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 <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 {{harv|Rognes|2008}}. 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 == * {{cite journal |last1=Mathew |first1=Akhil |last2=Meier |first2=Lennart |arxiv=1311.0514 |title=Affineness and chromatic homotopy theory |year=2015 |doi=10.1112/jtopol/jtv005 |volume=8 |issue=2 |journal=Journal of Topology |pages=476–528}} *{{citation | last = Rognes | first = John | doi = 10.1090/memo/0898 | issue = 898 | journal = Memoirs of the American Mathematical Society | mr = 2387923 | title = Galois extensions of structured ring spectra. Stably dualizable groups | volume = 192 | year = 2008| hdl = 21.11116/0000-0004-29CE-7 | hdl-access = free }}

== External links == *{{cite web |title=Homology of homotopy fixed point spectra |work=MathOverflow |date=June 30, 2012 |url=https://mathoverflow.net/q/101011 }}

Category:Algebraic topology Category:Spectra (topology) {{topology-stub}}