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

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).

This is a preview of subscription content, log in to check access.

Notes

  1. 1.

    The consequence of this technical assumption is that every thick \(\otimes \)-ideal of \(\mathcal {T} \) is radical, see [7, Rem. 4.3 and Prop. 4.4].

  2. 2.

    Note that when the tensor unit \(\mathbf {1}\in \mathcal {T} \) generates \(\mathcal {T} \) as a thick subcategory, then every thick subcategory of \(\mathcal {T} \) is also a thick \(\otimes \)-ideal. So there is no distinction between thick \(\otimes \)-ideals and thick subcategories of \({\mathrm {Sp}}^\omega \).

  3. 3.

    The fact that determining these inclusions is equivalent to knowing the topology follows from the second sentence of [11, Cor. 8.19].

  4. 4.

    For comparison with [26], observe that \(\varphi ^A(X)=\Phi ^A({\tilde{E}}A\wedge F(EA_+, i_*X))\) are the geometric fixed points of the Tate-construction of the A-spectrum \(i_*X\), where \(i_* X\) is the inflation of X.

  5. 5.

    Since we are now working p-locally, we will omit the second entry in the notation \({{\mathcal {P}}}(A',q,n)\) from the introduction, since q will always be p.

  6. 6.

    The construction of \({\mathcal O}_{{\mathrm {Sp}}\acute{\mathrm{e}}{\mathrm {t}}(R)}\) will partially be recalled during the proof of Lemma 3.11 below.

  7. 7.

    More is true: This is part of an equivalence between p-divisible commutative formal Lie groups over \(\pi _0(E)\) and connected p-divisible groups over \(\pi _0(E)\) [43, Prop. 1].

  8. 8.

    Poincaré duality for K(n)-cohomology makes it clear that the types of a finite complex and its dual agree.

References

  1. 1.

    Arone, G., Dwyer, W.G., Lesh, K.: Bredon homology of partition complexes. Doc. Math. 21, 1227–1268 (2016)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Arone, G., Lesh, K.: Fixed Points of Coisotropic Subgroups of \(\Gamma _{k}\) on Decomposition Spaces (2017). http://front.math.ucdavis.edu/1701.06070

  3. 3.

    Arone, G., Mahowald, M.: The Goodwillie tower of the identity functor and the unstable periodic homotopy of spheres. Invent. Math. 135(3), 743–788 (1999)

    MathSciNet  MATH  Article  Google Scholar 

  4. 4.

    Ando, M., Morava, J., Sadofsky, H.: Completions of \({\mathbf{Z}}/(p)\)-Tate cohomology of periodic spectra. Geom. Topol. 2, 145–174 (1998)

    MathSciNet  MATH  Article  Google Scholar 

  5. 5.

    Arone, G.: Iterates of the suspension map and Mitchell’s finite spectra with \(A_k\)-free cohomology. Math. Res. Lett. 5(4), 485–496 (1998)

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    Balmer, P.: Presheaves of triangulated categories and reconstruction of schemes. Math. Ann. 324(3), 557–580 (2002)

    MathSciNet  MATH  Article  Google Scholar 

  7. 7.

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

    MathSciNet  MATH  Article  Google Scholar 

  8. 8.

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

    MathSciNet  MATH  Article  Google Scholar 

  9. 9.

    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 

  10. 10.

    Benson, D.J., Iyengar, S.B., Krause, H.: Stratifying modular representations of finite groups. Ann. Math. (2) 174(3), 1643–1684 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  11. 11.

    Balmer, P., Sanders, B.: The spectrum of the equivariant stable homotopy category of a finite group. Invent. Math. 208(1), 283–326 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  12. 12.

    Douglas, C.L., Francis, J., Henriques, A.G., Hill, M.A. (eds.): Topological Modular Forms, Volume 201 of Mathematical Surveys and Monographs. American Mathematical Society, Providence (2014)

  13. 13.

    Goerss, P., Henn, H.-W., Mahowald, M., Rezk, C.: A resolution of the \(K(2)\)-local sphere at the prime 3. Ann. Math. (2) 162(2), 777–822 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  14. 14.

    Greenlees, J.P.C., Peter May, J.: Completions in algebra and topology. In: James, I.M. (ed.) Handbook of Algebraic Topology, pp. 255–276. North-Holland, Amsterdam (1995)

  15. 15.

    Greenlees, J.P.C., May, J.P.: Equivariant stable homotopy theory. In: James, I.M. (ed.) Handbook of Algebraic Topology, pp. 277–323. North-Holland, Amsterdam (1995)

  16. 16.

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

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Greenlees, J.P.C., Sadofsky, H.: The Tate spectrum of \(v_n\)-periodic complex oriented theories. Math. Z. 222(3), 391–405 (1996)

    MathSciNet  MATH  Article  Google Scholar 

  18. 18.

    Hahn, J.: On the Bousfield Classes of \(H_\infty \)-Ring Spectra (2016). http://front.math.ucdavis.edu/1612.041386

  19. 19.

    Hill, M.A., Hopkins, M.J., Ravenel, D.C.: On the nonexistence of elements of Kervaire invariant one. Ann. Math. (2) 184(1), 1–262 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  20. 20.

