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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 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 {{harv|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 \mathbb{Z}./2-Galois extension.

See also

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)

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