A commutative P^1-spectrum representing motivic cohomology over Dedekind domains

TL;DR

This paper constructs a structured motivic spectrum over Dedekind domains representing motivic cohomology, enabling new insights into arithmetic and geometric properties of schemes.

Contribution

It introduces a new motivic Eilenberg-MacLane spectrum with a highly structured multiplication over Dedekind domains, unifying rational and p-completed parts via an arithmetic square.

Findings

Realizes arithmetic motivic cohomology groups as Ext-groups in a triangulated category

Provides a spectrum compatible with base change and six functor formalism

Generalizes the Hopkins-Morel isomorphism and analyzes the dual motivic Steenrod algebra

Abstract

We construct a motivic Eilenberg-MacLane spectrum with a highly structured multiplication over smooth schemes over Dedekind domains which represents Levine's motivic cohomology. The latter is defined via Bloch's cycle complexes. Our method is by gluing p-completed and rational parts along an arithmetic square. Hereby the finite coefficient spectra are obtained by truncated \'etale sheaves (relying on the now proven Bloch-Kato conjecture) and a variant of Geisser's version of syntomic cohomology, and the rational spectra are the ones which represent Beilinson motivic cohomology. As an application the arithmetic motivic cohomology groups can be realized as Ext-groups in a triangulated category of Tate sheaves with integral coefficients. These can be modelled as representations of derived fundamental groups. Our spectrum is compatible with base change giving rise to a formalism of six functors for triangulated categories of motivic sheaves over general base schemes including the localization triangle. Further applications include a generalization of the Hopkins-Morel isomorphism and a structure result for the dual motivic Steenrod algebra in the case where the coefficient characteristic is invertible on the base scheme.