Generic motives and motivic cohomology of fields

TL;DR

This paper explores the structure of generic motives and their impact on the motivic cohomology of fields, providing new computations, conjectural frameworks, and insights into the infinite rank and growth of motivic cohomology groups.

Contribution

It introduces new computations of generic motives for curves and surfaces, proposes a conjectural framework for morphisms of generic motives, and offers a novel approach to understanding motivic cohomology of fields without regulator maps.

Findings

Motivic cohomology groups of nd  are often uncountable.

New computations suggest a conjectural framework for generic motives.

Motivic cohomology can have infinite rank, matching the cardinality of the base field.

Abstract

This paper investigates the structure of generic motives and their implications for the motivic cohomology of fields. Originating in Voevodsky's theory of motives and related to Beilinson's vision of a motivic $t$-structure, generic motives serve as pro-objects encoding essential information about cycles and cohomology. We present new computations of generic motives, focusing on curves and surfaces. These computations suggest a conjectural framework for morphisms of generic motives and highlight the central role of transcendental motives. We then focus on the motivic cohomology of fields, building on Borel's rank computation of K-theory and its relation to higher regulators. We provide a direct argument for determining the weights in the $\lambda$-structure of the K-theory of number fields, bypassing the need for regulator maps. We show that motivic cohomology groups are often of infinite rank, typically matching the cardinality of the base field. For instance, we prove that motivic cohomology groups of $\mathbb R$ and $\mathbb C$ are uncountable in many bi-degrees. Despite this, we propose a conjecture that complements the Beilinson-Soul\'e vanishing conjecture, suggesting that the growth of motivic cohomology is more controlled than these results may initially indicate.