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The Balmer spectrum of the equivariant homotopy category of a finite abelian group

  • Tobias Barthel
  • Markus Hausmann
  • Niko Naumann
  • Thomas Nikolaus
  • Justin Noel
  • Nathaniel Stapleton
Article
  • 135 Downloads

Abstract

For a finite abelian group A, we determine the Balmer spectrum of \({\mathrm {Sp}}_A^{\omega }\), the compact objects in genuine A-spectra. This generalizes the case \(A={\mathbb {Z}}/p{\mathbb {Z}}\) due to Balmer and Sanders (Invent Math 208(1):283–326, 2017), by establishing (a corrected version of) their \(\hbox {log}_p\)-conjecture for abelian groups. We also work out the consequences for the chromatic type of fixed-points and establish a generalization of Kuhn’s blue-shift theorem for Tate-constructions (Kuhn in Invent Math 157(2):345–370, 2004).

Notes

Acknowledgements

Markus Hausmann thanks Gregory Arone for helpful conversations on the subject of this paper. Niko Naumann thanks Neil Strickland for making unpublished notes of his available. We thank Paul Balmer, Beren Sanders, and the anonymous referee for pointing out inaccuracies in a preliminary draft of this paper. This work first began while Tobias Barthel, Thomas Nikolaus, and Nathaniel Stapleton were in Bonn and they thank the MPIM for its hospitality.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Tobias Barthel
    • 1
  • Markus Hausmann
    • 1
  • Niko Naumann
    • 2
  • Thomas Nikolaus
    • 3
  • Justin Noel
    • 4
  • Nathaniel Stapleton
    • 5
  1. 1.University of CopenhagenCopenhagenDenmark
  2. 2.NWF I - MathematikUniversity of RegensburgRegensburgGermany
  3. 3.University of MünsterMünsterGermany
  4. 4.Booking.comAmsterdamNetherlands
  5. 5.University of KentuckyLexingtonUSA

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