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] \,.

In other words, as a functor from Set to Ab

[]:SetAb \mathbb{Z}[-] \;\colon\; Set \longrightarrow Ab

this is a strong monoidal functor.

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


In homotopy theory/algebraic topology, the free abelian group construction is frequently applied degreewise to a simplicial set S S_\bullet which is (or is weakly equivalent to) the singular simplicial complex of a topological space — because the resulting chain complex (under the Dold-Kan correspondence) then computes the ordinary singular homology of that space. But since

  1. the product of simplicial sets is degreewise that in Set,

  2. the tensor product of simplicial abelian groups is degreewise the tensor product of abelian groups

the above Prop. implies that also the functor

[]:sSetsAb \mathbb{Z}[-] \;\colon\; sSet \longrightarrow sAb

from sSet to sAb is strong monoidal.


\bgein{proposition} \label{SubgroupsOfFreeAbelianGroupsAreFree} Assuming the axiom of choice, then every subgroup of a free abelian group (def. ) is itself a free abelian group. \end{proposition} (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 April 26, 2023 at 10:44:10. See the history of this page for a list of all contributions to it.