Homology, Homotopy and Applications

Volume 24 (2022)

Number 2

On the convergence of the orthogonal spectral sequence

Pages: 389 – 401

DOI: https://dx.doi.org/10.4310/HHA.2022.v24.n2.a20

Authors

Cesar Galindo (Instituto de Matemáticas, Ciudad Universitaria, Universidad Nacional Autónoma de México (UNAM), Ciudad México, Mexico)

Pablo Pelaez (Instituto de Matemáticas, Ciudad Universitaria, Universidad Nacional Autónoma de México (UNAM), Ciudad México, Mexico)

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 $\mathbb{A}^1$‑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 $E_1$‑term of Voevodsky’s slice spectral sequence.

Received 5 November 2021

Received revised 18 January 2022

Accepted 19 January 2022

Published 14 September 2022