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

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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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References

  1. Balmer, P.: The spectrum of prime ideals in tensor triangulated categories. J. Reine Angew. Math. 588, 149–168 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Balmer, P.: Spectra, spectra, spectra—tensor triangular spectra versus Zariski spectra of endomorphism rings. Algebric Geom. Topol. 10(3), 1521–1563 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Balmer, P.: Tensor triangular geometry. In: International Congress of Mathematicians, Hyderabad (2010), vol. II, pp. 85–112. Hindustan Book Agency (2010)

  4. Balmer, P.: Separability and triangulated categories. Adv. Math. 226(5), 4352–4372 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Balmer, P.: Separable extensions in tensor-triangular geometry and generalized Quillen stratification, p. 17 (2013). arXiv:1309.1808 [To appear in Ann. Sci. Éc. Norm. Supér. (4)] (preprint)

  6. Balmer, P.: Splitting tower and degree of tt-rings. Algebra Number Theory 8(3), 767–779 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Benson, D.J., Carlson, J.F., Rickard, J.: Thick subcategories of the stable module category. Fund. Math. 153(1), 59–80 (1997)

    MathSciNet  MATH  Google Scholar 

  8. Balmer, P., Dell’Ambrogio, I., Sanders, B.: Restriction to finite-index subgroups as étale extensions in topology, KK-theory and geometry. Algebraic Geom. Topol. 15(5), 3025–3047 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  9. Balmer, P., Dell’Ambrogio, I., Sanders, B.: Grothendieck-Neeman duality and the Wirth müller isomorphism. Compos. Math. 152(8), 1740–1776 (2016). doi:10.1112/S0010437X16007375

    Article  MathSciNet  Google Scholar 

  10. Balmer, P., Favi, G.: Generalized tensor idempotents and the telescope conjecture. Proc. Lond. Math. Soc. 102(6), 1161–1185 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Devinatz, E.S., Hopkins, M.J., Smith, J.H.: Nilpotence and stable homotopy theory. I. Ann. Math. 128(2), 207–241 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dress, A.: A characterisation of solvable groups. Math. Z. 110, 213–217 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  13. Friedlander, E.M., Pevtsova, J.: \(\Pi \)-supports for modules for finite group schemes. Duke Math. J. 139(2), 317–368 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Greenlees, J.P.C., May, J.P.: Generalized Tate cohomology. Mem. Am. Math. Soc. 113(543), viii+178 (1995)

  15. Greenlees, J.P.C.: Tate cohomology in axiomatic stable homotopy theory. In: Cohomological methods in homotopy theory (Bellaterra 1998), vol. 196 of Progr. Math., pp. 149–176. Birkhäuser, Basel (2001)

  16. Hill, M.A., Hopkins, M.J., Ravenel, D.C.: On the nonexistence of elements of Kervaire invariant one. Ann. Math. 184(1), 1–262 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hovey, M., Palmieri, J.H., Strickland, N.P.: Axiomatic stable homotopy theory. Mem. Am. Math. Soc. 128(610) (1997)

  18. Hovey, M., Sadofsky, H.: Tate cohomology lowers chromatic Bousfield classes. Proc. Am. Math. Soc. 124(11), 3579–3585 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hopkins, M.J., Smith, J.H.: Nilpotence and stable homotopy theory. II. Ann. Math. 148(1), 1–49 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  20. Joachimi, R.: Thick ideals in equivariant and motivic stable homotopy categories, p. 115 (2015). arXiv:1503.08456 (preprint)

  21. Kuhn, N.J.: Tate cohomology and periodic localization of polynomial functors. Invent. Math. 157(2), 345–370 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  22. Lewis, L.G., Jr., May, J.P., Steinberger, M.: Equivariant stable homotopy theory, vol. 1213 of Lecture Notes in Mathematics. Springer, Berlin (1986) (With contributions by J. E. McClure)

  23. Mandell, M.A., May, J.P.: Equivariant orthogonal spectra and \(S\)-modules. Mem. Am. Math. Soc. 159(755), x+108 (2002)

  24. Neeman, A.: The connection between the \(K\)-theory localization theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel. Ann. Sci. École Norm. Sup. 25(5), 547–566 (1992)

    MathSciNet  MATH  Google Scholar 

  25. Strickland, N.P.: Thick ideals of finite G-spectra. Unpublished notes (2012)

  26. Thomason, R.W.: The classification of triangulated subcategories. Compos. Math. 105(1), 1–27 (1997)

    Article  MathSciNet  MATH  Google Scholar 

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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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