Skip to main content
SpringerLink
Log in

Dualizing spheres for compact p-adic analytic groups and duality in chromatic homotopy

  • Published:
Inventiones mathematicae Aims and scope

Abstract

The primary goal of this paper is to study Spanier–Whitehead duality in the K(n)-local category. One of the key players in the K(n)-local category is the Lubin–Tate spectrum \(E_n\), whose homotopy groups classify deformations of a formal group law of height n, in the implicit characteristic p. It is known that \(E_n\) is self-dual up to a shift; however, that does not fully take into account the action of the automorphism group \({\mathbb {G}}_n\) of the formal group in question. In this paper we find that the \({\mathbb {G}}_n\)-equivariant dual of \(E_n\) is in fact \(E_n\) twisted by a sphere with a non-trivial (when \(n>1\)) action by \({\mathbb {G}}_n\). This sphere is a dualizing module for the group \({\mathbb {G}}_n\), and we construct and study such an object \(I_{{\mathcal {G}}}\) for any compact p-adic analytic group \({\mathcal {G}}\). If we restrict the action of \({\mathcal {G}}\) on \(I_{{\mathcal {G}}}\) to certain type of small subgroups, we identify \(I_{{\mathcal {G}}}\) with a specific representation sphere coming from the Lie algebra of \({\mathcal {G}}\). This is done by a classification of p-complete sphere spectra with an action by an elementary abelian p-group in terms of characteristic classes, and then a specific comparison of the characteristic classes in question. The setup makes the theory quite accessible for computations, as we demonstrate in the later sections of this paper, determining the K(n)-local Spanier–Whitehead duals of \(E_n^{hH}\) for select choices of p and n and finite subgroups H of \({\mathbb {G}}_n\).

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1

Similar content being viewed by others

Notes

  1. In later developments, an action of F on X is defined to be a map from BF to the classifying space of self-equivalences of X. The connection to our definition can be found in Proposition 4.2.4.4 of [51]; see also Remark 1.2.6.2 of the same source.

  2. What can be proved easily from [34] is that \(D_*(H_*X^{hG}) \cong [D_*H_*X]^G\) where \(D_*(-)\) is the graded Dieudonné module functor. The assertion here can be deduced from that.

  3. In fact, the same source explains that \(R^h\) has more equivariant multiplicative structure, but we don’t need that here.

  4. The map \(U(1) \rightarrow SU(2)\) is the inclusion of the maximal torus. The Weyl group is \(C_2\) and we have explicitly written down the canonical isomorphism \(H^*(BSU(2),{{{\mathbb {Z}}}}) \cong H^*(BU(1),{{{\mathbb {Z}}}})^{C_2}\).

References

  1. Adams, J.F.: Lectures on generalised cohomology. In: Category Theory, Homology Theory and their Applications, III (Battelle Institute Conference, Seattle, Washington, 1968, Vol. 3), pp. 1–138. Springer, Berlin (1969)

  2. Ando, M, Hopkins, M.J., Rezk, C.: Multiplicative orientations of \(\mathit{KO}\)-theory and of the spectrum of topological modular forms. http://www.math.uiuc.edu/~mando/papers/koandtmf.pdf

  3. Barwick, C.: Spectral Mackey functors and equivariant algebraic \(K\)-theory (I). Adv. Math. 304, 646–727 (2017). https://doi.org/10.1016/j.aim.2016.08.043

    Article  MathSciNet  MATH  Google Scholar 

  4. Barthel, T., Beaudry, A., Goerss, P.G., Stojanoska, V.: Constructing the determinant sphere using a Tate twist. Math. Z. (2021). https://doi.org/10.1007/s00209-021-02864-x

    Article  MATH  Google Scholar 

  5. Beaudry, A., Bobkova, I., Hill, M., Stojanoska, V.: Invertible \(K(2)\)-local \(E\)-modules in \(C_4\)-spectra. Algebr. Geom. Topol. 20(7), 3423–3503 (2020). https://doi.org/10.2140/agt.2020.20.3423

