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

Abstract

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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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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