On a nilpotence conjecture of J.P. May

We prove a conjecture of J.P. May concerning the nilpotence of elements in ring spectra with power operations, i.e., $H_{\mathrm{\infty }}$-ring spectra. Using an explicit nilpotence bound on the torsion elements in $K(n)$-local $H_{\mathrm{\infty }}$-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 $M{Spin}_{\ast }$ and $M{String}_{\ast }$, results about the behavior of the Adams spectral sequence for $E_{\mathrm{\infty }}$-ring spectra, and the non-existence of $E_{\mathrm{\infty }}$-ring structures on certain complex oriented ring spectra.