Tate blueshift and vanishing for Real oriented cohomology

Li, Guchuan; Lorman, Vitaly; Quigley, J.D.

doi:10.48550/arxiv.1910.06191

Cited by 1 publication

(3 citation statements)

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“…7]). The parametrized Tate construction was used to produce analogous results for real oriented cohomology theories in [LLQ19] and will be applied in forthcoming work of the first author with Chatham-Li-Lorman to study hyperreal oriented cohomology theories. The starting point for these computations is the observation that the parametrized Tate construction, instead of the ordinary Tate construction, can be used to access formal group law techniques for hyperreal oriented cohomology.…”

Section: Motivation and Applicationsmentioning

confidence: 99%

“…16.1]. This identification is used to prove parametrized Tate blueshift in [LLQ19] and to define C 2 -equivariant Mahowald invariants in [Qui21].…”

Section: Proof For Any Subgroupmentioning

confidence: 99%

“…, is analyzed in forthcoming work of the first author with Chatham-Li-Lorman. The n = 1 case was used in [LLQ19] to obtain a Tate splitting for real Johnson-Wilson theories and in [Qui21] to understand the C 2 -equivariant stable stems. These applications are outlined further in Remark 1.14.…”

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confidence: 99%

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On the parametrized Tate construction

Quigley¹,

Shah²

2021

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We introduce and study a genuine equivariant refinement of the Tate construction associated to an extension G of a finite group G by a compact Lie group K, which we call the parametrized Tate construction (−) t G K . Our main theorem establishes the coincidence of three conceptually distinct approaches to its construction when K is also finite: one via recollement theory for the K-free G-family, another via parametrized ambidexterity for G-local systems, and the last via parametrized assembly maps. We also show that (−) t G K uniquely admits the structure of a lax G-symmetric monoidal functor, thereby refining a theorem of Nikolaus and Scholze. Along the way, we apply a theorem of the second author to reprove a result of Ayala-Mazel-Gee-Rozenblyum on reconstructing a genuine G-spectrum from its geometric fixed points; our method of proof further yields a formula for the geometric fixed points of an F -complete G-spectrum for any G-family F . 4 We endow Fun G (B ψ G K, Sp G ) with the pointwise symmetric monoidal structure of Definition A.1. 5 We write this as Ψ K to distinguish it from the spectrum-valued functor of categorical K-fixed points (−) K : Sp G Sp.

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Section: Motivation and Applicationsmentioning

confidence: 99%

“…16.1]. This identification is used to prove parametrized Tate blueshift in [LLQ19] and to define C 2 -equivariant Mahowald invariants in [Qui21].…”

Section: Proof For Any Subgroupmentioning

confidence: 99%