In Hatcher's Algebraic Topology book on page 193 he defines a free resolution of an abelian group $H$ to be an exact sequence of the from
$... \rightarrow F_2 \rightarrow F_1 \rightarrow F_0 \rightarrow H \rightarrow 0$
where each $F_n$ is a free.
Later, in the second paragraph of page 195 he allows $F_0$ to be a free abelian group. My problem here is that free abelian groups are not necessarily free groups.
What is going on here?
Hatcher means free as a $\mathbf Z$-module, which is to say that there is a basis, much in the sense of linear algebra.
This is common, for example in the discussion of homology one would say the abelian group freely generated by simplices etc.