TY - JOUR

T1 - Dualizing spheres for compact p-adic analytic groups and duality in chromatic homotopy

AU - Beaudry, Agnès

AU - Goerss, Paul G.

AU - Hopkins, Michael J.

AU - Stojanoska, Vesna

N1 - Funding Information:
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.
Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2022/9

Y1 - 2022/9

N2 - 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 En, whose homotopy groups classify deformations of a formal group law of height n, in the implicit characteristic p. It is known that En is self-dual up to a shift; however, that does not fully take into account the action of the automorphism group Gn of the formal group in question. In this paper we find that the Gn-equivariant dual of En is in fact En twisted by a sphere with a non-trivial (when n> 1) action by Gn. This sphere is a dualizing module for the group Gn, and we construct and study such an object IG for any compact p-adic analytic group G. If we restrict the action of G on IG to certain type of small subgroups, we identify IG with a specific representation sphere coming from the Lie algebra of 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 EnhH for select choices of p and n and finite subgroups H of Gn.

AB - 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 En, whose homotopy groups classify deformations of a formal group law of height n, in the implicit characteristic p. It is known that En is self-dual up to a shift; however, that does not fully take into account the action of the automorphism group Gn of the formal group in question. In this paper we find that the Gn-equivariant dual of En is in fact En twisted by a sphere with a non-trivial (when n> 1) action by Gn. This sphere is a dualizing module for the group Gn, and we construct and study such an object IG for any compact p-adic analytic group G. If we restrict the action of G on IG to certain type of small subgroups, we identify IG with a specific representation sphere coming from the Lie algebra of 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 EnhH for select choices of p and n and finite subgroups H of Gn.

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U2 - 10.1007/s00222-022-01120-1

DO - 10.1007/s00222-022-01120-1

M3 - Article

AN - SCOPUS:85131664844

SN - 0020-9910

VL - 229

SP - 1301

EP - 1434

JO - Inventiones Mathematicae

JF - Inventiones Mathematicae

IS - 3

ER -