Embedding of category of twisted Chow-Witt correspondences into geometric stable $\A^1$-derived category over a field

TL;DR

This paper introduces the category of twisted Chow-Witt correspondences over a field and demonstrates its embedding into the geometric stable $ ext{A}^1$-derived category, confirming a conjecture of F. Morel on rational splitting of stable $ ext{A}^1$-cohomology.

Contribution

It defines the category of twisted Chow-Witt correspondences and proves its fully faithful embedding into the geometric stable $ ext{A}^1$-derived category over an infinite perfect field, also confirming Morel's conjecture.

Findings

Category of twisted Chow-Witt correspondences admits a fully faithful embedding into the geometric stable $ ext{A}^1$-derived category.

Confirmed F. Morel's conjecture on rational splitting of stable $ ext{A}^1$-cohomology.

Embedding holds after $ ext{Q}$-localization over an infinite perfect field.

Abstract

We introduce in this note the notion of the category of twisted Chow-Witt correspondences $CHW(k)$ over a field $k$ of characteristic different from $2$. Moreover, we show that over an infinite perfect field this category $CHW(k)$ admits a fully faithful embedding into the geometric stable $\A^1$-derived category $D_{\A^1,gm}(k)$ after taking $\Q$-localization. We also prove a conjecture of F. Morel about the rational splitting of stable $\A^1$-cohomology over an essentially smooth scheme $S$ over a field $k$ of $char(k) \neq 2$.