Nilpotence of $\eta$ in \'etale motivic spectra

Abstract

We show that every object of the stable \'etale 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 \'etale 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 $\mathrm{SH}_{\operatorname{\acute{e}t}}(S)$, i.e. that any positive degree self map of the unit is nilpotent.