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 $X^$. There is always
:$X^G \to X^,$
a map from the fixed point spectrum to a homotopy fixed point spectrum (because, by definition, $X^$ is the mapping spectrum $F(BG_+, X)^G$.)

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

The cofiber of $X_ \to X^$ 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
:$A \otimes_B A \to \prod_ A$
(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 ℤ./2-Galois extension.

See also

*Segal conjecture

References

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External links

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Algebraic topology
Homotopy theory

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