An algebraic model for free rational G-spectra

Abstract

We show that for any compact Lie group $G$ with identity component $N$ and component group $W=G/N$, the category of free rational $G$-spectra is equivalent to the category of torsion modules over the twisted group ring $H^{\ast }(BN)[W]$. This gives an algebraic classification of rational $G$-equivariant cohomology theories on free $G$-spaces and a practical method for calculating the groups of natural transformations between them. This uses the methods of arXiv:1101.2511, and some readers may find the simpler context of the present paper highlights the main thread of the argument.

Further minor expository changes and clarifications

References in corpus (4)

• An algebraic model for rational torus-equivariant spectra
• The homotopy theory of coalgebras over a comonad
• Morita Theory For Derived Categories: A Bicategorical Perspective
• The cellularization principle for Quillen adjunctions

Cited by in corpus (9)

• Grothendieck-Neeman duality and the Wirthmüller isomorphism
• Model Categories for Orthogonal Calculus
• An algebraic model for rational torus-equivariant spectra
• The Left Localization Principle, completions, and cofree $G$-spectra
• Flatness and Shipley's algebraicization theorem
• Rational equivariant cohomology theories with toral support
• Algebraic models of change of groups functors in (co)free rational equivariant spectra
• The localization of orthogonal calculus with respect to homology
• Rational local systems and connected finite loop spaces