Nilpotence of $\eta$ in étale motivic spectra

Abstract

We show that every object of the stable étale motivic homotopy category over any scheme is $\eta$-complete. In some cases we show that in fact the fourth power of $\eta$ is null, whereas the third power of $\eta$ is always nonvanishing, similar to the situation in topology. Moreover, we prove an étale version of May's nilpotence conjecture, that states that $H\mathbb{Z}\in \mathrm{Sp}$ detects the vanishing of ${\mathbf{E}}_{\infty }$-rings. We use this to show a version of Nishida's nilpotence theorem in ${\mathrm{SH}}_{\acute{\mathrm{e}}\mathrm{t}}(S)$, i.e. that any positive degree self map of the unit is nilpotent.

14 pages, comments welcome! v2: Added a section that HZ_et detects nilpotence, and that a version of Nishida's nilpotence theorem holds in SH_et