Tensor product: Difference between revisions
Jump to navigation
Jump to search
imported>Giovanni Antonio DiMatteo (creating page) |
imported>Joe Quick m (subpages) |
||
| (4 intermediate revisions by one other user not shown) | |||
| Line 1: | Line 1: | ||
{{subpages}} | |||
The tensor product is a bifunctor in the category of modules over a fixed ring <math>R</math>. In the subcategory of algebras over <math>R</math>, the tensor product is just the cofibered product over <math>R</math>. | The tensor product is a bifunctor in the category of modules over a fixed ring <math>R</math>. In the subcategory of algebras over <math>R</math>, the tensor product is just the cofibered product over <math>R</math>. | ||
==Definition== | ==Definition== | ||
The ''tensor product'' of two <math>R</math>-modules <math>M</math> and <math>M'</math>, denoted by <math>M\ | The ''tensor product'' of two <math>R</math>-modules <math>M</math> and <math>M'</math>, denoted by <math>M\otimes_R M'</math>, is an <math>R</math>-module <math>T</math> satisfying the universal property | ||
==Functoriality== | ==Functoriality== | ||
The functor <math>-\otimes_R M</math> is right-exact from the category of (right) <math>R-modules</math> to the category of <math>R</math>-modules. | |||
The derived functors <math>Tor^n_R(-,-)</math>. | |||
==Tensor products in linear algebra== | ==Tensor products in linear algebra== | ||
Latest revision as of 15:46, 23 December 2007
The tensor product is a bifunctor in the category of modules over a fixed ring . In the subcategory of algebras over , the tensor product is just the cofibered product over .
Definition
The tensor product of two -modules and , denoted by , is an -module satisfying the universal property
Functoriality
The functor is right-exact from the category of (right) to the category of -modules.
The derived functors .
