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 |