Periodic equivariant real \(K\)-theories have rational Tate theory
type: Item
instance of: scholarly article
title: Periodic equivariant real \(K\)-theories have rational Tate theory
zbMATH Open document ID: 0898.55004
DOI: 10.1007/BF02567950
author: Lisbeth Fajstrup
published in: Manuscripta Mathematica
publication date: 1998-11-01T00:00:00Z
full work available at URL: https://eudml.org/doc/156225
review text: There is a standard way to associate to a given equivariant cohomology theory its Tate cohomology [e.g., see \textit{J. P. Greenlees} and \textit{J. P. May}, Generalized Tate cohomology, Mem. Am. Math. Soc. 543 (1995; Zbl 0876.55003)]. For periodic \(G\)-equivariant \(K\)-theory with \(G\) finite, the associated Tate theory turned out to be a rational cohomology theory. In the present paper this is generalized to the case of periodic \(G\)-equivariant real \(K\)-theory. But now the fact, that the real structure and the \(G\)-action mix up, causes some difficulties. To set up an inductive proof, starting with [\textit{L. Fajstrup}, Trans. Am. Math. Soc. 347, No. 5, 1841-1846 (1995; Zbl 0854.55004)], the author uses and studies Karoubi's generalization [\textit{M. Karoubi}, Lect. Notes Math. 136, 187-253 (1970; Zbl 0197.19902)] of real \(K\)-theory and shows that these also have rational Tate theory. The paper ends with an explicit calculation of the Tate spectrum for \(\mathbb{Z}\times_\alpha \mathbb{Z}/p\)-equivariant real \(K\)-theory.
Mathematics Subject Classification ID: 55N15
Mathematics Subject Classification ID: 55R91
Mathematics Subject Classification ID: 55N91
Mathematics Subject Classification ID: 55P91
Mathematics Subject Classification ID: 55P42
Mathematics Subject Classification ID: 55R25
Mathematics Subject Classification ID: 55N20
zbMATH DE Number: 980690
zbMATH Keywords: equivariant \(K\)-theory
zbMATH Keywords: real \(K\)-theory
zbMATH Keywords: Tate cohomology
reviewed by: Karlheinz Knapp
MaRDI profile type: Publication
cites work: K-THEORY AND REALITY
cites work: Q5563439
cites work: Equivariant stable homotopy and Segal's Burnside ring conjecture
cites work: Tate Cohomology of Periodic K-Theory with Reality is Trivial
cites work: Q5594331
cites work: Equivariant stable homotopy theory. With contributions by J. E. McClure
cites work: Classifying spaces and spectral sequences
cites work: Q3943047
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description: scientific article; zbMATH DE number 980690