    Article  MathSciNet  MATH  Google Scholar 

  6. Barthel, T., Beaudry, A., Stojanoska, V.: Gross–Hopkins duals of higher real \(K\)-theory spectra. Trans. Am. Math. Soc. 372(5), 3347–3368 (2019). https://doi.org/10.1090/tran/7730

    Article  MathSciNet  MATH  Google Scholar 

  7. Behrens, M., Davis, D.G.: The homotopy fixed point spectra of profinite Galois extensions. Trans. Am. Math. Soc. 362(9), 4983–5042 (2010). https://doi.org/10.1090/S0002-9947-10-05154-8

    Article  MathSciNet  MATH  Google Scholar 

  8. Beaudry, A.: Towards the homotopy of the \(K(2)\)-local Moore spectrum at \(p=2\). Adv. Math. 306, 722–788 (2017). https://doi.org/10.1016/j.aim.2016.10.020

    Article  MathSciNet  MATH  Google Scholar 

  9. Behrens, M.: A modular description of the K(2)-local sphere at the prime 3. Topology 45(2), 343–402 (2006). https://doi.org/10.1016/j.top.2005.08.005

  10. Bobkova, I., Goerss, P.G.: Topological resolutions in K(2)-local homotopy theory at the prime 2. J. Top. 11, 918–957 (2018). https://doi.org/10.1112/topo.12076

    Article  MathSciNet  MATH  Google Scholar 

  11. Barwick, C., Glasman, S., Shah, J.: Spectral Mackey functors and equivariant algebraic \(K\)-theory. II. Tunis. J. Math. 2(1), 97–146 (2020). https://doi.org/10.2140/tunis.2020.2.97

    Article  MathSciNet  MATH  Google Scholar 

  12. Blumberg, A.J., Hill, M.A.: Operadic multiplications in equivariant spectra, norms, and transfers. Adv. Math. 285, 658–708 (2015). https://doi.org/10.1016/j.aim.2015.07.013

    Article  MathSciNet  MATH  Google Scholar 

  13. Blumberg, A.J., Hill, M.A.: \(G\)-symmetric monoidal categories of modules over equivariant commutative ring spectra. Tunis. J. Math. 2(2), 237–286 (2020). https://doi.org/10.2140/tunis.2020.2.237

    Article  MathSciNet  MATH  Google Scholar 

  14. Bobkova, I.: Spanier–Whitehead duality in the \(K(2)\)-local category at \(p=2\). Proc. Am. Math. Soc. 148(12), 5421–5436 (2020). https://doi.org/10.1090/proc/15078

    Article  MathSciNet  MATH  Google Scholar 

  15. Bousfield, A.K.: The localization of spectra with respect to homology. Topology 18(4), 257–281 (1979). https://doi.org/10.1016/0040-9383(79)90018-1

    Article  MathSciNet  MATH  Google Scholar 

  16. Bredon, G.E.: Introduction to Compact Transformation Groups. Pure and Applied Mathematics, Vol. 46. Academic Press, New York (1972)

  17. Bujard, C.: Finite subgroups of extended Morava stabilizer groups. arXiv:1206.1951 [math.AT]

  18. Gunnar Carlsson, G.B.: Segal’s burnside ring conjecture for \(({\bf Z}/2)^{k}\). Topology 22(1), 83–103 (1983). https://doi.org/10.1016/0040-9383(83)90046-0

    Article  MathSciNet  MATH  Google Scholar 

  19. Castellana, N.: On the \(p\)-compact groups corresponding to the \(p\)-adic reflection groups \(G(q, r, n)\). Trans. Am. Math. Soc. 358(7), 2799–2819 (2006). https://doi.org/10.1090/S0002-9947-06-04154-7

    Article  MathSciNet  MATH  Google Scholar 

  20. Clausen, D.: \(p\)-adic \(J\)-homomorphisms and a product formula. arXiv:1110.5851 [math.AT]

  21. Clausen, D.: Atiyah duality for \(p\)-adic lie groups. vol. 36, pp. 20–22. 2019. Abstracts from the workshop held August 4–10 (2019), Organized by Jesper Grodal, Michael Hill, and Birgit Richter, Oberwolfach Reports. https://www.mfo.de/occasion/1932/www_view. https://doi.org/10.4171/OWR/2019/36

