Convergence of Voevodsky's slice tower

TL;DR

This paper proves the convergence of Voevodsky's slice tower for finite spectra over certain fields, confirming a key conjecture in motivic homotopy theory.

Contribution

It demonstrates the convergence of the slice tower under specific conditions, advancing understanding of the structure of motivic spectra.

Findings

Slice tower converges for spectra over fields with finite cohomological dimension.

Filtration on homotopy sheaves is finite, exhaustive, and separated.

Partially verifies Voevodsky's conjecture on slice tower convergence.

Abstract

We consider Voevodsky's slice tower for a finite spectrum E in the motivic stable homotopy category over a perfect field k. In case k has finite cohomological dimension (in characteristic two, we also require that k is infinite), we show that the slice tower converges, in that the induced filtration on the bi-graded homotopy sheaves for each term in the tower for E is finite, exhaustive and separated at each stalk. This partially verifies a conjecture of Voevodsky.