Some comments on motivic nilpotence

Abstract

We discuss some results and conjectures related to the existence of the non-nilpotent motivic maps $η$ and $μ_9$. 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 $S[η^{-1},μ_9^{-1}]$ is closely related to higher Witt groups, where $S$ is the motivic sphere spectrum and $η$, $σ$ and $μ_9$ are explicit elements in $π_{**}(S)$.