Integral mixed motives in equal characteristic

TL;DR

This paper develops a framework for mixed motives over schemes in positive characteristic, establishing their properties and equivalences with classical categories, thus advancing the understanding of motives in algebraic geometry.

Contribution

It constructs and analyzes triangulated categories of mixed motives in equal characteristic, proving their properties and equivalences with classical motives via Eilenberg-MacLane spectra.

Findings

Formalism of six operations holds for these motives

Categories are equivalent to classical mixed motives over regular schemes

Motivic Eilenberg-MacLane spectra relate different categories of motives

Abstract

For noetherian schemes of finite dimension over a field of characteristic exponent $p$, we study the triangulated categories of $\mathbf{Z}[1/p]$-linear mixed motives obtained from cdh-sheaves with transfers. We prove that these have many of the expected properties. In particular, the formalism of the six operations holds in this context. When we restrict ourselves to regular schemes, we also prove that these categories of motives are equivalent to the more classical triangulated categories of mixed motives constructed in terms of Nisnevich sheaves with transfers. Such a program is achieved by comparing these various triangulated categories of motives with modules over motivic Eilenberg-MacLane spectra.