On \'etale motivic spectra and Voevodsky's convergence conjecture

TL;DR

This paper proves a new convergence result for the slice spectral sequence in motivic homotopy theory, verifying a variant of Voevodsky's conjecture and enabling a comprehensive understanding of the étale motivic stable category.

Contribution

It establishes a convergence result for the slice spectral sequence, verifies a derived form of Voevodsky's conjecture, and derives a Thomason-style étale descent for the motivic sphere spectrum.

Findings

Verified a derived convergence of the slice spectral sequence.

Established a Thomason-style étale descent for the motivic sphere spectrum.

Provided a complete description of the étale motivic stable category.

Abstract

We prove a new convergence result for the slice spectral sequence, following work by Levine and Voevodsky. This verifies a derived variant of Voevodsky's conjecture on convergence of the slice spectral sequence. This is, in turn, a necessary ingredient for our main theorem: a Thomason-style \'etale descent result for the Bott-inverted motivic sphere spectrum, which generalizes and extends previous \'etale descent results for special examples of motivic cohomology theories. Combined with first author's \'etale rigidity results, we obtain a complete structural description of the \'etale motivic stable category.