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

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 \({\mathbb {G}}_n\).

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Fig. 1

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 \(H^*(BSU(2),{{{\mathbb {Z}}}}) \cong H^*(BU(1),{{{\mathbb {Z}}}})^{C_2}\).

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Correspondence to Vesna Stojanoska.

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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.

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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

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Mathematics Subject Classification

  • 55P92
  • 55R25
  • 55R50
  • 55U30
  • 20E18
  • 22E41