On the parametrized Tate construction

2021/10/14 by J. D. Quigley, Jay Shah, Quigley, J. D. +1
Mathematics · #55P91 #Advanced Algebra and Geometry #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2110.07707

openalex publication_date 2021/10/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We introduce and study a genuine equivariant refinement of the Tate construction associated to an extension \widehatG 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 \widehatG-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.

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