free abelian group
Cohomology and Extensions
The free abelian group on a set is the abelian group whose elements are formal -linear combinations of elements of .
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.
Explicit descriptions of free abelian groups are discussed below.
In terms of formal linear combinations
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.
For Set, the group of formal linear combinations is the group whose underlying set is that of formal linear combinations, def. 2, 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. 3, on .
In terms of direct sums
For a set, the free abelian group is the direct sum in Ab of -copies of with itself:
Relation to formal linear combinations with coefficients
The definition 2 of formal linear combinations makes sense with coefficients in any abelian group , not necessarily the integers.
Revised on September 6, 2012 14:49:41
by Urs Schreiber