G-spectrum explained
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 . 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
./2-Galois extension.
See also
References
- Mathew . Akhil . Meier . Lennart . 1311.0514 . Affineness and chromatic homotopy theory . 2015 . 10.1112/jtopol/jtv005 . 8 . 2 . Journal of Topology . 476–528.
External links