Abstract
We argue that the very effective cover of hermitian K-theory in the sense of motivic homotopy theory is a convenient algebro-geometric generalization of the connective real topological K-theory spectrum. This means the very effective cover acquires the correct Betti realization, its motivic cohomology has the desired structure as a module over the motivic Steenrod algebra, and that its motivic Adams and slice spectral sequences are amenable to calculations.
Similar content being viewed by others
References
Adams, J.F.: Stable Homotopy and Generalised Homology. Chicago Lectures in Mathematics. The University of Chicago Press, Chicago (1974)
Bachmann, T.: The generalized slices of hermitian \(K\)-theory. J. Topol. 10(4), 1124–1144 (2017). https://doi.org/10.1112/topo.12032
Berthelot, P., Jussila, O., Grothendieck, A., Raynaud, M., Kleiman, S., Illusie, L.: Théorie des intersections et théorème de Riemann-Roch. Lecture Notes in Mathematics, vol. 225. Springer, Berlin (1971). Séminaire de Géométrie Algébrique du Bois-Marie 1966–1967 (SGA 6)
Cisinski, D.C.: Descente par éclatements en \(K\)-théorie invariante par homotopie. Ann. Math. (2) 177(2), 425–448 (2013). https://doi.org/10.4007/annals.2013.177.2.2
Greenlees, J.: Equivariant versions of real and complex connective \(K\)-theory. Homol. Homotopy Appl. 7(3), 63–82 (2005). https://doi.org/10.4310/HHA.2005.v7.n3.a5
Guillou, B.J., Hill, M.A., Isaksen, D.C., Ravenel, D.C.: The cohomology of \({C}_2\)-equivariant \(\cal{A}(1)\) and the homotopy of \(ko_{C_2}\). arXiv:1708:09568
Gutiérrez, J.J., Röndigs, O., Spitzweck, M., Østvær, P.: Motivic slices and coloured operads. J. Topol. 5(3), 727–755 (2012). https://doi.org/10.1112/jtopol/jts015
Heller, J., Ormsby, K.: Galois equivariance and stable motivic homotopy theory. Trans. Am. Math. Soc. 368(11), 8047–8077 (2016). https://doi.org/10.1090/tran6647
Hill, M.A.: Ext and the motivic Steenrod algebra over \(\mathbb{R}\). J. Pure Appl. Algebra 215(5), 715–727 (2011). https://doi.org/10.1016/j.jpaa.2010.06.017
Hornbostel, J.: \(\mathbb{A}^1\)-representability of hermitian \(K\)-theory and Witt groups. Topology 44(3), 661–687 (2005). https://doi.org/10.1016/j.top.2004.10.004
Hoyois, M.: From algebraic cobordism to motivic cohomology. J. Reine Angew. Math. 702, 173–226 (2015). https://doi.org/10.1515/crelle-2013-0038
Hoyois, M., Kelly, S., Østvær, P.: The motivic Steenrod algebra in positive characteristic. J. Eur. Math. Soc. (JEMS) 19(12), 3813–3849 (2017)
Isaksen, D.C., Shkembi, A.: Motivic connective \(K\)-theories and the cohomology of A(1). J. K-Theory 7(3), 619–661 (2011). https://doi.org/10.1017/is011004009jkt154
Karoubi, M.: Théorie de Quillen et homologie du groupe orthogonal. Ann. Math. (2) 112(2), 207–257 (1980). https://doi.org/10.2307/1971326
Kerz, M., Strunk, F.: On the vanishing of negative homotopy \(K\)-theory. J. Pure Appl. Algebra 221(7), 1641–1644 (2017). https://doi.org/10.1016/j.jpaa.2016.12.021
Levine, M.: The homotopy coniveau tower. J. Topol. 1(1), 217–267 (2008). https://doi.org/10.1112/jtopol/jtm004
Morel, F.: On the motivic \(\pi _0\) of the sphere spectrum. In: Axiomatic, enriched and motivic homotopy theory, NATO Sci. Ser. II Math. Phys. Chem., vol. 131, pp. 219–260. Kluwer, Dordrecht (2004)
Morel, F.: The stable \({\mathbb{A}}^{1}\)-connectivity theorems. \(K\)-Theory 35(1–2), 1–68 (2005). https://doi.org/10.1007/s10977-005-1562-7
Morel, F.: \(\mathbb{A}^1\)-Algebraic Topology Over a Field. Lecture Notes in Mathematics, vol. 2052. Springer, Heidelberg (2012)