  22. Cohen, F.R., Lada, T.J., May, J.P.: The Homology of Iterated Loop Spaces. Lecture Notes in Mathematics, Vol. 533. Springer, Berlin (1976)

  23. Cooke, G.: Replacing homotopy actions by topological actions. Trans. Am. Math. Soc. 237, 391–406 (1978). https://doi.org/10.2307/1997628

    Article  MathSciNet  MATH  Google Scholar 

  24. Dror, E., Dwyer, W.G., Kan, D.M.: Automorphisms of fibrations. Proc. Am. Math. Soc. 80(3), 491–494 (1980). https://doi.org/10.2307/2043747

    Article  MathSciNet  MATH  Google Scholar 

  25. Dixon, J.D., du Sautoy, M.P.F., Mann, A., Segal, D.: Analytic Pro-\(p\)-Groups. London Mathematical Society Lecture Note Series, vol. 157. Cambridge University Press, Cambridge (1991)

  26. Devinatz, E.S., Hopkins, M.J.: The action of the Morava stabilizer group on the Lubin–Tate moduli space of lifts. Am. J. Math. 117(3), 669–710 (1995). https://doi.org/10.2307/2375086

    Article  MathSciNet  MATH  Google Scholar 

  27. Devinatz, E.S., Hopkins, M.J.: Homotopy fixed point spectra for closed subgroups of the Morava stabilizer groups. Topology 43(1), 1–47 (2004)

    Article  MathSciNet  Google Scholar 

  28. Elmendorf, A.D., May, J.P.: Algebras Over Equivariant Sphere Spectra, vol. 116, pp. 139–149. (1997). Special Volume on the Occasion of the 60th Birthday of Professor Peter J. Freyd. https://doi.org/10.1016/S0022-4049(96)00080-1

  29. Fausk, H., Hu, P., May, J.P.: Isomorphisms between left and right adjoints. Theory Appl. Categ. 11(4), 107–131 (2003)

    MathSciNet  MATH  Google Scholar 

  30. Goerss, P.G., Hopkins, M.J.: Moduli spaces of commutative ring spectra. In: Structured Ring Spectra, Volume 315 of London Mathematics Society Lecture Note Series, pp. 151–200. Cambridge University Press, Cambridge (2004). https://doi.org/10.1017/CBO9780511529955.009

  31. 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). https://doi.org/10.4007/annals.2005.162.777

    Article  MathSciNet  MATH  Google Scholar 

  32. Goerss, P., Henn, H.-W., Mahowald, M., Rezk, C.: On Hopkins’ Picard groups for the prime 3 and chromatic level 2. J. Topol. 8(1), 267–294 (2015). https://doi.org/10.1112/jtopol/jtu024

    Article  MathSciNet  MATH  Google Scholar 

  33. Guillou, B., May, J.P.: Models of \(G\)-spectra as presheaves of spectra. arXiv e-prints arXiv:1110.3571 (2011)

  34. Goerss, P.G.: The homology of homotopy inverse limits. J. Pure Appl. Algebra 111(1–3), 83–122 (1996). https://doi.org/10.1016/0022-4049(95)00112-3

    Article  MathSciNet  MATH  Google Scholar 

  35. Grodal, J.: Algebraic Models for Finite \({G}\)-Spaces, vol. 4, pp. 2690–2692 (2007). Abstracts from the workshop held September 16–22 (2007), Organized by Paul Goerss, John Greenlees and Stefan Schwede, Oberwolfach Reports. Vol. 4, no. 4. https://doi.org/10.1287/moor.4.4.390

  36. Greenlees, J.P.C., Sadofsky, H.: The Tate spectrum of \(v_n\)-periodic complex oriented theories. Math. Z. 222(3), 391–405 (1996). https://doi.org/10.1007/PL00004264

    Article  MathSciNet  Google Scholar 

  37. Henn, H.-W.: On finite resolutions of \(K(n)\)-local spheres. In: Elliptic Cohomology, volume 342 of London Mathematics Society Lecture Note Series, pp. 122–169. Cambridge University Press, Cambridge (2007). https://doi.org/10.1017/CBO9780511721489.008

