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On very effective hermitian K-theory

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.

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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.

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Correspondence to Oliver Röndigs.

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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

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  • DOI: https://doi.org/10.1007/s00209-019-02302-z

Keywords

  • Hermitian K-theory
  • \(\mathbf {A}^1\)-homotopy theory
  • Slice filtration

Mathematics Subject Classification

  • 14F42