  A cycle is algebraically trivial if it can be exhibited as the difference of
two fibers in a family of cycles parameterized by a smooth scheme. Over an
algebraically closed field, it is a result of Weil that it suffices to consider
families of cycles parameterized by curves, or by abelian varieties. In this
paper, we extend these results to arbitrary base fields. The strengthening of
these results turns out to be a key step in our work elsewhere extending
Murre's results on algebraic representatives for varieties over algebraically
closed fields to arbitrary perfect fields.
