On the convergence of the orthogonal spectral sequence

TL;DR

This paper proves strong convergence of a specific orthogonal spectral sequence in Voevodsky's motivic category, provides examples of non-convergence, and establishes a criterion ensuring convergence for Voevodsky's slices.

Contribution

It demonstrates strong convergence of the orthogonal spectral sequence in the motivic setting and introduces a criterion for convergence in the context of Voevodsky's slices.

Findings

Strong convergence of the orthogonal spectral sequence in DM over a field k.

Examples where the spectral sequence is not strongly convergent.

A criterion ensuring convergence for Voevodsky's slices.

Abstract

We show that the orthogonal spectral sequence introduced by the second author is strongly convergent in Voevodsky's triangulated category of motives DM over a field k. In the context of the Morel-Voevodsky motivic stable homotopy category we provide concrete examples where the spectral sequence is not strongly convergent, and give a criterion under which the strong convergence still holds. This criterion holds for Voevodsky's slices, and as a consequence we obtain a spectral sequence which converges strongly to the E1-term of Voevodsky's slice spectral sequence.