Periodic equivariant real \(K\)-theories have rational Tate theory
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.
scientific article; zbMATH DE number 980690
scientific article