The spectrum of derived Mackey functors
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- by Irakli Patchkoria, Beren Sanders and Christian Wimmer PDF
- Trans. Amer. Math. Soc. 375 (2022), 4057-4105 Request permission
Abstract:
We compute the spectrum of the category of derived Mackey functors (in the sense of Kaledin) for all finite groups. We find that this space captures precisely the top and bottom layers (i.e. the height infinity and height zero parts) of the spectrum of the equivariant stable homotopy category. Due to this truncation of the chromatic information, we are able to obtain a complete description of the spectrum for all finite groups, despite our incomplete knowledge of the topology of the spectrum of the equivariant stable homotopy category. From a different point of view, we show that the spectrum of derived Mackey functors can be understood as the space obtained from the spectrum of the Burnside ring by “ungluing” closed points. In order to compute the spectrum, we provide a new description of Kaledin’s category, as the derived category of an equivariant ring spectrum, which may be of independent interest. In fact, we clarify the relationship between several different categories, establishing symmetric monoidal equivalences and comparisons between the constructions of Kaledin, the spectral Mackey functors of Barwick, the ordinary derived category of Mackey functors, and categories of modules over certain equivariant ring spectra. We also illustrate an interesting feature of the ordinary derived category of Mackey functors that distinguishes it from other equivariant categories relating to the behavior of its geometric fixed points.References
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Additional Information
- Irakli Patchkoria
- Affiliation: Institute of Mathematics, University of Aberdeen, Fraser Noble Building, Aberdeen, AB24 3UE, Scotland, United Kingdom
- MR Author ID: 986424
- Email: irakli.patchkoria@abdn.ac.uk
- Beren Sanders
- Affiliation: Mathematics Department, UC Santa Cruz, 95064 California
- MR Author ID: 1035411
- ORCID: 0000-0002-9550-6447
- Email: beren@ucsc.edu
- Christian Wimmer
- Affiliation: Bonn, Germany
- MR Author ID: 1296925
- ORCID: 0000-0002-3997-6671
- Email: wimmer@math.uni-bonn.de
- Received by editor(s): December 1, 2020
- Received by editor(s) in revised form: April 27, 2021
- Published electronically: March 31, 2022
- Additional Notes: The second-named author was supported by NSF grant DMS-1903429
- © Copyright 2022 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 375 (2022), 4057-4105
- MSC (2020): Primary 18G80, 55P91, 55U35
- DOI: https://doi.org/10.1090/tran/8485
- MathSciNet review: 4419053