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Motivic Brown-Peterson invariants of the rationals. (English) Zbl 1276.55023
Let $$BP\langle n\rangle$$ be the the family of motivic truncated Brown-Peterson spectra at the prime 2 over a characteristic 0 base field $$k$$. The authors perform some calculations for different motivic spectra related to $$BP$$. The main tool is the motivic Adams-Novikov spectral sequence. One of the main results is the calculation of the bigraded homotopy groups of the 2-completed $$BP\langle n\rangle$$. Another example is the following. Theorem. Let $$k$$ be a field with finite virtual cohomological dimension at 2. Let $$x_i, i\geq 1$$ denote the standard Lazard ring polynomial generators in degree $$i(1+\alpha)$$. Let $$BP=BP_k$$ and $$MGL=MGL_k$$ denote the 2-complete Brown-Peterson and algebraic cobordism spectra over $$k$$. Then there is an equivalence $\bigvee_{x_I}\sum^{|x_I|}BP\to MGL$ for a suitable collection of indices $$\{ I\}$$. This is the algebraic analogue of a theorem of Milnor and Novikov for the usual cobordism spectrum $$MGL$$.
The authors compute the rational homotopy group of $$MGL$$ and analyze the motivic Adams spectral sequence for $$BP$$ over the rationals and conditions for its convergence.

MSC:
 55T15 Adams spectral sequences 19D50 Computations of higher $$K$$-theory of rings 19E15 Algebraic cycles and motivic cohomology ($$K$$-theoretic aspects)
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References:
 [1] S Borghesi, Algebraic Morava $$K$$-theories, Invent. Math. 151 (2003) 381 · Zbl 1030.55003 · doi:10.1007/s00222-002-0257-4 [2] D Dugger, D C Isaksen, The motivic Adams spectral sequence, Geom. Topol. 14 (2010) 967 · Zbl 1206.14041 · doi:10.2140/gt.2010.14.967 [3] M A Hill, $$\mathrm{Ext}$$ and the motivic Steenrod algebra over $$\mathbbR$$, J. Pure Appl. Algebra 215 (2011) 715 · Zbl 1222.55014 · doi:10.1016/j.jpaa.2010.06.017 [4] M Hoyois, From algebraic cobordism to motivic cohomology · Zbl 1382.14006 · arxiv:1210.7182 [5] M Hoyois, S Kelly, P A Østvær, The motivic Steenrod algebra in positive characteristic (2012) · Zbl 1386.14087 · arxiv:1305.5690 [6] P Hu, On real-oriented Johnson-Wilson cohomology, Algebr. Geom. Topol. 2 (2002) 937 · Zbl 1025.55003 · doi:10.2140/agt.2002.2.937 · emis:journals/UW/agt/AGTVol2/agt-2-38.abs.html · eudml:123036 · arxiv:math/0211158 [7] P Hu, I Kriz, Some remarks on real and algebraic cobordism, $$K$$-Theory 22 (2001) 335 · Zbl 1032.55003 · doi:10.1023/A:1011196901303 [8] P Hu, I Kriz, K Ormsby, Convergence of the motivic Adams spectral sequence, J. $$K$$-Theory 7 (2011) 573 · Zbl 1309.14018 · doi:10.1017/is011003012jkt150 [9] P Hu, I Kriz, K Ormsby, Remarks on motivic homotopy theory over algebraically closed fields, J. K-Theory 7 (2011) 55 · Zbl 1248.14026 · doi:10.1017/is010001012jkt098 [10] J Milnor, The Steenrod algebra and its dual, Ann. of Math. 67 (1958) 150 · Zbl 0080.38003 · doi:10.2307/1969932 [11] J Milnor, Algebraic $$K$$-theory and quadratic forms, Invent. Math. 9 (1969/1970) 318 · Zbl 0199.55501 · doi:10.1007/BF01425486 · eudml:142017 [12] F Morel, An introduction to $$\mathbb A^1$$-homotopy theory (editors M