On a nilpotence conjecture of J.P. May

Abstract

We prove a conjecture of J.P. May concerning the nilpotence of elements in ring spectra with power operations, i.e., $H_\infty$-ring spectra. Using an explicit nilpotence bound on the torsion elements in $K(n)$-local $H_\infty$-algebras over $E_n$, we reduce the conjecture to the nilpotence theorem of Devinatz, Hopkins, and Smith. As corollaries we obtain nilpotence results in various bordism rings including $M\mathit{Spin}_*$ and $M\mathit{String}_*$, results about the behavior of the Adams spectral sequence for $E_\infty$-ring spectra, and the non-existence of $E_\infty$-ring structures on certain complex oriented ring spectra.