathbb {Z} /5\cong \mathbb {Z} /5[x,y]/(x^{3}+x-1)}
 
 and 
 
 
 
 
 Z
 
 [
 x
 ,
 y
 ]
 
 /
 
 (
 f
 )
 
 ⊗
 
 
 Z
 
 
 
 
 C
 
 ≅
 
 C
 
 [
 x
 ,
 y
 ]
 
 /
 
 (
 f
 )
 
 
 {\displaystyle \mathbb {Z} [x,y]/(f)\otimes _{\mathbb {Z} }\mathbb {C} \cong \mathbb {C} [x,y]/(f)}
 
.</li>
<li>Tensor products also can be used for taking <a href="/facts/Fiber_product/bC0QoX5U">products</a> of affine schemes over a field. For example, 
 
 
 
 
 C
 
 [
 
 x
 
 1
 
 
 ,
 
 x
 
 2
 
 
 ]
 
 /
 
 (
 f
 (
 x
 )
 )
 
 ⊗
 
 
 C
 
 
 
 
 C
 
 [
 
 y
 
 1
 
 
 ,
 
 y
 
 2
 
 
 ]
 
 /
 
 (
 g
 (
 y
 )
 )
 
 
 {\displaystyle \mathbb {C} [x_{1},x_{2}]/(f(x))\otimes _{\mathbb {C} }\mathbb {C} [y_{1},y_{2}]/(g(y))}
 
 is <a href="/facts/Isomorphic/pj8KgkxU">isomorphic</a> to the algebra 
 
 
 
 
 C
 
 [
 
 x
 
 1
 
 
 ,
 
 x
 
 2
 
 
 ,
 
 y
 
 1
 
 
 ,
 
 y
 
 2
 
 
 ]
 
 /
 
 (
 f
 (
 x
 )
 ,
 g
 (
 y
 )
 )
 
 
 {\displaystyle \mathbb {C} [x_{1},x_{2},y_{1},y_{2}]/(f(x),g(y))}
 
 which corresponds to an affine surface in 
 
 
 
 
 
 A
 
 
 
 C
 
 
 
 4
 
 
 
 
 {\displaystyle \mathbb {A} _{\mathbb {C} }^{4}}
 
 if f and g are not zero.</li>
<li>Given 
 
 
 
 R
 
 
 {\displaystyle R}
 
-algebras 
 
 
 
 A
 
 
 {\displaystyle A}
 
 and 
 
 
 
 B
 
 
 {\displaystyle B}
 
 whose underlying rings are <a href="/facts/Graded-commutative_ring/jPvVDpJC">graded-commutative rings</a>, the tensor product 
 
 
 
 A
 
 ⊗
 
 R
 
 
 B
 
 
 {\displaystyle 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'}
 
 for homogeneous 
 
 
 
 a
 
 
 {\displaystyle a}
 
, 
 
 
 
 
 a
 ′
 
 
 
 {\displaystyle a'}
 
, 
 
 
 
 b
 
 
 {\displaystyle b}
 
, and 
 
 
 
 
 b
 ′
 
 
 
 {\displaystyle b'}
 
.</li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Extension_of_scalars/RYGsTsYk">Extension of scalars</a></li>
<li><a href="/facts/Tensor_product_of_modules/K3oaFIS4">Tensor product of modules</a></li>
<li><a href="/facts/Tensor_product_of_fields/FklJQo5j">Tensor product of fields</a></li>
<li><a href="/facts/Linearly_disjoint/Totpn0fa">Linearly disjoint</a></li>
<li><a href="/facts/Multilinear_subspace_learning/B2mIlGQQ">Multilinear subspace learning</a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li>Kassel, Christian (1995), <a href="https://archive.org/details/quantumgroups0000kass">Quantum groups</a>, Graduate texts in mathematics, vol. 155, Springer, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-94370-1.</li>
<li>Lang, Serge (2002) [first published in 1993]. Algebra. Graduate Texts in Mathematics. Vol. 21. Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-95385-X.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Kassel (1995), p. 32. <a href="https://books.google.com/books?id=S1KE_pToY98C&pg=PA32&dq=%22we+put+an+algebra+structure+on+the+tensor+product%22" target="_blank">https://books.google.com/books?id=S1KE_pToY98C&pg=PA32&dq=%22we+put+an+algebra+structure+on+the+tensor+product%22</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Lang 2002, pp. 629–630. - Lang, Serge (2002) [first published in 1993]. Algebra. Graduate Texts in Mathematics. Vol. 21. Springer. ISBN 0-387-95385-X. <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Kassel (1995), p. 32. <a href="https://books.google.com/books?id=S1KE_pToY98C&pg=PA32&dq=%22Its+unit+is%22" target="_blank">https://books.google.com/books?id=S1KE_pToY98C&pg=PA32&dq=%22Its+unit+is%22</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Kassel (1995), p. 32. <a href="https://books.google.com/books?id=S1KE_pToY98C&pg=PA32&dq=%22get+algebra+morphisms%22" target="_blank">https://books.google.com/books?id=S1KE_pToY98C&pg=PA32&dq=%22get+algebra+morphisms%22</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
</ol>

Tensor product of algebras open-in-new

Tensor product of algebras