The free abelian group on a set is the abelian group whose elements are formal -linear combinations of elements of .
Let
be the forgetful functor from the category Ab of abelian groups, to the category Set of sets. This has a left adjoint free construction:
This is the free abelian group functor. For Set, the free abelian group Ab is the free object on 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 is representable by an abelian group , then we may say is a free abelian group on . A specific choice of isomorphism
corresponds, via the Yoneda lemma, to a function which exhibits , or rather its image under this function, as a specific basis of . If is so equipped with such a universal arrow , then it is harmless to call “the” free abelian group on .
Explicit descriptions of free abelian groups are discussed below.
A formal linear combination of elements of a set is a function
such that only finitely many of the values are non-zero.
Identifying an element with the function which sends to and all other elements to 0, this is written as
In this expression one calls the coefficient of in the formal linear combination.
Definition of formal linear combinations makes sense with coefficients in any abelian group , not necessarily the integers.
For Set, the group of formal linear combinations is the group whose underlying set is that of formal linear combinations, def. , and whose group operation is the pointwise addition in :
The free abelian group on is, up to isomorphism, the group of formal linear combinations, def. , of elements of .
For a set, the free abelian group is the direct sum in Ab of -copies of with itself:
The free abelian group of a Cartesian product of sets is naturally isomorphic to the tensor product of the free abelian groups of the factors:
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.
The free abelian group on the singular simplicial complex of a topological space consists of the singular chains on .
For a ring and a set, the tensor product of abelian groups is the free module over on the basis . If is a field, then this is the vector space over with basis .
For a ring, the tensor product of abelian groups is the abelian group underlying the ring of polynomials over .
Textbook accounts:
Last revised on May 30, 2022 at 12:37:06. See the history of this page for a list of all contributions to it.