Karoubi, A O Kuku, C Pedrini), ICTP Lect. Notes XV, Abdus Salam Int. Cent. Theoret. Phys. (2004) 357 [13] F Morel, The stable $$\mathbbA^1$$-connectivity theorems, $$K$$-Theory 35 (2005) · Zbl 1117.14023 · doi:10.1007/s10977-005-1562-7 [14] F Morel, V Voevodsky, $$\mathbfA^1$$-homotopy theory of schemes, Inst. Hautes Études Sci. Publ. Math. (1999) 45 · Zbl 0983.14007 · doi:10.1007/BF02698831 · numdam:PMIHES_1999__90__45_0 · eudml:104163 [15] N Naumann, M Spitzweck, P A Østvær, Existence and uniqueness of $$E_{\infty}$$-structures on motivic $$K$$-theory spectra · Zbl 1326.14054 · arxiv:1010.3944 [16] N Naumann, M Spitzweck, P A Østvær, Chern classes, $$K$$-theory and Landweber exactness over nonregular base schemes (editors R de Jeu, J D Lewis), Fields Inst. Commun. 56, Amer. Math. Soc. (2009) 307 · Zbl 1190.14022 [17] N Naumann, M Spitzweck, P A Østvær, Motivic Landweber exactness, Doc. Math. 14 (2009) 551 · Zbl 1230.55005 · emis:journals/DMJDMV/vol-14/20.html · eudml:227052 [18] J Neukirch, A Schmidt, K Wingberg, Cohomology of number fields, Grundl. Math. Wissen. 323, Springer (2008) · Zbl 1136.11001 [19] K M Ormsby, Motivic invariants of $$p$$-adic fields, J. $$K$$-Theory 7 (2011) 597 · Zbl 1258.14025 · doi:10.1017/is011004017jkt153 · arxiv:1002.5007 [20] K Ormsby, P A Østvær, Motivic invariants of low-dimensional fields, in preparation · Zbl 1304.55008 [21] D Quillen, On the formal group laws of unoriented and complex cobordism theory, Bull. Amer. Math. Soc. 75 (1969) 1293 · Zbl 0199.26705 · doi:10.1090/S0002-9904-1969-12401-8 [22] D C Ravenel, Complex cobordism and stable homotopy groups of spheres, Pure and Applied Mathematics 121, Academic Press (1986) · Zbl 0608.55001 [23] J Rognes, C Weibel, Two-primary algebraic $$K$$-theory of rings of integers in number fields, J. Amer. Math. Soc. 13 (2000) 1 · Zbl 0934.19001 · doi:10.1090/S0894-0347-99-00317-3 [24] O Röndigs, P A Østvær, Motives and modules over motivic cohomology, C. R. Math. Acad. Sci. Paris 342 (2006) 751 · Zbl 1097.14016 · doi:10.1016/j.crma.2006.03.013 [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] M Spitzweck, Relations between slices and quotients of the algebraic cobordism spectrum, Homology, Homotopy Appl. 12 (2010) 335 · Zbl 1209.14019 · doi:10.4310/HHA.2010.v12.n2.a11 · intlpress.com [27] A A Suslin, On the $$K$$-theory of local fields (editors E M Friedlander, M Karoubi), J. Pure Appl. Algebra 34 (1984) 301 · Zbl 0548.12009 · doi:10.1016/0022-4049(84)90043-4 [28] 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 [29] V Voevodsky, $$\mathbfA^1$$-homotopy theory, Doc. Math. Extra Vol. I (1998) 579 · Zbl 0907.19002 · emis:journals/DMJDMV/xvol-icm/00/Voevodsky.MAN.html [30] 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 [31] 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 [32] V Voevodsky, Motivic Eilenberg-Maclane spaces, Publ. Math. Inst. Hautes Études Sci. (2010) 1 · Zbl 1227.14025 · doi:10.1007/s10240-010-0024-9 [33] N Yagita, Applications of Atiyah-Hirzebruch spectral sequences for motivic cobordism, Proc. London Math. Soc. 90 (2005) 783 · Zbl 1086.55005 · doi:10.1112/S0024611504015084
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