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 $F(B{G}_{+},X{)}^G$).

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

$A{\otimes }_{B}A\to \prod _{g\in G}A$

(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.

See also

• Segal conjecture

References

• Template:Cite journal
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External links

• Template:Cite web