Vol. 5, No. 2, 2020

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On modules over motivic ring spectra

Elden Elmanto and Håkon Kolderup

Vol. 5 (2020), No. 2, 327–355
Abstract

We provide an axiomatic framework that characterizes the stable -categories that are module categories over a motivic spectrum. This is done by invoking Lurie’s -categorical version of the Barr–Beck theorem. As an application, this gives an alternative approach to Röndigs and Østvær’s theorem relating Voevodsky’s motives with modules over motivic cohomology and to Garkusha’s extension of Röndigs and Østvær’s result to general correspondence categories, including the category of Milnor–Witt correspondences in the sense of Calmès 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 Déglise.

Keywords
motivic homotopy theory, generalized motivic cohomology, Milnor–Witt K-theory, Barr–Beck–Lurie theorem, $\infty$-categories
Mathematical Subject Classification 2010
Primary: 14F40, 14F42
Secondary: 19E15, 55P42, 55P43, 55U35
Milestones
Received: 12 June 2019
Revised: 6 October 2019
Accepted: 22 October 2019
Published: 20 June 2020
Authors
Elden Elmanto
Harvard University
Cambridge, MA
United States
Håkon Kolderup
University of Oslo
Oslo
Norway