In algebraic topology, a G-spectrum is a spectrum with an action of a (finite) group.
Let X be a spectrum with an action of a finite group G. The important notion is that of the homotopy fixed point set . There is always
a map from the fixed point spectrum to a homotopy fixed point spectrum (because, by definition, is the mapping spectrum ).
Example: acts on the complex K-theory KU by taking the conjugate bundle of a complex vector bundle. Then , the real K-theory.
The cofiber of is called the Tate spectrum of X.
This notion is due to J. Rognes . Let A be an E<sub>âÂÂ</sub>-ring with an action of a finite group G and B = A<sup>hG</sup> its invariant subring. Then B â A (the map of B-algebras in E<sub>âÂÂ</sub>-sense) is said to be a G-Galois extension if the natural map
(which generalizes in the classical setup) is an equivalence. The extension is faithful if the Bousfield classes of A, B over B are equivalent.
Example: KO â KU is a ./2-Galois extension.