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

TL;DR

This paper proves that certain algebraic structures cannot be realized as spectra, resolving Ravenel's 1983 problem by showing nonexistence of nontrivial algebraic extensions of the sphere spectrum.

Contribution

It establishes a topological nonrealizability theorem for specific $BP_*$-modules, demonstrating the nonexistence of algebraic extensions of the sphere spectrum beyond trivial cases.

Findings

Certain graded $BP_*$-modules cannot be realized as $BP$-homology of spectra.

Algebraic extensions of the sphere spectrum do not exist except trivially.

The result applies even when modules admit $BP_*BP$-comodule structures.

Abstract

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.