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Dualizing spheres for compact p-adic analytic groups and duality in chromatic homotopy.
- Source :
- Inventiones Mathematicae; Sep2022, Vol. 229 Issue 3, p1301-1434, 134p
- Publication Year :
- 2022
-
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 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 . [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 00209910
- Volume :
- 229
- Issue :
- 3
- Database :
- Complementary Index
- Journal :
- Inventiones Mathematicae
- Publication Type :
- Academic Journal
- Accession number :
- 158366365
- Full Text :
- https://doi.org/10.1007/s00222-022-01120-1