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VANISHING IN STABLE MOTIVIC HOMOTOPY SHEAVES

Part of: (Co)homology theory

Published online by Cambridge University Press:  05 March 2018

KYLE ORMSBY ,
OLIVER RÖNDIGS  and
PAUL ARNE ØSTVÆR
KYLE ORMSBY
Affiliation:
Reed College, USA; [email protected]
OLIVER RÖNDIGS
Affiliation:
Universität Osnabrück, Germany; [email protected]
PAUL ARNE ØSTVÆR
Affiliation:
University of Oslo, Norway; [email protected]

Abstract

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We determine systematic vanishing regions for the bigraded homotopy sheaves of the motivic sphere spectrum over a field of characteristic different from two.

MSC classification

Primary: 14F42: Motivic cohomology; motivic homotopy theory
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 6 , 2018 , e3
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2018 This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.

References

Andrews, M. and Miller, H., ‘Inverting the Hopf map’, J. Topol. 10(4) (2017), 1145–1168.CrossRefGoogle Scholar
Asok, A. and Fasel, J., ‘Algebraic vector bundles on spheres’, J. Topol. 7(3) (2014), 894–926.Google Scholar
Bachmann, T., ‘Motivic and real etale stable homotopy theory’, Preprint, 2017, arXiv:1608.08855v2.Google Scholar
Dugger, D. and Isaksen, D. C., ‘The motivic Adams spectral sequence’, Geom. Topol. 14(2) (2010), 967–1014.CrossRefGoogle Scholar
Dugger, D. and Isaksen, D. C., ‘ $\mathbb{Z}/2$ -equivariant and $\mathbb{R}$ -motivic stable stems’, Preprint, 2016, arXiv:1603.09305.Google Scholar
Dugger, D. and Isaksen, D. C., ‘Low-dimensional Milnor–Witt stems over ℝ’, Ann. K-Theory 2(2) (2017), 175–210.Google Scholar
Gheorghe, B. and Isaksen, D. C., ‘The structure of motivic homotopy groups’, Bol. Soc. Mat. Mexicana (3) 23(1) (2017), 389–397.CrossRefGoogle Scholar
Guillou, B. J. and Isaksen, D. C., ‘The 𝜂-inverted ℝ-motivic sphere’, Algebr. Geom. Topol. 16(5) (2016), 3005–3027.Google Scholar
Heller, J. and Ormsby, K., ‘Galois equivariance and stable motivic homotopy theory’, Trans. Amer. Math. Soc. 368(11) (2016), 8047–8077.Google Scholar
Heller, J. and Ormsby, K., ‘The stable Galois correspondence for real closed fields’, Preprint, 2017, arXiv:1701.09099.Google Scholar
Hornbostel, J. and Yagunov, S., ‘Rigidity for henselian local rings and A1 -representable theories’, Math. Z. 255(2) (2007), 437–449.Google Scholar
Hoyois, M., ‘From algebraic cobordism to motivic cohomology’, J. Reine Angew. Math. 702 (2015), 173–226.CrossRefGoogle Scholar
Hu, P., Kriz, I. and Ormsby, K., ‘Convergence of the motivic Adams spectral sequence’, J. K-Theory 7(3) (2011), 573–596.CrossRefGoogle Scholar
Hu, P., Kriz, I. and Ormsby, K., ‘Remarks on motivic homotopy theory over algebraically closed fields’, J. K-Theory 7(1) (2011), 55–89.CrossRefGoogle Scholar
Levine, M., ‘Convergence of Voevodsky’s slice tower’, Doc. Math. 18 (2013), 907–941.Google Scholar
Merkurjev, A. S. and Suslin, A. A., ‘The group K 3 for a field’, Izv. Akad. Nauk SSSR Ser. Mat. 54(3) (1990), 522–545.Google Scholar
Morel, F., ‘Suite spectrale d’Adams et invariants cohomologiques des formes quadratiques’, C. R. Acad. Sci. Paris Sér. I Math. 328(11) (1999), 963–968.Google Scholar
