  We prove that every flat chain with finite mass in $\mathbb{R}^d$ with
coefficients in a normed abelian group $G$ is the restriction of a normal
$G$-current to a Borel set. We deduce a characterization of real flat chains
with finite mass in terms of a pointwise relation between the associated
measure and vector field. We also deduce that any codimension-one real flat
chain with finite mass can be written as an integral of multiplicity-one
rectifiable currents, without loss of mass.
  Given a Lipschitz homomorphism $\phi:\tilde G\to G$ between two groups, we
then study the associated map $\pi$ between flat chains in $\mathbb{R}^d$ with
coefficients in $\tilde G$ and $G$ respectively. In the case $\tilde
G=\mathbb{R}$ and $G=\mathbb{S}^1$, we prove that if $\phi$ is surjective, so
is the restriction of $\pi$ to the set of flat chains with finite mass of
dimension $0$, $1$, $d-1$, $d$.
