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On the Balmer spectrum for compact Lie groups

Part of: Homotopy theory

Published online by Cambridge University Press:  14 November 2019

Tobias Barthel ,
J. P. C. Greenlees  and
Markus Hausmann
Tobias Barthel
Affiliation:
Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 København Ø, Denmark email tbarthel@math.ku.dk, barthel.tobi@gmail.com
J. P. C. Greenlees
Affiliation:
Warwick Mathematics Institute, Zeeman Building, Coventry CV4 7AL, UK email john.greenlees@warwick.ac.uk
Markus Hausmann
Affiliation:
Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 København Ø, Denmark email hausmann@math.ku.dk, hausmann@math.uni-bonn.de
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Abstract

We study the Balmer spectrum of the category of finite $G$-spectra for a compact Lie group $G$, extending the work for finite $G$ by Strickland, Balmer–Sanders, Barthel–Hausmann–Naumann–Nikolaus–Noel–Stapleton and others. We give a description of the underlying set of the spectrum and show that the Balmer topology is completely determined by the inclusions between the prime ideals and the topology on the space of closed subgroups of $G$. Using this, we obtain a complete description of this topology for all abelian compact Lie groups and consequently a complete classification of thick tensor ideals. For general compact Lie groups we obtain such a classification away from a finite set of primes $p$.

Keywords

equivariant spectrathick tensor idealsBalmer spectrumspace of subgroups

MSC classification

Primary: 55P42: Stable homotopy theory, spectra 55P91: Equivariant homotopy theory
Type
Research Article
Information
Compositio Mathematica , Volume 156 , Issue 1 , January 2020 , pp. 39 - 76
Copyright
© The Authors 2019 

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Footnotes

1

Current address: Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany

2

Current address: Mathematical Institute, Rheinische Friedrich-Wilhelms-Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany

TB was supported by the Danish National Research Foundation Grant DNRF92 and the European Unions Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 751794. JPCG is grateful to the EPSRC for support from EP/P031080/1. MH was supported by the Danish National Research Foundation Grant DNRF92.

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