tensor product

Motivation

Tensor products are a way to systematically study bilinear maps of modules. They end up having a lot of uses that come from the fact that multilinear functions appear in a lot of places. Tensor products were first introduced to study the following question: Does there exist an -module such that

Thus, tensor products are a way to use what we know of homomorphisms of modules (which are linear maps) to study multi-linear maps.

Universal Property

A tensor product of and is an -module together with an -bilinear map which satisfies the following universal property:

For any -bilinear map , there exists a unique -linear map which makes the following diagram commute:

Uniqueness

If it exists, is unique up to unique isomorphism.

Proof

Say are two tensor products. By the universal property of , there exists a unique -module homomorphism such that the following diagram commutes:

Also,

uniquely. Applying again, the universal property shows that and are mutually inverse.

Therefore, we denote the unique tensor product as .

Existence (formal construction)

Commutative Case

We can construct an explicit module and bilinear map the satisfies the universal property above where is commutative.

Let be the free R-module on the set . Thus, an arbitrary element of takes the form

Let be the -submodule generated by the following elements:

3. .

The canonical map

is a tensor product

Proof

First, is bilinear:

by property 2 of . Also,

by property 4. The same idea follows using properties 1 and 3 in the first position.

Now, we just need to show it satisfies the universal property. Let be -bilinear. Define an -linear map

This determines uniquely based on the universal property of free modules. Note that

and

by bilinearity of . This proves that is in the kernel of . By the universal property of quotients, there exists a unique -linear map such that the following diagram commutes

By the universal properties, is unique.

Notation

We write for the class of in . Thus, the canonical map

The relations 1-4 in which descend to then read

Non-commutative case

Note that the multiplication relationship above gives an issue for a non-commutative ring. For example, consider if both

that would imply that but is not commutative.

To fix this, we say that for

• - right -module
• - left -module

Then we define

A priori, this is just an abelian group, if in addition is a -bimodule, then is a left -module with the action

This works and respects the multiplication relation 3 above because

Consistency of notation

This construction gives the symbols

in which the middle indices of “cancel out”.

Properties

Tensor products as functors

Let be a morphism of -modules. This induces a morphism of -modules as follows

is bilinear. Hence, by the universal property of the tensor product, there exists a unique -linear map such that the following diagram commutes:

Thus, by the universal property, induces the identity map

Therefore, for each -module , we get a functor

We can also define a second functor

where

Tensor-Hom adjunction

For -modules , there is a canonical -module isomorphism

Proof

Let , interpreted as an -bilinear map

Then . This defines a map

This is -linear.

Similarly, given , define an -linear map

This defines a map

which is inverse to .

Other properties

The tensor functor is right exact. This plays an important role in homological algebra.

Example (non-commutative case)

Let be a ring homomorphism, then is a -bimodule. The actions are defined as

If is a left -module, then

is a left -module.

This gives a functor

which is sometimes called the restriction on .