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Invertible $K(2)$–local $E$–modules in $C_4$–spectra

Agnès Beaudry, Irina Bobkova, Michael Hill and Vesna Stojanoska

Algebraic & Geometric Topology 20 (2020) 3423–3503

We compute the Picard group of the category of K(2)–local module spectra over the ring spectrum EhC4, where E is a height 2 Morava E–theory and C4 is a subgroup of the associated Morava stabilizer group. This group can be identified with the Picard group of K(2)–local E–modules in genuine C4–spectra. We show that in addition to a cyclic subgroup of order 32 generated by E S1, the Picard group contains a subgroup of order 2 generated by E S7+σ, where σ is the sign representation of the group C4. In the process, we completely compute the RO(C4)–graded Mackey functor homotopy fixed point spectral sequence for the C4–spectrum E.

chromatic homotopy theory, Morava E–theory, Picard groups, higher real K–theory
Mathematical Subject Classification 2010
Primary: 55P42, 55Q91
Secondary: 20J06, 55M05, 55P60, 55Q51
Received: 14 January 2019
Revised: 16 September 2019
Accepted: 22 October 2019
Published: 29 December 2020
Agnès Beaudry
Department of Mathematics
University of Colorado Boulder
Boulder, CO
United States
Irina Bobkova
Department of Mathematics
Texas A&M University
College Station, TX
United States
Michael Hill
Department of Mathematics
University of California, Los Angeles
Los Angeles, CA
United States
Vesna Stojanoska
Department of Mathematics
University of Illinois at Urbana–Champaign
Urbana, IL
United States