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 ${\displaystyle X^{hG}}$. There is always

${\displaystyle X^{G}\to X^{hG},}$

a map from the fixed point spectrum to a homotopy fixed point spectrum (because, by definition, ${\displaystyle X^{hG}}$ is the mapping spectrum ${\displaystyle F(BG_{+},X)^{G}}$.)

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

The cofiber of ${\displaystyle X_{hG}\to X^{hG}}$ 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 BA (the map of B-algebras in E-sense) is said to be a G-Galois extension if the natural map

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

(which generalizes ${\displaystyle x\otimes y\mapsto (g(x)y)}$ in the classical setup) is an equivalence. The extension is faithful if the Bousfield classes of A, B over B are equivalent.

Example: KOKU is a ℤ./2-Galois extension.

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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