Naumann, N., Spitzweck, M., Østvær, P.: Motivic Landweber exactness. Doc. Math. 14, 551–593 (2009)
Panin, I., Pimenov, K., Röndigs, O.: On Voevodsky’s algebraic \(K\)-theory spectrum. In: Algebraic Topology, Abel Symp., vol. 4, pp. 279–330. Springer, Berlin (2009)
Röndigs, O., Østvær, P.: The multiplicative structure on the slices of hermitian \({K}\)-theory and Witt-theory. Homol. Homotopy Appl. 18(1), 373–380 (2016). https://doi.org/10.4310/HHA.2016.v18.n1.a20
Röndigs, O., Østvær, P.: Slices of hermitian \({K}\)-theory and Milnor’s conjecture on quadratic forms. Geom. Topol. 20(2), 1157–1212 (2016)
Röndigs, O., Spitzweck, M., Østvær, P.: The motivic Hopf map solves the homotopy limit problem for \(K\)-theory. Doc. Math. 23, 1405–1424 (2018)
Röndigs, O., Spitzweck, M., Østvær, P.: The first stable homotopy groups of motivic spheres. Ann. Math. (2) 189(1), 1–74 (2019). https://doi.org/10.4007/annals.2019.189.1.1
Spitzweck, M.: Relations between slices and quotients of the algebraic cobordism spectrum. Homol. Homotopy Appl. 12(2), 335–351 (2010)
Spitzweck, M.: A commutative \({\bf P}^1\)-spectrum representing motivic cohomology over Dedekind domains. Mém. Soc. Math. Fr. Nouv. Sér. 157, 110 (2018)
Spitzweck, M., Østvær, P.A.: Motivic twisted \(K\)-theory. Algebr. Geom. Topol. 12(1), 565–599 (2012). https://doi.org/10.2140/agt.2012.12.565
Voevodsky, V.: \(\mathbf{A}^1\)-homotopy theory. In: Proceedings of the International Congress of Mathematicians, vol. I (Berlin, 1998), Extra Vol. I, pp. 579–604 (1998) (electronic)
Voevodsky, V.: Open problems in the motivic stable homotopy theory. I. In: Motives, Polylogarithms and Hodge Theory, Part I (Irvine, CA, 1998), Int. Press Lect. Ser., vol. 3, pp. 3–34. Int. Press, Somerville (2002)
Voevodsky, V.: A possible new approach to the motivic spectral sequence for algebraic \(K\)-theory. In: Recent Progress in Homotopy Theory (Baltimore, MD, 2000), Contemp. Math., vol. 293, pp. 371–379. Amer. Math. Soc., Providence (2002)
Voevodsky, V.: Reduced power operations in motivic cohomology. Publ. Math. Inst. Hautes Études Sci. 98, 1–57 (2003). https://doi.org/10.1007/s10240-003-0009-z
Acknowledgements
Mark Behrens asked about a motivic version of \(\mathrm {ko}\) during Bert Guillou’s ECHT talk in Spring 2017. His question prompted a first version of this paper. We also acknowledge Tom Bachmann’s related work in [2]. Theorem 17 was first obtained in a different way during the first author’s visit to Universität Osnabrück in November 2016. Work on this paper took place at Institut Mittag-Leffler in Spring 2017, where the first author held a postdoctoral fellowship financed by Vergstiftelsen, and the Hausdorff Research Institute for Mathematics in Summer 2017. We thank both institutions for excellent working conditions and support. The authors acknowledge support from the RCN programme “Motivic Hopf equations”. Ananyevskiy is supported by RFBR Grants 15-01-03034 and 16-01-00750, and by “Native towns”, a social investment program of PJSC “Gazprom Neft”. Röndigs receives support from the DFG priority programme “Homotopy theory and algebraic geometry”. Østvær is supported by a Friedrich Wilhelm Bessel Research Award from the Humboldt Foundation. Finally, we express our gratitude to the referee for helpful comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Ananyevskiy, A., Röndigs, O. & Østvær, P.A. On very effective hermitian K-theory. Math. Z. 294, 1021–1034 (2020). https://doi.org/10.1007/s00209-019-02302-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-019-02302-z