Tensor product: Difference between revisions
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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>. | ||
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==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== | ||
[[Category: | [[Category:Suggestion Bot Tag]] | ||
Latest revision as of 16:00, 25 October 2024
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 .
