Convergence of Voevodsky's slice tower

Abstract

We consider Voevodsky's slice tower for a finite spectrum $\mathcal{E}$ in the motivic stable homotopy category over a perfect field $k$. In case $k$ has finite cohomological dimension, we show that the slice tower converges, in that the induced filtration on the bi-graded homotopy sheaves ${\Pi }_{a,b}{f}_n\mathcal{E}$ is finite, exhaustive and separated at each stalk (after inverting the exponential characteristic of $k$). This partially verifies a conjecture of Voevodsky.