On the motive of a commutative algebraic group

TL;DR

This paper establishes a canonical Kunneth decomposition for the motive of a commutative algebraic group, demonstrating its compatibility with the group law and exploring applications to weight filtrations, Weil cohomology, and 1-motives.

Contribution

It introduces a canonical Kunneth decomposition for motives of commutative group schemes and analyzes its behavior under group law, with applications to various cohomological theories.

Findings

Decomposition behaves under group law like in cohomology

Applications to weight filtration and Weil cohomology

Insights into 1-motives

Abstract

We prove a canonical Kunneth decomposition for the motive of a commutative group scheme over a field. Moreover, we show that this decomposition behaves under the group law just as in cohomology. We also deduce applications of the decomposition to the existence of a weight filtration, computation of any Weil cohomology theory and study of 1-motives.