Cohomology and Extensions
With duals for objects
With duals for morphisms
Special sorts of products
In higher category theory
For and two abelian groups, their tensor product is a new abelian group which is such that a group homomorphism is equivalently a bilinear map out of and .
Equivalently this means explicitly:
For two abelian groups, their tensor product of abelian groups is the abelian group which is the quotient of the free group on the product (direct sum) by the relations
for all and .
In words: it is the group whose elements are presented by pairs of elements in and and such that the group operation for one argument fixed is that of the other group in the other argument.
The following relates the tensor product to bilinear functions. It is a definition or a proposition dependening on whether one takes the notion of bilinear function to be defined before or after that of tensor product of abelian groups.
A function of underlying sets is a bilinear function precisely if it factors by the morphism of 2 through a group homomorphism out of the tensor product:
Monoidal category structure
To see that is the unit object, consider for any abelian group the map
which sends for
Due to the quotient relation defining the tensor product, the element on the left is also equal to
This shows that is in fact an isomorphism.
The tensor product of abelian groups distributes over the direct sum of abelian groups
A monoid in is equivalently a ring.
Let be a monoid in . The fact that the multiplication
is bilinear means by the above that for all we have
This is precisely the distributivity law of the ring.
For positive we write for the cyclic group of order , as usual.
For and positive, we have
where denotes the greatest common divisor.
A proof is spelled out for instance as (Conrad, theorem 4.1).
An exposition is in
- Collin Roberts, Introduction to the tensor product (pdf)
and, in the further generality of the tensor product of modules, in
- Keith Conrad, Tensor products (pdf)