  We investigate the Tukey order in the class of $F_\sigma$ ideals of subsets
of $\omega$. We show that no nontrivial $F_\sigma$ ideal is Tukey below a
$G_\delta$ ideal of compact sets. We introduce the notions of flat ideals and
gradually flat ideals. We prove a dichotomy theorem for flat ideals isolating
gradual flatness as the side of the dichotomy that is structurally good. We
give diverse characterizations of gradual flatness among flat ideals using
Tukey reductions and games. For example, we show that gradually flat ideals are
precisely those flat ideals that are Tukey below the ideal of density zero
sets.
