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tensor product (Definition)

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 $ R$ be a commutative ring, and let $ A, B$ be $ R$-modules. There exists an $ R$-module $ A\otimes B$, called the tensor product of $ A$ and $ B$ over $ R$, 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 $ R$-module homomorphism
$\displaystyle \phi: A\times B\rightarrow C,$
lifts to a unique $ R$-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 $ R$-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 $ R$ clear, the tensor product $ A\otimes B$ is sometimes written as $ A\otimes_R B$.

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

  1. $ R\otimes M\cong M$,
  2. $ M\otimes N\cong N\otimes M$,
  3. $ (L\otimes M) \otimes N\cong L\otimes (M\otimes N)$
  4. $ (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 $ R$-modules $ M$, the functor $ (-) \otimes M$ is left-adjoint to the functor $ \mathrm{Hom}(M,-)$.



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"tensor product" is owned by rmilson. [ full author list (3) ]
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See Also: module, outer multiplication


Attachments:
tensor product (vector spaces) (Definition) by rmilson
properties of tensor product (Derivation) by polarbear
tensor product of algebras (Definition) by CWoo
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Cross-references: functor, categories, language, categorical, relations, obvious, generated by, lifts, universal property, isomorphism, homomorphism, canonical, commutative ring, tensor, second fundamental forms, differential geometry, operation, origin, linear map, equivalent, objects, bilinear maps, vector spaces, modules, multiplication, bilinear
There are 17 references to this entry.

This is version 9 of tensor product, born on 2002-02-17, modified 2008-01-03.
Object id is 2043, canonical name is TensorProduct.
Accessed 13788 times total.

Classification:
AMS MSC: 13-00 (Commutative rings and algebras :: General reference works )
 18-00 (Category theory; homological algebra :: General reference works )
Pending Errata and Addenda
None.
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Tensor Product by SGL on 2006-12-19 03:41:52
I have a quick question about the tensor product of two R-modules A,B.
If, say, B is also an R'-module (R,R' commutative rings with identity) then an R'-module structure can be defined on A*B (Spanier, Algebraic Topology). My problem is it doesn't say how this is defined and i haven't seen this in any other book. If i define it in the most natural way, say if x belongs to A*B then x is a finite sum of elements of the form a(i)*b(i) then let xr'=a(i)*(b(i)r'); but i can't proove that this is well defined in general since the elements of the form a*b do not form a basis for A*B in general.
[ reply | up ]
A \otimes_R B by iwnbap on 2004-08-23 06:29:16
Perhaps the entry should mention the $\otimes_R$ syntax?

[ reply | up ]
Operators on tensor product spaces by dsiska on 2004-07-20 09:21:37
I have a question: what is the meaining of
$A \otimes A$ defined on $\widehat{L^2}(\mathbb{R}^2)$, which is just the space of symmetrix $L^2$ functions on $\mathbb{R}^2$ and $A$ is defined as: $A=-\frac{d^2}{dx^2} + x^2 + 1$, i.e. a densly defined linear operator?

Would it help that for Hermite functions $e_n$ we have
$A e_n = (2n_2)e_n$, that is Hermite functions are a basis of $L^2(\mathbb{R})?

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