Some comments on motivic nilpotence

TL;DR

This paper explores the properties of motivic maps and spectra, establishing a power operations theory, demonstrating the failure of a motivic analogue of a classical theorem, and relating certain spectra to higher Witt groups.

Contribution

It develops a theory of power operations for motivic $H_{}$-spectra and analyzes the non-nilpotent motivic maps $$ and $$, revealing new structural insights.

Findings

Naive motivic unstable Kahn-Priddy theorem fails.

Motivic $T$-spectrum $S[^{-1},^{-1}]$ relates to higher Witt groups.

Established a power operations framework for motivic $H_{}$-spectra.

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 $\pi_{**}(S)$.