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

Ravenel's algebraic extensions of the sphere spectrum do not exist