Nilpotence phenomena over ${\mathbb{E}}_{\infty }$-rings in positive characteristic

Abstract

We study the behavior of nilpotent classes in ${\mathbb{E}}_{\infty }$-rings over finite fields. In characteristic $p=2$, we show that any nilpotent class can be killed by a nilpotence detecting map of ${\mathbb{E}}_{\infty }$-rings. As a consequence, we prove the existence of ``residue fields'', namely that any non-zero ${\mathbb{E}}_{\infty }$-ring over ${\mathbb{F}}_2$ admits a map to a separably closed graded field. Conversely, for any odd prime $p$, we construct non-zero ${\mathbb{E}}_{\infty }$-rings over ${\mathbb{F}}_p$ which admit nilpotent classes that cannot be killed by any non-zero map of ${\mathbb{E}}_{\infty }$-rings, a phenomenon that does not occur classically or at any finite height.

arXiv comment28 pages, comments welcome