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The motivic Adams spectral sequence. (English) Zbl 1206.14041
Since F. Morel’s and V. Voevodsky’s fundamental work on motivic homotopy theory [Publ. Math., Inst. Hautes Étud. Sci. 90, 45–143 (1999; Zbl 0983.14007)], many tools from classical algebraic topology have been transferred to \(\mathbb{A}^1\)-homotopy theory. One of these tools is the classical mod 2-Adams spectral sequence which allows to deduce information on the stable homotopy groups of spheres. Its motivic analog that provides information on the bigraded family of stable homotopy groups of the motivic sphere spectrum has first been described by F. Morel [C. R. Acad. Sci., Paris, Sér. I, Math. 328, No. 11, 963–968 (1999; Zbl 0937.19002)]. The authors continue this study by providing the first extensive calculations over an algebraically closed field of characteristic zero. To clarify the picture a little bit, let \(F\) be a field of characteristic zero, let \(M_2\) denote the bigraded motivic cohomology of \(F\) with \(\mathbb{Z}/2\)-coefficients and let \(A\) be the mod 2 motivic Steenrod algebra over \(F\), studied by V. Voevodsky [Publ. Math., Inst. Hautes Étud. Sci. 98, 59–104 (2003; Zbl 1057.14028)]. The motivic Adams spectral sequence is the trigraded spectral sequence with \[ E_2^{s,t,u}=\mathrm{Ext}_A^{s,(t+s,u)}(M_2,M_2) \] and differentials \(d_r:E_r^{s,t,u} \to E_r^{s+r,t-1,u}\). The most remarkable and fascinating aspect of the paper under review is that it turns around the flow of information between classical and motivic theory. The authors show how the motivic trigraded version of the Adams spectral sequence offers new insights into the classical version. This uncloses a new perspective with potentially far reaching and ground breaking consequences for the interaction between classical algebraic topology and motivic homotopy theory. Apart from this exchange of information the motivic Adams spectral sequence also reveals new phenomena that do not occur in the classical theory. The crucial point is that there is the additional motivic weight \(u\) that makes the story tri- and not only bigraded. Although this looks (and is at many points) more complicated, this extra information also makes some purely topological phenomena easier to analyze. On their way, the authors provide many interesting calculations in the motivic Steenrod algebra and on motivic Massey products. An appendix completes this very interesting and nicely written paper with several charts of spectral sequences.

MSC:
14F42 Motivic cohomology; motivic homotopy theory
55T15 Adams spectral sequences
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