G spectrum - Alchetron, The Free Social Encyclopedia

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 . There is always

X G → X h G ,

a map from the fixed point spectrum to a homotopy fixed point spectrum (because, by definition, X h G is the mapping spectrum F ( B G + , X ) G .)

Example: 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 , the real K-theory.

The cofiber of X h G → X h G 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 = 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

A ⊗ B A → ∏ g ∈ G A

(which generalizes x ⊗ y ↦ ( 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./2-Galois extension.

References

G-spectrum Wikipedia

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