On the Integral Part of A-Motivic Cohomology

TL;DR

This paper explores the integral part of A-motivic cohomology over global fields of positive characteristic, defining and comparing model and adic versions, and revealing their divergence in general.

Contribution

It introduces the definitions of model and adic versions of A-motivic cohomology and demonstrates their non-coincidence, unlike the conjectural expectations from number field cases.

Findings

Model version is contained in adic version.

Model and adic approaches do not match in general.

Introduces regulated extensions of mixed Anderson A-motives.

Abstract

The deepest arithmetic invariants attached to an algebraic variety defined over a number field $F$ are conjecturally captured by the integral part of its motivic cohomology. There are essentially two ways of defining it when $X$ is a smooth projective variety: one is via the $K$-theory of a regular model, the other is through its $\ell$-adic realization. Both approaches are conjectured to coincide. This paper initiates the study of motivic cohomology for global fields of positive characteristic, hereafter named $A$-motivic cohomology, where classical mixed motives are replaced by mixed Anderson $A$-motives. Our main objective is to set the definitions of the model version and the $\ell$-adic version of the integral part of $A$-motivic cohomology, using Gardeyn's notion of maximal models of $A$-motives as the analogue of regular models of varieties. Our main result states that the model version is contained in the $\ell$-adic version. As opposed to what is expected in the number field setting, we show that the two approaches do not match in general. We conclude this work by introducing the submodule of regulated extensions of mixed Anderson $A$-motives, for which we expect the two approaches to match, and solve some particular cases of this expectation.