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tensor product
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(Definition)
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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).
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 $$\otimes: A\times B\rightarrow A\otimes B,$$ distinguished, up to isomorphism, by the following
universal property. Every bilinear $R$ -module homomorphism $$\phi: A\times B\rightarrow C,$$ 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:
The tensor product $A\otimes B$ can be constructed by taking the free $R$ -module generated by all formal symbols $$a\otimes b,\quad a\in A,\;b\in B,$$ and quotienting by the obvious bilinear relations:
Note. In order to make the base ring $R$ clear, the tensor product $A\otimes B$ is sometimes written as $A\otimes_R B$ .
Basic properties. Let $R$ be a commutative ring and $L,M,N$ be $R$ -modules, then, as modules, we have the following isomorphisms:
- $R\otimes M\cong M$ ,
- $M\otimes N\cong N\otimes M$ ,
- $(L\otimes M) \otimes N\cong L\otimes (M\otimes N)$
- $(L\oplus M)\otimes N \cong (L\otimes N) \oplus (M\otimes N)$
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 $\mathrm{Hom}(M,-)$ .
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"tensor product" is owned by rmilson. [ full author list (3) ]
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Cross-references: functor, categories, language, categorical, relations, obvious, generated by, lifts, universal property, isomorphism, homomorphism, canonical, commutative ring, tensor, second fundamental forms, differential geometry, operation, origin, linear map, equivalent, objects, bilinear maps, vector spaces, modules, multiplication, bilinear
There are 11 references to this entry.
This is version 9 of tensor product, born on 2002-02-17, modified 2008-01-03.
Object id is 2043, canonical name is TensorProduct.
Accessed 17424 times total.
Classification:
| AMS MSC: | 13-00 (Commutative rings and algebras :: General reference works ) | | | 18-00 (Category theory; homological algebra :: General reference works ) |
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Pending Errata and Addenda
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