Generic motives and motivic cohomology of fields

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.