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Tensor Product

Summary. The tensor product is a formal bilinear multiplication of two modules or vector spaces. In essence, it permits us to replace bilinear maps from two such objects by an equivalent linear map from the tensor product of the two objects. The origin of this operation lies in classic differential geometry and physics, which had need of multiply indexed geometric objects such as the first and second fundamental forms, and the stress tensor -- see Tensor Product (Classical).

Definition (Standard). Let be a commutative ring, and let be -modules. There exists an -module $A\otimes B$ , called the tensor product of and over , together with a canonical bilinear homomorphism

$\displaystyle \otimes: A\times B\rightarrow A\otimes B,$

distinguished, up to isomorphism, by the following universal property. Every bilinear

-module homomorphism

$\displaystyle \phi: A\times B\rightarrow C,$

lifts to a unique

-module homomorphism

$\displaystyle \tilde{\phi}: A\otimes B\rightarrow C,$

such that

$\displaystyle \phi(a,b) = \tilde{\phi}(a\otimes b)$

for all $a\in A,\; b\in B.$ Diagramatically:

$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{\ar[dr]^(.55)\phi \ar[r]^\otimes A\times B& A\otimes B \ar@{-->}[d]^(.4){\exists !\, \tilde{\phi}}\\ &C} } \end{xy}$

The tensor product $A\otimes B$ can be constructed by taking the free -module generated by all formal symbols

$\displaystyle a\otimes b,\quad a\in A,\;b\in B,$

and quotienting by the obvious bilinear relations:

$\displaystyle (a_1+a_2)\otimes b$	$\displaystyle = a_1\otimes b + a_2\otimes b,\quad$	$\displaystyle a_1,a_2\in A,\; b\in B$
$\displaystyle a\otimes(b_1+b_2)$	$\displaystyle = a\otimes b_1 + a\otimes b_2,\quad$	$\displaystyle a\in A,\;b_1,b_2\in B$
$\displaystyle r(a\otimes b)$	$\displaystyle = (ra)\otimes b= a\otimes (rb)\quad$	$\displaystyle a\in A,\;b\in B,\; r\in R$

Note. In order to make the base ring clear, the tensor product $A\otimes B$ is sometimes written as $A\otimes_R B$ .

Basic properties. Let be a commutative ring and be -modules, then, as modules, we have the following isomorphisms:

$R\otimes M\cong M$ ,
$M\otimes N\cong N\otimes M$ ,
$(L\otimes M) \otimes N\cong L\otimes (M\otimes N)$
$(L\oplus M)\otimes N \cong (L\otimes N) \oplus (M\otimes N)$

Definition (Categorical). Using the language of categories, all of the above can be expressed quite simply by stating that for all -modules , the functor $(-) \otimes M$ is left-adjoint to the functor $\mathrm{Hom}(M,-)$ .

1	Types Of Homomorphisms
1	Category
1	Isomorphism
1	Functor
1	Linear Transformation
1	Vector Space
1	Filter Basis
1	Obvious
1	Operation
1	Language
1	Commutative Ring
1	Module
1	Multiplication
1	Generating Set Of A Group
2	Relation Between Objects
2	Origin
2	Canonical
3	Universal Property
3	Neutral Geometry
3	Tensor
5	Differential Geometry
5	Bilinear Map
8	Lifts Transversals
9	Tensor Product Classical
11	Second Fundamental Form