Motives and modules over motivic cohomology

In this Note we summarize the main results and techniques in our homotopical algebraic approach to motives. A major part of this work relies on highly structured models for motivic stable homotopy theory. For any noetherian and separated base scheme of finite Krull dimension these frameworks give rise to a homotopy theoretic meaningful study of modules over motivic cohomology. When the base scheme is $\mathrm{Spec}(k)$, for k a field of characteristic zero, the corresponding homotopy category is equivalent to Voevodsky's big category of motives. To cite this article: O. Röndigs, P.A. Østvær, C. R. Acad. Sci. Paris, Ser. I 342 (2006).