A Serre-type spectral sequence for motivic cohomology

TL;DR

This paper develops a Serre-type spectral sequence for motivic cohomology, enabling new computations of motives related to algebraic groups and Severi-Brauer varieties.

Contribution

It introduces a novel spectral sequence for motivic cohomology associated with bisimplicial schemes, facilitating explicit motive calculations.

Findings

Spectral sequence for motivic cohomology constructed.

Computed motivic cohomology of Nisnevich classifying spaces.

Explicit description of motives of Severi-Brauer varieties.

Abstract

In this paper, we construct and study a Serre-type spectral sequence for motivic cohomology associated to a map of bisimplicial schemes with motivically cellular fiber. Then, we show how to apply it in order to approach the computation of the motivic cohomology of the Nisnevich classifying space of projective general linear groups. This naturally yields an explicit description of the motive of a Severi-Brauer variety in terms of twisted motives of its \v{C}ech simplicial scheme.