Nilpotence in E_n Algebras

TL;DR

This paper explores nilpotence detection in the homotopy of _n-algebras, revealing limitations of classical criteria and constructing examples with non-nilpotent torsion at various chromatic heights.

Contribution

It demonstrates that classical nilpotence detection via the Hurewicz homomorphism does not always hold for _n-algebras and provides explicit counterexamples at different heights.

Findings

Constructed _{2n-1}-algebras with non-nilpotent p^n-torsion

Showed the bound is sharp at height 1 using Bousfield-Kuhn functor and Rezk's logarithm

Discussed the situation at height 2 and limitations of classical detection methods

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 $2$.