    Hopkins, M.J., Kuhn, N.J., Ravenel, D.C.: Generalized group characters and complex oriented cohomology theories. J. Am. Math. Soc. 13(3), 553–594 (2000)

    MathSciNet  MATH  Article  Google Scholar 

  21. 21.

    Hopkins, M.J., Lurie, J.: Ambidexterity in \(K(n)\)-Local Stable Homotopy Theory (2013). http://www.math.harvard.edu/~lurie/papers/Ambidexterity.pdf

  22. 22.

    Hopkins, M.J.: Global methods in homotopy theory. In: Homotopy Theory-Proceedings Durham Symposium 1985. Cambridge University Press, Cambridge (1987)

  23. 23.

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

    MathSciNet  MATH  Article  Google Scholar 

  24. 24.

    Hopkins, M.J., Smith, J.H.: Nilpotence and stable homotopy theory. II. Ann. Math. (2) 148(1), 1–49 (1998)

    MathSciNet  MATH  Article  Google Scholar 

  25. 25.

    Joachimi, R.: Thick Ideals in Equivariant and Motivic Stable Homotopy Categories (2015). http://front.math.ucdavis.edu/1503.08456

  26. 26.

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

    MathSciNet  MATH  Article  Google Scholar 

  27. 27.

    Laumon, G., Moret-Bailly, L.: Champs algébriques, volume 39 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer, Berlin (2000)

  28. 28.

    Lewis, L.G., May, J.P., Steinberger, M.: Equivariant Stable Homotopy Theory, Volume 1213 of Lecture Notes in Mathematics. Springer, Berlin (1986)

  29. 29.

    Lurie, J.: Chromatic Homotopy Theory (2010). http://www.math.harvard.edu/~lurie/252x.html

  30. 30.

    Lurie, J.: Spectral Schemes (2011). http://www.math.harvard.edu/~lurie/papers/DAG-VII.pdf

  31. 31.

    Lurie, J.: Spectral Algebraic Geometry (2017). http://www.math.harvard.edu/~lurie/papers/SAG-rootfile.pdf

  32. 32.

    Mitchell, S.A.: Finite complexes with \(A(n)\)-free cohomology. Topology 24(2), 227–246 (1985)

    MathSciNet  MATH  Article  Google Scholar 

  33. 33.

    Mathew, A., Meier, L.: Affineness and chromatic homotopy theory. J. Topol. 8(2), 476–528 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  34. 34.

    Mathew, A., Naumann, N., Noel, J.: Derived Induction and Restriction Theory (2015). http://front.math.ucdavis.edu/1507.06867

  35. 35.

    Mathew, A., Naumann, N., Noel, J.: Nilpotence and descent in equivariant stable homotopy theory. Adv. Math. 305, 994–1084 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  36. 36.

    Neeman, A.: The chromatic tower for \(D(R)\). Topology 31(3), 519–532 (1992). With an appendix by Marcel Bökstedt

    MathSciNet  MATH  Article  Google Scholar 

  37. 37.

    Neeman, A: Triangulated Categories, Volume 148 of Annals of Mathematics Studies. Princeton University Press, Princeton (2001)

  38. 38.

    Ravenel, D.C.: Nilpotence and Periodicity in Stable Homotopy Theory, Volume 128 of Annals of Mathematics Studies. Princeton University Press, Princeton (1992). Appendix C by Jeff Smith

  39. 39.

    Rezk, C.: Notes on the Hopkins–Miller theorem. In: Homotopy Theory via Algebraic Geometry and Group Representations (Evanston, IL, 1997), Volume 220 of Contemporary Mathematics, pp. 313–366. American Mathematical Society (1997)

  40. 40.

    Stapleton, N.: Transchromatic generalized character maps. Algebr. Geom. Topol. 13(1), 171–203 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  41. 41.

    Strickland, N.P.: Finite subgroups of formal groups. J. Pure Appl. Algebra 121(2), 161–208 (1997)

    MathSciNet  MATH  Article  Google Scholar 

  42. 42.

    Stroilova, O.: The Generalized Tate Construction (2012). http://web.mit.edu/stroilo/www/main_no_cover.pdf

  43. 43.

    Tate, J.T.: \(p\)-Divisible groups. In: Proceedings of the Conference on Local Fields (Driebergen, 1966), pp. 158–183. Springer, Berlin (1967)

  44. 44.

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

    MathSciNet  MATH  Article  Google Scholar 

  45. 45.

    Verdier, J.-L.: Des catégories dérivées des catégories abéliennes. Astérisque (239), xii+253 (1996). With a preface by Luc Illusie, Edited and with a note by Georges Maltsiniotis (1997)

Download references

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.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Niko Naumann.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Justin Noel was partially supported by the DFG Grants: NO 1175/1-1 and SFB 1085—Higher Invariants, Regensburg. Niko Naumann and Nathaniel Stapleton were also partially supported by the SFB 1085—Higher Invariants, Regensburg. Tobias Barthel and Markus Hausmann were supported by the DNRF92.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Barthel, T., Hausmann, M., Naumann, N. et al. The Balmer spectrum of the equivariant homotopy category of a finite abelian group. Invent. math. 216, 215–240 (2019). https://doi.org/10.1007/s00222-018-0846-5

Download citation