×

zbMATH — the first resource for mathematics

Ext and the motivic Steenrod algebra over \(\mathbb R\). (English) Zbl 1222.55014
The author presents some computations of Ext over the dual mod 2 motivic Steenrod algebra \({\mathcal A}\), or rather its sub-Hopf algebras \(E(n)\) and \({\mathcal A}(1)\), over \({\mathbb R}\), as a preamble to future work of computing motivic stable homotopy groups using the Adams spectral sequence over \({\mathbb R}\). The method is through a descent style, Bockstein spectral sequence to utilize some corresponding results over \({\mathbb C}\). The calculations are very delicate.
At the end, the author relates such motivic calculations to the classical ones related to \({\mathbb Z}/2\)-equivariant homotopy, \(BP\langle n\rangle\), and \(ko\).

MSC:
55T15 Adams spectral sequences
14F42 Motivic cohomology; motivic homotopy theory
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Borghesi, Simone, Algebraic Morava \(K\)-theories, Invent. math., 151, 2, 381-413, (2003), MR1953263 (2003m:55006) · Zbl 1030.55003
[2] Simone Borghesi, Cohomology operations and algebraic geometry, in: Proceedings of the Nishida Fest (Kinosaki 2003), Geom. Topol. Monogr., vol. 10, Geom. Topol. Publ., Coventry, 2007, pp. 75-115. MR2402778 (2009m:14024). · Zbl 1104.14014
[3] Dugger, Daniel, An atiyah – hirzebruch spectral sequence for \(K R\)-theory, \(K\)-theory, 35, 3-4, 213-256, (2005), MR2240234 (2007g:19004) · Zbl 1109.14024
[4] Dugger, Daniel; Isaksen, Daniel C., Motivic cell structures, Algebr. geom. topol., 5, 615-652, (2005), (electronic). MR2153114 (2007c:55015) · Zbl 1086.55013
[5] Daniel Dugger, Daniel C. Isaksen, The motivic Adams spectral sequence, arxiv:0901.1632, 2009. · Zbl 1206.14041
[6] Michael A. Hill, Michael J. Hopkins, Douglas C. Ravenel, On the non-existence of elements of Kervaire invariant one, arXiv:0908.3724. · Zbl 1373.55023
[7] Michael J. Hopkins, Fabien Morel, On the zero slice of \(M G L\) and \(S^0\) (in preparation).
[8] Hu, Po; Kriz, Igor, Real-oriented homotopy theory and an analogue of the adams – novikov spectral sequence, Topology, 40, 2, 317-399, (2001), MR1808224 (2002b:55032) · Zbl 0967.55010
[9] Hu, Po; Kriz, Igor, Some remarks on real and algebraic cobordism, J. \(K\)-theory, 22, 4, 335-366, (2001), MR1847399 (2002g:19004) · Zbl 1032.55003
[10] Po Hu, Igor Kriz, Kyle Ormsby, Convergence of the motivic Adams spectral sequence, Available on the \(K\)-theory, arxiv:0962, 2010. · Zbl 1247.14020
[11] Hu, Po; Kriz, Igor; Ormsby, Kyle, Some remarks on motivic homotopy theory over algebraically closed fields, J. \(K\)-theory, (2010) · Zbl 1248.14026
[12] Daniel Isaksen, Armira Shkembi, Motivic connective K-theories and the cohomology of \(A(1)\), arXiv:1002.2368, 2010. · Zbl 1266.14015
[13] Morel, Fabien, Suite spectrale d’adams et invariants cohomologiques des formes quadratiques, C.R. acad. sci. Paris Sér. I math., 328, 11, 963-968, (1999), MR1696188 (2000d:11056) · Zbl 0937.19002
[14] Morel, Fabien, On the motivic \(\pi_0\) of the sphere spectrum, (), 219-260, MR2061856 (2005e:19002) · Zbl 1130.14019
[15] Morel, Fabien, The stable \(\mathbb{A}^1\)-connectivity theorems, J. \(K\)-theory, 35, 1-2, 1-68, (2005), MR2240215 (2007d:14041) · Zbl 1117.14023
[16] Morel, Fabien; Voevodsky, Vladimir, \(\mathbf{A}^1\)-homotopy theory of schemes, Publ. math. inst. hautes études sci., 1999, 90, 45-143, (2001), MR1813224 (2002f:14029) · Zbl 0983.14007
[17] Kyle M. Ormsby, Motivic invariants of \(p\)-adic fields, 2010 (submitted for publication). · Zbl 1258.14025
[18] Ravenel, Douglas C., Complex cobordism and stable homotopy groups of spheres, () · Zbl 1073.55001
[19] Armira Shkembi, The cohomology of A(1) and motivic connective theories, Ph.D. Thesis, Wayne State University, 2009. · Zbl 1266.14015
[20] Suslin, A. A., The beĭlinson spectral sequence for the \(K\)-theory of the field of real numbers, Mat. metody i fiz.-mekh. polya, 28, 51-52, (1988), 105. MR967267 (89i:12005)
[21] V. Voevodsky, Motivic Eilenberg-McLane spaces, K-theory Archive. · Zbl 1227.14025
[22] Voevodsky, Vladimir, Motivic cohomology with \(\mathbf{Z} / 2\)-coefficients, Publ. math. inst. hautes études sci., 98, 59-104, (2003), MR2031199 (2005b:14038b) · Zbl 1057.14028
[23] Voevodsky, Vladimir, Reduced power operations in motivic cohomology, Publ. math. inst. hautes études sci., 98, 1-57, (2003), MR2031198 (2005b:14038a) · Zbl 1057.14027
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.