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Fixed point adjunctions for equivariant module spectra. (English) Zbl 1297.55013

This paper provides a useful general context for two well-known results. The first result is that the category of \(G\)-spectra over a normal subgroup \(N\) is equivalent to the category of \(G/N\)-spectra. The second is the Eilenberg-Moore spectral sequence regarding the calculation of the cohomology of a free \(G\)-space \(X\) from \(H^*(X/G)\) considered as a \(H^*(BG)\)-module. The general context is that both of these results are instances of where a fixed point-inflation adjunction induces a Quillen equivalence of model categories.
The paper contains numerous specializations of the main results and has a good number of interesting examples. It is another illustration of the importance of cellularization (right Bousfield localisation). Additionally, there are applications to a larger project on classifying rational (genuine) \(G\)-spectra in terms of an algebraic model, where \(G\) is a compact Lie group. Specifically the Quillen equivalence of the final section is central to the latest work on this project.

MSC:

55P91 Equivariant homotopy theory in algebraic topology
55N91 Equivariant homology and cohomology in algebraic topology
55P42 Stable homotopy theory, spectra
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