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 . There is always

a map from the fixed point spectrum to a homotopy fixed point spectrum (because, by definition, is the mapping spectrum .)

Example: acts on the complex K-theory KU by taking the conjugate bundle of a complex vector bundle. Then , the real K-theory.

The cofiber of 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

(which generalizes 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
  • Mathew, Akhil; Meier, Lennart (2015). "Affineness and chromatic homotopy theory". Journal of Topology. 8 (2): 476–528. arXiv:1311.0514. doi:10.1112/jtopol/jtv005.
  • Rognes, John (2008), "Galois extensions of structured ring spectra. Stably dualizable groups", Memoirs of the American Mathematical Society, 192 (898), doi:10.1090/memo/0898, hdl:21.11116/0000-0004-29CE-7, MR 2387923


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