Some comments on motivic nilpotence

Abstract

We discuss some results and conjectures related to the existence of the non-nilpotent motivic maps $\eta$ and $\mu_9$. To this purpose, we establish a theory of power operations for motivic $H_{\infty}$-spectra. Using this, we show that the naive motivic analogue of the unstable Kahn-Priddy theorem fails. Over the complex numbers, we show that the motivic $T$-spectrum $S[\eta^{-1},\mu_9^{-1}]$ is closely related to higher Witt groups, where $S$ is the motivic sphere spectrum and $\eta$, $\sigma$ and $\mu_9$ are explicit elements in $\pi_{**}(S)$.