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• W3205957029 abstract "We introduce and study a genuine equivariant refinement of the Tate construction associated to an extension $widehat{G}$ 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 $widehat{G}$-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 $mathcal{F}$-complete $G$-spectrum for any $G$-family $mathcal{F}$." @default.
• W3205957029 created "2021-10-25" @default.
• W3205957029 creator A5040388325 @default.
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• W3205957029 date "2021-10-14" @default.
• W3205957029 modified "2023-10-13" @default.
• W3205957029 title "On the parametrized Tate construction" @default.
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• W3205957029 doi "https://doi.org/10.48550/arxiv.2110.07707" @default.
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