group theory

Contents

Idea

The free abelian group $\mathbb{Z}[S]$ on a set $S$ is the abelian group whose elements are formal $\mathbb{Z}$-linear combinations of elements of $S$.

Definition

Let

$U : Ab \to Set$

be the forgetful functor from the category Ab of abelian groups, to the category Set of sets. This has a left adjoint free construction:

$\mathbb{Z}[-] : Set \to Ab \,.$
Definition

This is the free abelian group functor. For $S \in$ Set, the free abelian group $\mathbb{Z}[S] \in$ Ab is the free object on $S$ with respect to this free/forgetful adjunction.

Explicit descriptions of free abelian groups are discussed below.

Properties

In terms of formal linear combinations

Definition

A formal linear combination of elements of a set $S$ is a function

$a : S \to \mathbb{Z}$

such that only finitely many of the values $a_s \in \mathbb{Z}$ are non-zero.

Identifying an element $s \in S$ with the function $S \to \mathbb{Z}$ which sends $s$ to $1 \in \mathbb{Z}$ and all other elements to 0, this is written as

$a = \sum_{s \in S} a_s \cdot s \,.$

In this expression one calls $a_s \in \mathbb{Z}$ the coefficient of $s$ in the formal linear combination.

Definition

For $S \in$ Set, the group of formal linear combinations $\mathbb{Z}[S]$ is the group whose underlying set is that of formal linear combinations, def. 2, and whose group operation is the pointwise addition in $\mathbb{Z}$:

$(\sum_{s \in S} a_s \cdot s) + (\sum_{s \in S} b_s \cdot s) = \sum_{s \in S} (a_s + b_s) \cdot s \,.$
Proposition

The free abelian group on $S \in Set$ is, up to isomorphism, the group of formal linear combinations, def. 3, on $S$.

In terms of direct sums

Proposition

For $S$ a set, the free abelian group $\mathbb{Z}[S]$ is the direct sum in Ab of ${|S|}$-copies of $\mathbb{Z}$ with itself:

$\mathbb{Z}[S] \simeq \oplus_{s \in S} \mathbb{Z} \,.$

Relation to formal linear combinations with coefficients

The definition 2 of formal linear combinations makes sense with coefficients in any abelian group $A$, not necessarily the integers.

$A[S] \coloneqq \mathbb{Z}[S] \otimes A \,.$

Examples

• The free abelian group on the singular simplicial complex of a topological space $X$ consists of the singular chains on $X$.

• For $R$ a ring and $S$ a set, the tensor product of abelian groups $\mathbb{Z}[S] \otimes R$ is the free module over $R$ on the basis $S$. If $R = k$ is a field, then this is the vector space over $k$ with basis $S$.

• For $R$ a ring, the tensor product of abelian groups $\mathbb{Z}[\mathbb{N}]\otimes R$ is the abelian group underlying the ring of polynomials over $R$.

Revised on September 6, 2012 14:49:41 by Urs Schreiber (131.174.188.45)