Morel, F., ‘On the motivic 𝜋0 of the sphere spectrum’, inAxiomatic, Enriched and Motivic Homotopy Theory, NATO Sci. Ser. II Math. Phys. Chem., 131 (Kluwer Academic Publishers, Dordrecht, 2004), 219–260.CrossRefGoogle Scholar
Morel, F., ‘The stable A1 -connectivity theorems’, K-Theory 35(1–2) (2005), 1–68.Google Scholar
Morel, F., A1 -Algebraic Topology over a Field, Lecture Notes in Mathematics, 2052 (Springer, Heidelberg, 2012).CrossRefGoogle Scholar
Novikov, S. P., ‘Methods of algebraic topology from the point of view of cobordism theory’, Izv. Akad. Nauk SSSR Ser. Mat. 31 (1967), 855–951.Google Scholar
Ormsby, K., Computations in stable motivic homotopy theory. ProQuest LLC, Ann Arbor, MI, PhD Thesis, University of Michigan, 2010.Google Scholar
Ormsby, K. M. and Østvær, P. A., ‘Stable motivic 𝜋1 of low-dimensional fields’, Adv. Math. 265 (2014), 97–131.CrossRefGoogle Scholar
Ravenel, D. C., Complex Cobordism and Stable Homotopy Groups of Spheres, Pure and Applied Mathematics, 121 (Academic Press, Inc., 1986).Google Scholar
Röndigs, O., ‘On the $\unicode[STIX]{x1D702}$ -inverted sphere’. Proceedings of the International Colloquium on $K$ -Theory, TIFR, to appear, Preprint, 2016, arXiv:1602.08798.Google Scholar
Röndigs, O., Spitzweck, M. and Østvær, P. A., ‘The first stable homotopy groups of motivic spheres’, Preprint, 2016, arXiv:1604.00365.Google Scholar
Scheiderer, C., Real and Étale Cohomology, Lecture Notes in Mathematics, 1588 (Springer, Berlin, 1994).Google Scholar
Serre, J.-P., Galois Cohomology, Springer Monographs in Mathematics (Springer, Berlin, 2002).Google Scholar
Suslin, A. and Voevodsky, V., ‘Bloch–Kato conjecture and motivic cohomology with finite coefficients’, inThe Arithmetic and Geometry of Algebraic Cycles (Banff, AB, 1998), NATO Sci. Ser. C Math. Phys. Sci., 548 (Kluwer Academic Publishers, Dordrecht, 2000), 117–189.Google Scholar
Suslin, A. and Voevodsky, V., ‘Singular homology of abstract algebraic varieties’, Invent. Math. 123(1) (1996), 61–94.CrossRefGoogle Scholar
Toda, H., Composition Methods in Homotopy Groups of Spheres, Annals of Mathematics Studies, 49 (Princeton University Press, Princeton, N.J., 1962).Google Scholar
Voevodsky, V., ‘ A 1 -homotopy theory’, inProceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998), Extra Vol. I (1998), 579–604.Google Scholar
Voevodsky, V., ‘Open problems in the motivic stable homotopy theory. I’, inMotives, Polylogarithms and Hodge Theory, Part I (Irvine, CA, 1998), Int. Press Lect. Ser., 3 (International Press, Somerville, MA, 2002), 3–34.Google Scholar
Voevodsky, V., ‘Motivic cohomology with Z/2-coefficients’, Publ. Math. Inst. Hautes Études Sci. 98 (2003), 59–104.CrossRefGoogle Scholar
Voevodsky, V., ‘On motivic cohomology with Z/l-coefficients’, Ann. of Math. (2) 174(1) (2011), 401–438.Google Scholar
Wilson, G. M., ‘The $\unicode[STIX]{x1D702}$ -inverted motivic sphere over the rationals’, Algebr. Geom. Topol. In preparation, arXiv:1708.06523.Google Scholar
Wilson, G. M. and Østvær, P. A., ‘Two-complete stable motivic stems over finite fields’, Algebr. Geom. Topol. 17(2) (2017), 1059–1104.CrossRefGoogle Scholar
Zahler, R., ‘The Adams–Novikov spectral sequence for the spheres’, Ann. of Math. (2) 96 (1972), 480–504.Google Scholar