  38. Hewett, T.: Finite subgroups of division algebras over local fields. J. Algebra 173(3), 518–548 (1995). https://doi.org/10.1006/jabr.1995.1101

    Article  MathSciNet  MATH  Google Scholar 

  39. Hopkins, M.J., Gross, B.H.: The rigid analytic period mapping, Lubin–Tate space, and stable homotopy theory. Bull. Am. Math. Soc. (N.S.) 30(1), 76–86 (1994). https://doi.org/10.1090/S0273-0979-1994-00438-0

    Article  MathSciNet  MATH  Google Scholar 

  40. Heard, D., Li, G., Shi, X.D.: Picard groups and duality for real Morava \(E\)-theories. Algebr. Geom. Topol. 21(6), 2703–2760. https://doi.org/10.2140/agt.2021.21.2703

  41. Hill, M.A., Meier, L.: The \(C_2\)-spectrum \({{\rm Tmf}}_1(3)\) and its invertible modules. Algebr. Geom. Topol. 17(4), 1953–2011 (2017). https://doi.org/10.2140/agt.2017.17.1953

    Article  MathSciNet  MATH  Google Scholar 

  42. Heard, D., Mathew, A., Stojanoska, V.: Picard groups of higher real \(K\)-theory spectra at height \(p-1\). Compos. Math. 153(9), 1820–1854 (2017). https://doi.org/10.1112/S0010437X17007242

    Article  MathSciNet  MATH  Google Scholar 

  43. Hopkins, M.J.: Equivariant dual of Morava \(E\)-theory. Homotopy Theory: Tools and Applications. https://www.youtube.com/playlist?list=PLEdRA1j39FG6mbWyYVendxnE-lfQ3K9a6 (2017)

  44. Hopkins, M.J., Smith, J.H.: Nilpotence and stable homotopy theory. II. Ann. Math. (2) 148(1), 1–49 (1998). https://doi.org/10.2307/120991

    Article  MathSciNet  MATH  Google Scholar 

  45. Hovey, M., Strickland, N.P.: Morava \(K\)-theories and localisation. Mem. Am. Math. Soc. 139(666), viii+100 (1999). https://doi.org/10.1090/memo/0666

    Article  MathSciNet  MATH  Google Scholar 

  46. Hahn, J., Shi, X.L.D.: Real orientations of Lubin–Tate spectra. Invent. Math. 221(3), 731–776 (2020). https://doi.org/10.1007/s00222-020-00960-z

    Article  MathSciNet  MATH  Google Scholar 

  47. Klein, J.R.: The dualizing spectrum of a topological group. Math. Ann. 319(3), 421–456 (2001). https://doi.org/10.1007/PL00004441

    Article  MathSciNet  MATH  Google Scholar 

  48. Lannes, J.: Sur les espaces fonctionnels dont la source est le classifiant d’un \(p\)-groupe abélien élémentaire. Inst. Hautes Études Sci. Publ. Math. 75, 135–244 (1992). With an appendix by Michel Zisman. http://www.numdam.org/item?id=PMIHES_1992__75__135_0

  49. Lazard, M.: Groupes analytiques \(p\)-adiques. Inst. Hautes Études Sci. Publ. Math. 26, 389–603 (1965)

  50. Lewis, L.G., May, J.P., Steinberger, M., McClure, J.E.: Equivariant Stable Homotopy Theory, Volume 1213 of Lecture Notes in Mathematics. Springer, Berlin: With contributions by. J. E. McClure (1986). https://doi.org/10.1007/BFb0075778

  51. Lurie, J.: Higher Topos Theory, Volume 170 of Annals of Mathematics Studies. Princeton University Press, Princeton (2009). https://doi.org/10.1515/9781400830558

  52. Lannes, J., Zarati, S.: Sur les \({\mathscr {U}}\)-injectifs. Ann. Sci. École Norm. Sup. (4) 19(2), 303–333 (1986)

  53. May, J.P.: Equivariant Homotopy and Cohomology Theory, Volume 91 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI (1996). With contributions by M. Cole, G. Comezaña, S. Costenoble, A. D. Elmendorf, J. P. C. Greenlees, L. G. Lewis, Jr., R. J. Piacenza, G. Triantafillou, and S. Waner. https://doi.org/10.1090/cbms/091

