  In this paper we prove a topological nonrealizability theorem: certain
classes of graded $BP_*$-modules are shown to never occur as the $BP$-homology
of a spectrum. Many of these $BP_*$-modules admit the structure of
$BP_*BP$-comodules, meaning that their topological nonrealizability does not
follow from earlier results like Landweber's filtration theorem. As a
consequence we solve Ravenel's 1983 problem on the existence of "algebraic
extensions of the sphere spectrum": algebraic extensions of the sphere spectrum
do not exist, except in trivial cases.
