Rigidity and exotic models for v1-local G-equivariant stable homotopy

Irakli Patchkoria; Constanze Roitzheim

doi:10.1007/s00209-019-02364-z

Rigidity and exotic models for _v1-local G-equivariant stable homotopy

Irakli Patchkoria, Constanze Roitzheim^* (Corresponding Author)

^*Corresponding author for this work

Mathematical Science

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Abstract

We prove that the v1-local G-equivariant stable homotopy category for G
a finite group has a unique G-equivariant model at p = 2. This means that at the
prime 2 the homotopy theory of G-spectra up to fixed point equivalences on K-theory is uniquely determined by its triangulated homotopy category and basic Mackey structure. The result combines the rigidity result for K-local spectra of the second author with the equivariant rigidity result for G-spectra of the first author. Further, when the prime p is at least 5 and does not divide the order of G, we provide an algebraic exotic model as well as a G-equivariant exotic model for the v1-local G-equivariant stable homotopy category, showing that for primes p ≥ 5 equivariant rigidity fails in general.

Original language	English
Pages (from-to)	839-875
Number of pages	37
Journal	Mathematische Zeitschrift
Volume	295
Early online date	12 Aug 2019
DOIs	https://doi.org/10.1007/s00209-019-02364-z
Publication status	Published - 1 Jun 2020

Keywords

ALGEBRAIC MODEL
K-THEORY
CATEGORIES
SPECTRA
EQUIVALENCES
LOCALIZATION
FUNCTORS
RESPECT

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Cite this

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title = "Rigidity and exotic models for v1-local G-equivariant stable homotopy",

abstract = "We prove that the v1-local G-equivariant stable homotopy category for Ga finite group has a unique G-equivariant model at p = 2. This means that at theprime 2 the homotopy theory of G-spectra up to fixed point equivalences on K-theory is uniquely determined by its triangulated homotopy category and basic Mackey structure. The result combines the rigidity result for K-local spectra of the second author with the equivariant rigidity result for G-spectra of the first author. Further, when the prime p is at least 5 and does not divide the order of G, we provide an algebraic exotic model as well as a G-equivariant exotic model for the v1-local G-equivariant stable homotopy category, showing that for primes p ≥ 5 equivariant rigidity fails in general.",

keywords = "ALGEBRAIC MODEL, K-THEORY, CATEGORIES, SPECTRA, EQUIVALENCES, LOCALIZATION, FUNCTORS, RESPECT",

author = "Irakli Patchkoria and Constanze Roitzheim",

note = "The first author acknowledges the support of the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92), the German Research Foundation Schwerpunktprogramm 1786 and the Shota Rustaveli National Science Foundation Grant 217–614. He also thanks the Hausdorff Research Institute for Mathematics in Bonn for their hospitality. Finally the first author would like to thank Gijs Heuts, Akhil Mathew, Stefan Schwede and Christian Wimmer for helpful conversations. The second author thanks the Department of Mathematical Sciences in Copenhagen and the Hausdorff Research Institute for Mathematics in Bonn for their hospitality. Furthermore, the second author would like to thank David Barnes, Anna Marie Bohmann, John Greenlees and Mike Hill for useful discussions. We thank the referee for useful comments. ",

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doi = "10.1007/s00209-019-02364-z",

language = "English",

volume = "295",

pages = "839--875",

journal = "Mathematische Zeitschrift",

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AU - Patchkoria, Irakli

AU - Roitzheim, Constanze

N1 - The first author acknowledges the support of the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92), the German Research Foundation Schwerpunktprogramm 1786 and the Shota Rustaveli National Science Foundation Grant 217–614. He also thanks the Hausdorff Research Institute for Mathematics in Bonn for their hospitality. Finally the first author would like to thank Gijs Heuts, Akhil Mathew, Stefan Schwede and Christian Wimmer for helpful conversations. The second author thanks the Department of Mathematical Sciences in Copenhagen and the Hausdorff Research Institute for Mathematics in Bonn for their hospitality. Furthermore, the second author would like to thank David Barnes, Anna Marie Bohmann, John Greenlees and Mike Hill for useful discussions. We thank the referee for useful comments.

PY - 2020/6/1

Y1 - 2020/6/1

N2 - We prove that the v1-local G-equivariant stable homotopy category for Ga finite group has a unique G-equivariant model at p = 2. This means that at theprime 2 the homotopy theory of G-spectra up to fixed point equivalences on K-theory is uniquely determined by its triangulated homotopy category and basic Mackey structure. The result combines the rigidity result for K-local spectra of the second author with the equivariant rigidity result for G-spectra of the first author. Further, when the prime p is at least 5 and does not divide the order of G, we provide an algebraic exotic model as well as a G-equivariant exotic model for the v1-local G-equivariant stable homotopy category, showing that for primes p ≥ 5 equivariant rigidity fails in general.

AB - We prove that the v1-local G-equivariant stable homotopy category for Ga finite group has a unique G-equivariant model at p = 2. This means that at theprime 2 the homotopy theory of G-spectra up to fixed point equivalences on K-theory is uniquely determined by its triangulated homotopy category and basic Mackey structure. The result combines the rigidity result for K-local spectra of the second author with the equivariant rigidity result for G-spectra of the first author. Further, when the prime p is at least 5 and does not divide the order of G, we provide an algebraic exotic model as well as a G-equivariant exotic model for the v1-local G-equivariant stable homotopy category, showing that for primes p ≥ 5 equivariant rigidity fails in general.

KW - ALGEBRAIC MODEL

KW - K-THEORY

KW - CATEGORIES

KW - SPECTRA

KW - EQUIVALENCES

KW - LOCALIZATION

KW - FUNCTORS

KW - RESPECT

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UR - http://www.mendeley.com/research/rigidity-exotic-models-v1-v-1-local-gequivariant-stable-homotopy-theory

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JO - Mathematische Zeitschrift

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SN - 0025-5874

ER -

Rigidity and exotic models for _v1-local G-equivariant stable homotopy

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Rigidity and exotic models for v1-local G-equivariant stable homotopy

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Rigidity and exotic models for _v1-local G-equivariant stable homotopy