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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.

14F42 Motivic cohomology; motivic homotopy theory
55T15 Adams spectral sequences
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[1] J F Adams, A finiteness theorem in homological algebra, Proc. Cambridge Philos. Soc. 57 (1961) 31 · Zbl 0097.16003
[2] M Artin, G A., J L Verdier, Théorie des topos et cohomologie étale des schémas. Tome 1, 2, 3, Lecture Notes in Math. 269, 270, 305, Springer (1972-3)
[3] T Bauer, Computation of the homotopy of the spectrum \tttmf (editors N Iwase, T Kohno, R Levi, D Tamaki, J Wu), Geom. Topol. Monogr. 13, Geom. Topol. Publ. (2008) 11 · Zbl 1147.55005 · doi:10.2140/gtm.2008.13.11 · arxiv:math/0311328
[4] S Bloch, Algebraic cycles and higher \(K\)-theory, Adv. in Math. 61 (1986) 267 · Zbl 0608.14004 · doi:10.1016/0001-8708(86)90081-2
[5] A K Bousfield, The localization of spectra with respect to homology, Topology 18 (1979) 257 · Zbl 0417.55007 · doi:10.1016/0040-9383(79)90018-1
[6] D Dugger, D C Isaksen, Topological hypercovers and \(\mathbb A^1\)-realizations, Math. Z. 246 (2004) 667 · Zbl 1055.55016 · doi:10.1007/s00209-003-0607-y
[7] D Dugger, D C Isaksen, Motivic cell structures, Algebr. Geom. Topol. 5 (2005) 615 · Zbl 1086.55013 · doi:10.2140/agt.2005.5.615 · emis:journals/UW/agt/AGTVol5/agt-5-26.abs.html · eudml:125949 · arxiv:math/0310190
[8] M A Hill, \(\Ext\) and the motivic Steenrod algebra over \(\mathbbR\) · Zbl 1222.55014 · doi:10.1016/j.jpaa.2010.06.017
[9] M J Hopkins, M A Hill, D C Ravenel, On the nonexistence of elements of Kervaire invariant one · Zbl 1366.55007 · arxiv:0908.3724
[10] M Hovey, B Shipley, J Smith, Symmetric spectra, J. Amer. Math. Soc. 13 (2000) 149 · Zbl 0931.55006 · doi:10.1090/S0894-0347-99-00320-3
[11] P Hu, I Kriz, Some remarks on Real and algebraic cobordism, \(K\)-Theory 22 (2001) 335 · Zbl 1032.55003 · doi:10.1023/A:1011196901303
[12] P Hu, I Kriz, K Ormsby, Remarks on motivic homotopy theory over algebraically closed fields, \(K\)-Theory (2010) · Zbl 1248.14026 · doi:10.1017/is010001012jkt098
[13] J F Jardine, Motivic symmetric spectra, Doc. Math. 5 (2000) 445 · Zbl 0969.19004 · emis:journals/DMJDMV/vol-05/15.html · eudml:121235
[14] J P May, The cohomology of restricted Lie algebras and of Hopf algebras, Bull. Amer. Math. Soc. 71 (1965) 372 · Zbl 0134.19103 · doi:10.1090/S0002-9904-1965-11300-3
[15] J P May, A general algebraic approach to Steenrod operations, Lecture Notes in Math. 168, Springer (1970) 153 · Zbl 0242.55023
[16] C Mazza, V Voevodsky, C Weibel, Lecture notes on motivic cohomology, Clay Math. Monogr. 2, Amer. Math. Soc. (2006) · Zbl 1115.14010
[17] J Milnor, The Steenrod algebra and its dual, Ann. of Math. \((2)\) 67 (1958) 150 · Zbl 0080.38003 · doi:10.2307/1969932
[18] F Morel, Suite spectrale d’Adams et invariants cohomologiques des formes quadratiques, C. R. Acad. Sci. Paris Sér. I Math. 328 (1999) 963 · Zbl 0937.19002 · doi:10.1016/S0764-4442(99)80306-1
[19] F Morel, On the motivic \(\pi_0\) of the sphere spectrum (editor J P C Greenlees), NATO Sci. Ser. II Math. Phys. Chem. 131, Kluwer Acad. Publ. (2004) 219 · Zbl 1130.14019
[20] F Morel, The stable \(\mathbbA^1\)-connectivity theorems, \(K\)-Theory 35 (2005) 1 · Zbl 1117.14023 · doi:10.1007/s10977-005-1562-7
[21] F Morel, V Voevodsky, \(\mathbfA^1\)-homotopy theory of schemes, Inst. Hautes Études Sci. Publ. Math. (1999) · Zbl 0983.14007 · doi:10.1007/BF02698831 · numdam:PMIHES_1999__90__45_0 · eudml:104163
[22] R E Mosher, M C Tangora, Cohomology operations and applications in homotopy theory, Harper & Row (1968) · Zbl 0153.53302
[23] I Panin, K Pimenov, O Röndigs, A universality theorem for Voevodsky’s algebraic cobordism spectrum · Zbl 1162.14013 · doi:10.4310/HHA.2008.v10.n2.a11 · intlpress.com
[24] D C Ravenel, Complex cobordism and stable homotopy groups of spheres, Pure and Applied Math. 121, Academic Press (1986) · Zbl 0608.55001
[25] O Röndigs, P A Østvær, Modules over motivic cohomology, Adv. Math. 219 (2008) 689 · Zbl 1180.14015 · doi:10.1016/j.aim.2008.05.013
[26] J P Serre, Cohomologie modulo \(2\) des complexes d’Eilenberg-MacLane, Comment. Math. Helv. 27 (1953) 198 · Zbl 0052.19501 · doi:10.1007/BF02564562 · eudml:139063
[27] M C Tangora, On the cohomology of the Steenrod algebra, Math. Z. 116 (1970) 18 · Zbl 0198.28202 · doi:10.1007/BF01110185 · eudml:171367
[28] M C Tangora, Some Massey products in \(\mathrm{Ext}\) (editors E M Friedlander, M E Mahowald), Contemp. Math. 158, Amer. Math. Soc. (1994) 269 · Zbl 0798.55014
[29] G Vezzosi, Brown-Peterson spectra in stable \(\mathbb A^1\)-homotopy theory, Rend. Sem. Mat. Univ. Padova 106 (2001) 47 · Zbl 1165.14308 · numdam:RSMUP_2001__106__47_0 · eudml:108568
[30] V Voevodsky, Motivic Eilenberg-Mac Lane spaces · arxiv:0805.4432
[31] V Voevodsky, Motivic cohomology with \(\mathbfZ/2\)-coefficients, Publ. Math. Inst. Hautes Études Sci. (2003) 59 · Zbl 1057.14028 · doi:10.1007/s10240-003-0010-6 · numdam:PMIHES_2003__98__59_0 · eudml:104197
[32] V Voevodsky, Reduced power operations in motivic cohomology, Publ. Math. Inst. Hautes Études Sci. (2003) 1 · Zbl 1057.14027 · doi:10.1007/s10240-003-0009-z · numdam:PMIHES_2003__98__1_0 · eudml:104196
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