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Rational \(\mathrm{SO}(2)\)-equivariant spectra. (English) Zbl 1369.55005
The homotopy theory of \(G\)-spectra, for a compact Lie group \(G\), is extremely complicated. However, after rationalization it simplifies considerably and one can hope to obtain a purely algebraic model, which has been the subject of much prior work by many authors, including those of the current paper. For example, when \(G\) is a finite group, then rational \(G\)-spectra become equivalent to the derived category of Mackey functors (which also simplifies after rationalization and becomes semisimple). In general, one expects that rational \(G\)-spectra can be modeled by the differential objects in an abelian category of finite homological dimension. This was studied in detail for \(G = S^1\) (at least at the level of homotopy categories) by Greenlees, and for the torus a general comparison (including a Quillen equivalence) has been given by Greenlees-Shipley. The present paper focuses on the special case \(G = S^1\), and provides a Quillen equivalence with a clearly algebraic model category. This case allows simplifications from that of a higher-dimensional torus, and the Quillen equivalence produced is symmetric monoidal (unlike in the work of Greenlees-Shipley). The basic strategy is to decompose via isotropy separation on the unit into finite subgroups and geometric fixed points. The authors use here a previous result of Greenlees-Shipley that identifies the category of module spectra over the dual of EF\(_+\) (where EF is the classifying space of finite subgroups) as nonequivariant module spectra over the fixed points. They combine that with the geometric fixed points using general arguments about gluing model categories. Finally, they use the equivalence between rational spectra and chain complexes, which takes module spectra to differential modules.

55N91 Equivariant homology and cohomology in algebraic topology
55P42 Stable homotopy theory, spectra
55P60 Localization and completion in homotopy theory
55P91 Equivariant homotopy theory in algebraic topology
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