On Modules Over Motivic Ring Spectra

TL;DR

This paper develops an axiomatic framework for module categories over motivic spectra using $mbda$-categorical techniques, providing new insights into motivic homotopy theory and extending key comparison theorems.

Contribution

It introduces an axiomatic characterization of module categories over motivic spectra via Lurie's Barr--Beck theorem, and extends important comparison results to broader schemes.

Findings

Provides an alternative proof of Rd6ndigs and d8stve6r's theorem.

Extends comparison results to regular Noetherian schemes over a field.

Generalizes the framework to include Milnor-Witt correspondences.

Abstract

In this note, we provide an axiomatic framework that characterizes the stable $\infty$-categories that are module categories over a motivic spectrum. This is done by invoking Lurie's $\infty$-categorical version of the Barr--Beck theorem. As an application, this gives an alternative approach to R\"ondigs and \O stv\ae r's theorem relating Voevodsky's motives with modules over motivic cohomology, and to Garkusha's extension of R\"ondigs and \O stv\ae r's result to general correspondence categories, including the category of Milnor-Witt correspondences in the sense of Calm\`es and Fasel. We also extend these comparison results to regular Noetherian schemes over a field (after inverting the residue characteristic), following the methods of Cisinski and D\'eglise.