free abelian group



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.



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 \,.

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.


In terms of formal linear combinations


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.


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 \,.

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


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 \,.


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