  Let $X$ be a normal proper variety over a perfect field $k$. We describe
abelian coverings of X in terms of the functor $\underline{\rm HDiv}_X$ of
principal relative Cartier divisors on $X$. If the base field $k$ is finite,
the geometric Galois group of the maximal abelian extension of the function
field of $X$ is given by the $k$-valued points of the Cartier dual of the
completion of $\underline{\rm HDiv}_X$. As another application, we present the
geometric abelianized fundamental group of the affine $n$-space over a finite
field by the group of big Witt vectors in $n$ variables, a generalization of
the (usual) big Witt vectors.
