G-spectrum

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 formula_1. There is always
a map from the fixed point spectrum to a homotopy fixed point spectrum (because, by definition, formula_1 is the mapping spectrum formula_4.)
Example: formula_5 acts on the complex "K"-theory "KU" by taking the conjugate bundle of a complex vector bundle. Then formula_6, the real "K"-theory.
The cofiber of formula_7 is called the Tate spectrum of "X".
"G"-Galois extension in the sense of Rognes.
This notion is due to J. Rognes . Let "A" be an E∞-ring with an action of a finite group "G" and "B" = "A""hG" its invariant subring. Then "B" → "A" (the map of "B"-algebras in E∞-sense) is said to be a "G-Galois extension" if the natural map
(which generalizes formula_9 in the classical setup) is an equivalence. The extension is faithful if the Bousfield classes of "A", "B" over "B" are equivalent.
Example: "KO" → "KU" is a ℤ./2-Galois extension.