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

• 成果类型: Article
• 署名作者: Beaudry, Agnes; Goerss, Paul G.; Hopkins, Michael J.; Stojanoska, Vesna
• 署名单位: University of Colorado System; University of Colorado Boulder; Northwestern University; Harvard University; University of Illinois System; University of Illinois Urbana-Champaign
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-022-01120-1
• 发表日期: 2022
• 页码: 1301-1434
• 关键词: fixed-point spectra; k(2)-local sphere; picard-groups; conjecture; SUBGROUPS; ALGEBRAS; SPACE

摘要

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 G(n) of the formal group in question. In this paper we find that the 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 G(n). This sphere is a dualizing module for the group G(n), and we construct and study such an object I-G for any compact p-adic analytic group g. If we restrict the action of G on I-G to certain type of small subgroups, we identify I-G 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 E-n(hH) for select choices of p and n and finite subgroups H of G(n).