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 h G {\displaystyle X^{hG}} . There is always

X G → X h G , {\displaystyle X^{G}\to X^{hG},}

a map from the fixed point spectrum to a homotopy fixed point spectrum (because, by definition, X h G {\displaystyle X^{hG}} is the mapping spectrum F ( B G + , X ) G {\displaystyle F(BG_{+},X)^{G}} ). Example: Z / 2 {\displaystyle \mathbb {Z} /2} acts on the complex K-theory KU by taking the conjugate bundle of a complex vector bundle. Then K U h Z / 2 = K O {\displaystyle KU^{h\mathbb {Z} /2}=KO} , the real K-theory. The cofiber of X h G → X h G {\displaystyle 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 (Rognes 2008). 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 ⊗ B A → ∏ g ∈ G A {\displaystyle A\otimes _{B}A\to \prod _{g\in G}A}

(which generalizes x ⊗ y ↦ ( g ( x ) y ) {\displaystyle 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 Z {\displaystyle \mathbb {Z} } ./2-Galois extension.

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