nLab tensor product of abelian groups

group theory

Cohomology and Extensions

Monoidal categories

monoidal categories

With traces

• trace

• traced monoidal category?

Contents

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 $A,B↦A\otimes B$ 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) $A×B$ by the relations

• $\left({a}_{1},b\right)+\left({a}_{2},b\right)\sim \left({a}_{1}+{a}_{2},b\right)$

• $\left(a,{b}_{1}\right)+\left(a,{b}_{2}\right)\sim \left(a,{b}_{1}+{b}_{2}\right)$

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 $\left(0,b\right)\sim 0$ and $\left(a,0\right)\sim 0$ 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

$A×B\stackrel{\otimes }{\to }A\otimes B\phantom{\rule{thinmathspace}{0ex}}.$A \times B \stackrel{\otimes}{\to} A \otimes B \,.

On elements this sends $\left(a,b\right)$ 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 $f:A×B\to C$ 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:

$f:A×B\stackrel{\otimes }{\to }A\otimes B\stackrel{\varphi }{\to }C\phantom{\rule{thinmathspace}{0ex}}.$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 $\left(\mathrm{Ab},\otimes \right)$ is the additive group of integers $ℤ$.

Proof

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

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

which sends for $n\in ℕ\subset ℤ$

$\left(a,n\right)↦n\cdot a≔{\underset{⏟}{a+a+\cdots +a}}_{n\phantom{\rule{thickmathspace}{0ex}}\mathrm{summands}}\phantom{\rule{thinmathspace}{0ex}}.$(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

$\left(a,n\right)=\left(a,{\underset{⏟}{1+1\cdots +1}}_{n\phantom{\rule{thickmathspace}{0ex}}\mathrm{summands}}\right)={\underset{⏟}{\left(a,1\right)+\left(a,1\right)+\cdots +\left(a,1\right)}}_{n\phantom{\rule{thickmathspace}{0ex}}\mathrm{summands}}\phantom{\rule{thinmathspace}{0ex}}.$(a, n) = (a, \underbrace{1 + 1 \cdots + 1}_{n\; summands}) = \underbrace{ (a,1) + (a,1) + \cdots + (a,1) }_{n\; summands} \,.

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

Proposition

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

$A\otimes {\oplus }_{s\in S}{B}_{s}\simeq {\oplus }_{s\in S}\left(A\otimes {B}_{s}\right)\phantom{\rule{thinmathspace}{0ex}}.$A \otimes \oplus_{s \in S} B_s \simeq \oplus_{s \in S} ( A \otimes B_s ) \,.

Monoids

Proposition

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

Proof

Let $\left(A,\cdot \right)$ be a monoid in $\left(\mathrm{Ab},\otimes \right)$. 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

$\left({a}_{1}+{a}_{2}\right)\cdot b={a}_{1}\cdot b+{a}_{2}\cdot b$(a_1 + a_2) \cdot b = a_1 \cdot b + a_2 \cdot b

and

$b\cdot \left({a}_{1}+{a}_{2}\right)=b\cdot {a}_{1}+b\cdot {a}_{2}\phantom{\rule{thinmathspace}{0ex}}.$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 $n\in ℕ$ positive we write ${ℤ}_{n}$ for the cyclic group of order $n$, as usual.

Example

For $a,b\in ℕ$ and positive, we have

${ℤ}_{a}\otimes {ℤ}_{b}\simeq {ℤ}_{\left(a,b\right)}\phantom{\rule{thinmathspace}{0ex}},$\mathbb{Z}_a \otimes \mathbb{Z}_b \simeq \mathbb{Z}_{(a,b)} \,,

where $\left(-,-\right)$ denotes the greatest common divisor?.

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

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)

Revised on September 22, 2013 01:45:26 by Anonymous Coward (75.92.72.80)