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

TL;DR

This paper proves that in the stable étale motivic homotopy category, all objects are η-complete, and establishes étale analogues of classical nilpotence theorems, revealing deep structural properties of motivic spectra.

Contribution

It demonstrates η-completeness of objects, establishes η-nilpotence results, and proves an étale version of May's nilpotence conjecture, extending classical topology results to motivic homotopy theory.

Findings

Every object is η-complete in the étale motivic homotopy category.

The fourth power of η is null in some cases, while the third power is always nonvanishing.

An étale version of May's nilpotence conjecture is proven, and a version of Nishida's nilpotence theorem is established.

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.