On the rational motivic homotopy category

TL;DR

This paper investigates the structure of the rational motivic stable homotopy category, establishing fundamental properties and relationships with other motivic theories over general base schemes.

Contribution

It proves key properties like absolute purity, duality, and SL-orientation for SH_Q, and relates it to Milnor-Witt motives and Chow-Witt groups, advancing the understanding of rational motivic homotopy theory.

Findings

Proved absolute purity and duality for SH_Q

Established canonical SL-orientation of SH_Q

Connected SH_Q with rational Milnor-Witt motives and Chow-Witt groups

Abstract

We study the structure of the rational motivic stable homotopy category over general base schemes. Our first class of results concerns the six operations: we prove absolute purity, stability of constructible objects, and Grothendieck-Verdier duality for SH_Q. Next, we prove that SH_Q is canonically SL-oriented; we compare SH_Q with the category of rational Milnor-Witt motives; and we relate the rational bivariant A^1-theory to Chow-Witt groups. These results are derived from analogous statements for the minus part of SH[1/2].