nLab
free abelian group

Contents

Idea

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

Definition

Let

U:AbSet 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:

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

This is the free abelian group functor. For SS \in Set, the free abelian group [S]\mathbb{Z}[S] \in Ab is the free object on SS 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 SS is a function

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

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

Identifying an element sSs \in S with the function SS \to \mathbb{Z} which sends ss to 11 \in \mathbb{Z} and all other elements to 0, this is written as

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

In this expression one calls a sa_s \in \mathbb{Z} the coefficient of ss in the formal linear combination.

Definition

For SS \in Set, the group of formal linear combinations [S]\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}:

( sSa ss)+( sSb ss)= sS(a s+b s)s. (\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 SSetS \in Set is, up to isomorphism, the group of formal linear combinations, def. 3, on SS.

In terms of direct sums

Proposition

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

[S] sS. \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 AA, not necessarily the integers.

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

Examples

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