In algebraic topology, a G-spectrum is a

spectrum A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors i ...

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^

. There is always :

X^G \to X^,

a map from the fixed point spectrum to a homotopy fixed point spectrum (because, by definition,

X^

is the mapping spectrum

F(BG_+, X)^G

.) Example:

\mathbb/2

acts on the complex ''K''-theory ''KU'' by taking the

conjugate bundle In mathematics, a complex vector bundle is a vector bundle whose fibers are complex vector spaces. Any complex vector bundle can be viewed as a real vector bundle through the restriction of scalars. Conversely, any real vector bundle ''E'' can be ...

of a

complex vector bundle In mathematics, a complex vector bundle is a vector bundle whose fibers are complex vector spaces. Any complex vector bundle can be viewed as a real vector bundle through the restriction of scalars. Conversely, any real vector bundle ''E'' can be ...

. Then

KU^ = KO

, the real ''K''-theory. The cofiber of

X_ \to X^

is called the

Tate spectrum Tate is an institution that houses, in a network of four art galleries, the United Kingdom's national collection of British art, and international modern and contemporary art. It is not a government institution, but its main sponsor is the U ...

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'' = ''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 :

A \otimes_B A \to \prod_ A

(which generalizes

x \otimes y \mapsto (g(x) y)

in the classical setup) is an equivalence. The extension is faithful if the

Bousfield class In algebraic topology, the Bousfield class of, say, a spectrum ''X'' is the set of all (say) spectra ''Y'' whose smash product with ''X'' is zero: X \otimes Y = 0. Two objects are Bousfield equivalent if their Bousfield classes are the same. The no ...

es of ''A'', ''B'' over ''B'' are equivalent. Example: ''KO'' → ''KU'' is a ℤ./2-Galois extension.

References

* *

External links

* Algebraic topology Homotopy theory {{topology-stub

''G''-Galois extension in the sense of Rognes

See also

References

External links