Nilpotence in E_n Algebras

Abstract

Nilpotence in the homotopy of ${\mathbb{E}}_{\infty }$-ring spectra is detected by the classical $H\mathbb{Z}$-Hurewicz homomorphism. Inspired by questions of Mathew, Noel, and Naumann, we investigate the extent to which this criterion holds in the homotopy of ${\mathbb{E}}_n$-ring spectra. 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 $2$.

Comments welcome

References in corpus (1)

• Power operations for Morava E-theory of height 2 at the prime 2