# 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) (http://planetmath.org/TensorProductClassical).

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

lifts to a unique $R$-module homomorphism

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

such that

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

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

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

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

 $a\otimes b,\quad a\in A,\;b\in B,$
 $\displaystyle(a_{1}+a_{2})\otimes b$ $\displaystyle=a_{1}\otimes b+a_{2}\otimes b,$ $\displaystyle a_{1},a_{2}\in A,\;b\in B$ $\displaystyle a\otimes(b_{1}+b_{2})$ $\displaystyle=a\otimes b_{1}+a\otimes b_{2},$ $\displaystyle a\in A,\;b_{1},b_{2}\in B$ $\displaystyle r(a\otimes b)$ $\displaystyle=(ra)\otimes b=a\otimes(rb)$ $\displaystyle a\in A,\;b\in B,\;r\in R$

Note.

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

1. 1.

$R\otimes M\cong M$,

2. 2.

$M\otimes N\cong N\otimes M$,

3. 3.

$(L\otimes M)\otimes N\cong L\otimes(M\otimes N)$

4. 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 $\mathrm{Hom}(M,-)$.

Title tensor product TensorProduct 2013-03-22 12:21:26 2013-03-22 12:21:26 rmilson (146) rmilson (146) 12 rmilson (146) Definition msc 13-00 msc 18-00 Module OuterMultiplication