Holmstrom Motivic stable homotopy theory

Voevodsky’s Open problems paper, in Voevodsky folder.

http://mathoverflow.net/questions/27239/relation-between-motivic-homotopy-category-and-the-derived-category-of-motives contains nice answer by SimonPL

arXiv:0907.1510 Periodizable motivic ring spectra. from arXiv Front: math.AG by Markus Spitzweck. We show that the cellular objects in the module category over a motivic E infinity ring spectrum E can be described as the module category over a graded topological spectrum if E is strongly periodizable in our language. A similar statement is proven for triangulated categories of motives. Since MGL is strongly periodizable we obtain topological incarnations of motivic Landweber spectra. Under some categorical assumptions the unit object of the model category for triangulated motives is as well strongly periodizable giving motivic cochains whose module category models integral triangulated categories of Tate motives.

arXiv:1002.2368 Motivic connective K-theories and the cohomology of A(1) from arXiv Front: math.KT by Daniel C. Isaksen, Armira Shkembi We make some computations in stable motivic homotopy theory over Spec \mathbb{C}, completed at 2. Using homotopy fixed points and the algebraic K-theory spectrum, we construct a motivic analogue of the real K-theory spectrum KO. We also establish a theory of connective covers to obtain a motivic version of ko. We establish an Adams spectral sequence for computing motivic ko-homology. The E_2-term of this spectral sequence involves Ext groups over the subalgebra A(1) of the motivic Steenrod algebra. We make several explicit computations of these E_2-terms in interesting special cases.

See articles by Pelaez, including Functoriality of the slice filtration (arxiv)

http://mathoverflow.net/questions/16310/why-does-one-invert-g-m-in-the-construction-of-the-motivic-stable-homotopy-cate

arXiv:1206.3645 Noncommutative Motives I: A Universal Characterization of the Motivic Stable Homotopy Theory of Schemes from arXiv Front: math.AG by Marco Robalo Let $\V$ be a symmetric monoidal model category and let $X$ be an object in $\V$. From this we can construct a new symmetric monoidal model category $Sp^{\Sigma}(\V,X)$ of symmetric spectra objects in $\V$ with respect to $X$, together with a left Quillen monoidal map $\V\to Sp^{\Sigma}(\V,X)$ sending $X$ to an invertible object. In this paper we use the recent developments in the subject of Higher Algebra to understand the nature of this construction. Every symmetric monoidal model category has an underlying symmetric monoidal $(\infty,1)$-category and the first notion should be understood as a mere “presentation” of the second. Our main result is the characterization of the underlying symmetric monoidal $\infty$-category of $Sp^{\Sigma}(\V,X)$, by means of a universal property inside the world of symmetric monoidal $(\infty,1)$-categories. In the process we also describe the link between the construction of ordinary spectra and the one of symmetric spectra. As a corollary, we obtain a precise universal characterization for the motivic stable homotopy theory of schemes with its symmetric monoidal structure. This characterization trivializes the problem of finding motivic monoidal realizations and opens the way to compare the motivic theory of schemes with other motivic theories.

nLab page on Motivic stable homotopy theory