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 \( F(BG_{+},X)^{G}.) \)
Example: \( \mathbb {Z} /2 \) acts on the complex K-theory KU by taking the conjugate bundle of a complex vector bundle. Then \(KU^{{h{\mathbb {Z}}/2}}=KO \), 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 (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
\( A\otimes _{B}A\to \prod _{{g\in G}}A \)
(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 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
External links
"Homology of homotopy fixed point spectra". MathOverflow. June 30, 2012.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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