Английская Википедия:G-spectrum

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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 <math>X^{hG}</math>. There is always

<math>X^G \to X^{hG},</math>

a map from the fixed point spectrum to a homotopy fixed point spectrum (because, by definition, <math>X^{hG}</math> is the mapping spectrum <math>F(BG_+, X)^G</math>.)

Example: <math>\mathbb{Z}/2</math> acts on the complex K-theory KU by taking the conjugate bundle of a complex vector bundle. Then <math>KU^{h\mathbb{Z}/2} = KO</math>, the real K-theory.

The cofiber of <math>X_{hG} \to X^{hG}</math> is called the Tate spectrum of X.

G-Galois extension in the sense of Rognes

This notion is due to J. Rognes Шаблон: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

<math>A \otimes_B A \to \prod_{g \in G} A</math>

(which generalizes <math>x \otimes y \mapsto (g(x) y)</math> 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 <math>\mathbb{Z}</math>./2-Galois extension.

See also

• Segal conjecture

References

• Шаблон:Cite journal
• Шаблон:Citation

External links

• Шаблон:Cite web

Шаблон:Topology-stub