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

There is always a map from the fixed point spectrum to a homotopy fixed point spectrum (because, by definition,

acts on the complex K-theory KU by taking the conjugate bundle of a complex vector bundle.

is called the Tate spectrum of X.

Let A be an E∞-ring with an action of a finite group G and B = AhG its invariant subring.

in the classical setup) is an equivalence.

The extension is faithful if the Bousfield classes of A, B over B are equivalent.

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