nLab
tensor product of abelian groups

Idea

For $A$ and $B$ two abelian groups, their tensor product $A\otimes B$ is a new abelian group which is such that a group homomorphism $A\otimes B\to C$ is equivalently a bilinear map out of $A$ and $B$.

Definition

Definition

Let Ab be the collection of abelian groups, regarded as a multicategory whose multimorphisms are the multilinear maps $A_1,\cdots ,A_n\to B$.

The tensor product $A,B\mapsto A\otimes B$ in this multicategory is the tensor product of abelian groups.

Equivalently this means explicitly:

Definition

For $A,B$ two abelian groups, their tensor product of abelian groups is the abelian group $A\otimes B$ which is the quotient of the free group on the product (direct sum) $A\times B$ by the relations

• $(a_1,b)+(a_2,b)\sim (a_1+a_2,b)$

• $(a,b_1)+(a,b_2)\sim (a,b_1+b_2)$

for all $a,a_1,a_2\in A$ and $b,b_1,b_2\in B$.

In words: it is the group whose elements are presented by pairs of elements in $A$ and $B$ and such that the group operation for one argument fixed is that of the other group in the other argument.

Remark

The 0-ary relations $(0,b)\sim 0$ and $(a,0)\sim 0$ follow automatically; one needs them explicitly only if one generalises to abelian monoids.

Remark

By definition of the free construction and the quotient there is a canonical function of the underlying sets

$$A\times B\stackrel{\otimes }{\to }A\otimes B.$$A \times B \stackrel{\otimes}{\to} A \otimes B \,.

On elements this sends $(a,b)$ to the equivalence class that it represents under the above equivalence relations.

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.

Definition/Proposition

A function of underlying sets $f:A\times B\to C$ is a bilinear function precisely if it factors by the morphism of 2 through a group homomorphism $\varphi :A\otimes B\to C$ out of the tensor product:

$$f:A\times B\stackrel{\otimes }{\to }A\otimes B\stackrel{\varphi }{\to }C.$$f : A \times B \stackrel{\otimes}{\to} A \otimes B \stackrel{\phi}{\to} C \,.

Properties

Monoidal category structure

Proposition

Equipped with the tensor product $\otimes$ of def. 2 Ab becomes a monoidal category.

The unit object in $(\mathrm{Ab},\otimes )$ is the additive group of integers $\mathbb{Z}$.

Proof

To see that $\mathbb{Z}$ is the unit object, consider for any abelian group $A$ the map

$$A\otimes \mathbb{Z}\to A$$A \otimes \mathbb{Z} \to A

which sends for $n\in \mathbb{N}\subset \mathbb{Z}$

$$(a,n)\mapsto n\cdot a≔{\underbrace{a+a+\cdots +a}}_{n\mathrm{summands}}.$$(a, n) \mapsto n \cdot a \coloneqq \underbrace{a + a + \cdots + a}_{n\;summands} \,.

Due to the quotient relation defining the tensor product, the element on the left is also equal to

$$(a,n)=(a,{\underbrace{1+1\cdots +1}}_{n\mathrm{summands}})={\underbrace{(a,1)+(a,1)+\cdots +(a,1)}}_{n\mathrm{summands}}.$$(a, n) = (a, \underbrace{1 + 1 \cdots + 1}_{n\; summands}) = \underbrace{ (a,1) + (a,1) + \cdots + (a,1) }_{n\; summands} \,.

This shows that $A\otimes \mathbb{Z}\to A$ is in fact an isomorphism.

Proposition

The tensor product of abelian groups distributes over the direct sum of abelian groups

$$A\otimes {\oplus }_{s\in S}{B}_s\simeq {\oplus }_{s\in S}(A\otimes B_s).$$A \otimes \oplus_{s \in S} B_s \simeq \oplus_{s \in S} ( A \otimes B_s ) \,.

Monoids

Proposition

A monoid in $(\mathrm{Ab},\otimes )$ is equivalently a ring.

Proof

Let $(A,\cdot )$ be a monoid in $(\mathrm{Ab},\otimes )$. The fact that the multiplication

$$\cdot :A\otimes A\to A$$\cdot : A \otimes A \to A

is bilinear means by the above that for all $a_1,a_2,b\in A$ we have

$$(a_1+a_2)\cdot b=a_1\cdot b+a_2\cdot b$$(a_1 + a_2) \cdot b = a_1 \cdot b + a_2 \cdot b

and

$$b\cdot (a_1+a_2)=b\cdot a_1+b\cdot a_2.$$b \cdot (a_1 + a_2) = b \cdot a_1 + b \cdot a_2 \,.

This is precisely the distributivity law of the ring.

Examples

For $n\in \mathbb{N}$ positive we write ${\mathbb{Z}}_n$ for the cyclic group of order $n$, as usual.

Example

For $a,b\in \mathbb{N}$ and positive, we have

$${\mathbb{Z}}_a\otimes {\mathbb{Z}}_b\simeq {\mathbb{Z}}_{\mathrm{LCM}(a,b)},$$\mathbb{Z}_a \otimes \mathbb{Z}_b \simeq \mathbb{Z}_{LCM(a,b)} \,,

where $\mathrm{LCM}(-,-)$ denotes the least common multiple.

A proof is spelled out for instance as (Conrad, theorem 4.1).

Related concepts

• tensor product of chain complexes

• tensor product of modules

References

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)