  54. Miller, H.: The Sullivan conjecture on maps from classifying spaces. Ann. Math. (2) 120(1), 39–87 (1984). https://doi.org/10.2307/2007071

    Article  MathSciNet  MATH  Google Scholar 

  55. Madsen, Ib., Milgram, R.J.: The Classifying Spaces for Surgery and Cobordism of Manifolds, Volume 92 of Annals of Mathematics Studies. Princeton University Press, Princeton (1979)

  56. Morava, J.: Noetherian localisations of categories of cobordism comodules. Ann. Math. (2) 121(1), 1–39 (1985). https://doi.org/10.2307/1971192

    Article  MathSciNet  MATH  Google Scholar 

  57. Milnor, J.W., Stasheff, J.D.: Characteristic Classes. Annals of Mathematics Studies, No. 76. Princeton University Press, Princeton (1974)

  58. Mathew, A., Stojanoska, V.: The Picard group of topological modular forms via descent theory. Geom. Topol. 20(6), 3133–3217 (2016). https://doi.org/10.2140/gt.2016.20.3133

    Article  MathSciNet  MATH  Google Scholar 

  59. Pstrągowski, P.: Chromatic homotopy theory is algebraic when \(p>n^2+n+1\). Adv. Math. 391, Paper No. 107958 (2021). https://doi.org/10.1016/j.aim.2021.107958

  60. 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, Providence (1998). https://doi.org/10.1090/conm/220/03107

  61. Rognes, J.: Galois extensions of structured ring spectra. Stably dualizable groups. Mem. Am. Math. Soc. 192(898), viii+137 (2008). https://doi.org/10.1090/memo/0898

    Article  MathSciNet  MATH  Google Scholar 

  62. Serre, J.-P.: Galois Cohomology. Springer, Berlin (1997). Translated from the French by Patrick Ion and revised by the author. https://doi.org/10.1007/978-3-642-59141-9

  63. Strickland, N.P.: Gross–Hopkins duality. Topology 39(5), 1021–1033 (2000). https://doi.org/10.1016/S0040-9383(99)00049-X

    Article  MathSciNet  MATH  Google Scholar 

  64. Symonds, P., Weigel, T.: Cohomology of \(p\)-adic analytic groups. In: New Horizons in Pro-\(p\) Groups, Volume 184 of Program in Mathematics, pp. 349–410. Birkhäuser, Boston (2000)

  65. Tsuchiya, A.: Characteristic classes for spherical fiber spaces. Nagoya Math. J. 43:1–39 (1971)

  66. Washington, L.C.: Elliptic Curves. Discrete Mathematics and Its Applications (Boca Raton). Chapman & Hall/CRC, Boca Raton (2003). Number Theory and Cryptography

  67. Wu, W.-T.: On Pontrjagin classes II. Acta Math. Sin. 4, 171–199 (1954)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Colorado Boulder, Boulder, USA

    Agnès Beaudry

  2. Department of Mathematics, Northwestern University, Evanston, USA

    Paul G. Goerss

  3. Department of Mathematics, Harvard University, Cambridge, USA

    Michael J. Hopkins

  4. Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, USA

    Vesna Stojanoska

Authors
  1. Agnès Beaudry
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Paul G. Goerss
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Michael J. Hopkins
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Vesna Stojanoska
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vesna Stojanoska.

Additional information

Publisher's Note

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

This material is based upon work supported by the National Science Foundation under grants No. DMS–1510417, DMS–1812122 and DMS–1906227. The authors also thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the program Homotopy Harnessing Higher Structures. This work was supported by EPSRC Grant Number EP/R014604/1.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Beaudry, A., Goerss, P.G., Hopkins, M.J. et al. Dualizing spheres for compact p-adic analytic groups and duality in chromatic homotopy. Invent. math. 229, 1301–1434 (2022). https://doi.org/10.1007/s00222-022-01120-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00222-022-01120-1

Mathematics Subject Classification

Search

Navigation