Description of mixed motives

TL;DR

This paper proposes a framework for explicitly describing mixed motives assuming a standard conjecture, explores their properties, and connects them to schemes and cohomology theories, including $ ext{l}$-adic realizations.

Contribution

It introduces a new approach to describe mixed motives explicitly under a conjecture and constructs 2-motives unconditionally, advancing the understanding of their properties.

Findings

Provides an explicit description of mixed motives assuming the Künneth conjecture.

Associates mixed motives to schemes over a field, serving as a universal cohomology theory.

Constructs 2-motives unconditionally and discusses their properties.

Abstract

Assuming the K\"unneth type standard conjecture, we propose a way to describe objects of mixed motives explicitly. We study their formal properties, and we associate mixed motives to schemes smooth and separated over a field. This serves as a universal cohomology theory. We also discuss $\ell$-adic realizations, and we discuss an unconditional construction of 2-motives and their properties.