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G-spectrum

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Source updated March 26, 2024

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.

01G-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 = 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

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

(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.

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G-spectrum

01G-Galois extension in the sense of Rognes

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