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The spectrum of the equivariant stable homotopy category of a finite group

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We study the spectrum of prime ideals in the tensor-triangulated category of compact equivariant spectra over a finite group. We completely describe this spectrum as a set for all finite groups. We also make significant progress in determining its topology and obtain a complete answer for groups of square-free order. For general finite groups, we describe the topology up to an unresolved indeterminacy, which we reduce to the case of p-groups. We then translate the remaining unresolved question into a new chromatic blue-shift phenomenon for Tate cohomology. Finally, we draw conclusions on the classification of thick tensor ideals.

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Acknowledgments

We are very grateful to Neil Strickland, for the reasons explained above. We also thank John Greenlees and Mike Hill for several stimulating discussions.

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Authors and Affiliations

  1. Mathematics Department, UCLA, Los Angeles, CA, 90095-1555, USA

    Paul Balmer

  2. Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100, Copenhagen Ø, Denmark

    Beren Sanders

Authors
  1. Paul Balmer
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  2. Beren Sanders
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Correspondence to Paul Balmer.

Additional information

P. Balmer: Supported by NSF Grant DMS-1303073.

B. Sanders: Supported by the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92).

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Balmer, P., Sanders, B. The spectrum of the equivariant stable homotopy category of a finite group. Invent. math. 208, 283–326 (2017). https://doi.org/10.1007/s00222-016-0691-3

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  • DOI: https://doi.org/10.1007/s00222-016-0691-3

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