nLab 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 \colon Ab \longrightarrow 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}[-] \colon Set \longrightarrow 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.

Of course, this notion is meant to be invariant under isomorphism: it doesn’t depend on the left adjoint chosen. Thus, if a functor of the form hom Set(S,U):AbSet\hom_{Set}(S, U-): Ab \to Set is representable by an abelian group AA, then we may say AA is a free abelian group on SS. A specific choice of isomorphism

hom Ab(A,)hom Set(S,U)\hom_{Ab}(A, -) \cong \hom_{Set}(S, U-)

corresponds, via the Yoneda lemma, to a function SUAS \to U A which exhibits SS, or rather its image under this function, as a specific basis of AA. If AA is so equipped with such a universal arrow SUAS \to U A, then it is harmless to call AA “the” free abelian group on SS.

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.


Definition 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 \,.

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. , 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. , of elements of SS.


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} \,.

Basic properties


The free abelian group of a Cartesian product S×ZS \times Z of sets S,TSetsS, T \,\in\, Sets is naturally isomorphic to the tensor product of the free abelian groups of the factors:

[S×T][S][T]. \mathbb{Z}[S \times T] \;\simeq\; \mathbb{Z}[S] \otimes \mathbb{Z}[T] \,.

This follows, for instance, from the above expression (Prop. ) of free abelian groups as groups of formal linear combinations.



Assuming the axiom of choice, then every subgroup of a free abelian group (def. ) is itself a free abelian group.

(e.g. Lang 02, Appendix 2 §2, page 880) For a full proof see at principal ideal domain this theorem.


Prop. implies that (assuming AC) every abelian group admits a free resolution of length 2, hence with trivial syzygies. See there.



Textbook accounts:

  • Serge Lang, Algebra, Graduate Texts in Mathematics 211 (Revised third ed.), Springer. 2002

Last revised on May 30, 2022 at 12:37:06. See the history of this page for a list of all contributions to it.