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 be a commutative ring, and let be -modules. There exists an -module , called the
tensor product of and over , together with a canonical
bilinear homomorphism
distinguished, up to isomorphism, by the following universal
property
.
Every bilinear -module homomorphism
lifts to a unique -module homomorphism
such that
for all Diagramatically:
The tensor product can be constructed by taking the free -module generated by all formal symbols
and quotienting by the obvious bilinear relations:
Note.
Basic . Let be a commutative ring and be -modules, then, as modules, we have the following isomorphisms:
-
1.
,
-
2.
,
-
3.
-
4.
Definition (Categorical). Using the language of categories
, all
of the above can be expressed quite simply by stating that for all
-modules , the functor
is left-adjoint to the
functor .
Title | tensor product |
---|---|
Canonical name | TensorProduct |
Date of creation | 2013-03-22 12:21:26 |
Last modified on | 2013-03-22 12:21:26 |
Owner | rmilson (146) |
Last modified by | rmilson (146) |
Numerical id | 12 |
Author | rmilson (146) |
Entry type | Definition |
Classification | msc 13-00 |
Classification | msc 18-00 |
Related topic | Module |
Related topic | OuterMultiplication |