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 $$X^{hG}$$. There is always
 * $$X^G \to X^{hG},$$

a map from the fixed point spectrum to a homotopy fixed point spectrum (because, by definition, $$X^{hG}$$ is the mapping spectrum $$F(BG_+, X)^G$$).

Example: $$\mathbb{Z}/2$$ acts on the complex K-theory KU by taking the conjugate bundle of a complex vector bundle. Then $$KU^{h\mathbb{Z}/2} = KO$$, the real K-theory.

The cofiber of $$X_{hG} \to X^{hG}$$ 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 = AhG 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
 * $$A \otimes_B A \to \prod_{g \in G} A$$

(which generalizes $$x \otimes y \mapsto (g(x) y)$$ 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 $$\mathbb{Z}$$./2-Galois extension.