# E-motives and motivic stable homotopy

## TL;DR

This paper develops the theory of pure E-motives within motivic homotopy theory, constructing twisted E-cohomology and embedding Chow-Witt motives into the geometric stable A^1-derived category, advancing the understanding of motivic categories.

## Contribution

It introduces the category of pure E-motives for a motivic ring spectrum and embeds Chow-Witt motives into the geometric stable A^1-derived category.

## Key findings

- Construction of pure E-motives using six functors formalism.
- Definition of twisted E-cohomology in the motivic setting.
- Fully faithful embedding of Chow-Witt motives into the geometric stable A^1-derived category.

## Abstract

We introduce in this work the notion of the category of pure $\mathbf{E}$-Motives, where $\mathbf{E}$ is a motivic strict ring spectrum and construct twisted $\mathbf{E}$-cohomology by using six functors formalism of J. Ayoub. In particular, we construct the category of pure Chow-Witt motives $CHW(k)_{\mathbb{Q}}$ over a field $k$ and show that this category admits a fully faithful embedding into the geometric stable $\mathbb{A}^1$-derived category $D_{\mathbb{A}^1,gm}(k)_{\mathbb{Q}}$.