Invertible K(2)–local E–modules in C4–spectra

Beaudry, Agnès; Bobkova, Irina; Hill, Michael; Stojanoska, Vesna

doi:10.2140/agt.2020.20.3423

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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

DOI: 10.2140/agt.2020.20.3423

Abstract

We compute the Picard group of the category of $K (2)$ –local module spectra over the ring spectrum $E^{h C_{4}}$ , where $E$ is a height $2$ Morava $E$ –theory and $C_{4}$ 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 $C_{4}$ –spectra. We show that in addition to a cyclic subgroup of order $32$ generated by $E \land S^{1}$ , the Picard group contains a subgroup of order $2$ generated by $E \land S^{7 + σ}$ , where $σ$ is the sign representation of the group $C_{4}$ . In the process, we completely compute the $RO (C_{4})$ –graded Mackey functor homotopy fixed point spectral sequence for the $C_{4}$ –spectrum $E “$ .

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Keywords

chromatic homotopy theory, Morava E–theory, Picard groups, higher real K–theory

Mathematical Subject Classification 2010

Primary: 55P42, 55Q91

Secondary: 20J06, 55M05, 55P60, 55Q51

References

Publication

Received: 14 January 2019

Revised: 16 September 2019

Accepted: 22 October 2019

Published: 29 December 2020

Authors

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

Volume 20, issue 7 (2020)