Birational motives, II: Triangulated birational motives

TL;DR

This paper develops birational versions of Voevodsky's triangulated motives, connecting them with pure birational motives and providing a new interpretation of unramified cohomology through higher derived functors.

Contribution

It introduces triangulated birational motives and relates them to existing pure birational motives, expanding the theoretical framework of motives.

Findings

Establishment of birational triangulated motives

Connection with pure birational motives

Interpretation of unramified cohomology via higher derived functors

Abstract

We develop birational versions of Voevodsky's triangulated categories of motives over a field, and relate them with the pure birational motives studied in arXiv:0902.4902 [math.AG]. We also get an interpretation of unramified cohomology in this framework, leading to "higher derived functors of unramified cohomology".