Variations on a Nilpotence Theorem of Hopkins and Mahowald

Abstract

It is a theorem of Hopkins and Mahowald that the nilpotence of $p$-torsion classes in $\mathbb{E}2$-ring spectra can be detected by the classical $H\mathbb{F}p$-Hurewicz homomorphism. On the other hand, the May Nilpotence Conjecture, recently proven by Mathew, Naumann, and Noel, shows that the nilpotence of a not-necessarily $p$-torsion class in an $\mathbb{E}\infty$-ring may be detected by the $H\mathbb{Z}$-Hurewicz homomorphism. Inspired by questions of Mathew, Naumann, and Noel, we investigate nilpotence in the homotopy of $\mathbb{E}n$-ring spectra with $2<n<\infty$. For all odd primes $p$ and all chromatic heights $h$, we use the Cohen-Moore-Neisendorfer theorem to construct examples of $K(h)$-local, $\mathbb{E}{2n-1}$-algebras with non-nilpotent $p^n$-torsion. We exploit the interaction of the Bousfield-Kuhn functor on odd spheres and Rezk's logarithm to show that our bound is sharp at height $1$, and remark on the situation at height $2$. We then turn our attention to the case $n=\infty$, where we refine the May Nilpotence Conjecture to the stronger statement that any $K(h)$-acyclic $\mathbb{E}\infty$-ring must also be $K(h+1)$-acyclic. This affirms a conjecture of Mark Hovey, and may be viewed as a restriction on the sorts of mixed characteristics that can appear in $\mathbb{E}_\infty$-ring spectra. Finally, we discuss an ongoing project, joint with Dylan Wilson, to extend the original Hopkins--Mahowald theorem to an equivariant setting.