E-motives and motivic stable homotopy

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 $D_{\mathbb{A}^1,gm}(k)_{\mathbb{Q}}$.