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A monoidal algebraic model for rational $$\mathrm{SO}(2)$$-spectra. (English) Zbl 1371.55008
Summary: The category of rational $$\mathrm{SO}(2)$$-equivariant spectra admits an algebraic model. That is, there is an abelian category $${\mathcal A}$$($$SO(2)$$) whose derived category is equivalent to the homotopy category of rational $$\mathrm{SO}(2)$$-equivariant spectra. An important question is: does this algebraic model capture the smash product of spectra?
The category $${\mathcal A}(\mathrm{SO}(2))$$ is known as Greenlees’ standard model, it is an abelian category that has no projective objects and is constructed from modules over a non-Noetherian ring. As a consequence, the standard techniques for constructing a monoidal model structure cannot be applied. In this paper a monoidal model structure on $${\mathcal A}\mathrm{SO}(2))$$ is constructed and the derived tensor product on the homotopy category is shown to be compatible with the smash product of spectra. The method used is related to techniques developed by the author in earlier joint work with Roitzheim. That work constructed a monoidal model structure on Franke’s exotic model for the $$K_{(p)}$$-local stable homotopy category.
A monoidal Quillen equivalence to a simpler monoidal model category $$R^{\bullet}$$-mod that has explicit generating sets is also given. Having monoidal model structures on $${\mathcal A}\mathrm{SO}(2))$$ and $$R^{\bullet}$$-mod removes a serious obstruction to constructing a series of monoidal Quillen equivalences between the algebraic model and rational $$\mathrm{SO}(2)$$-equivariant spectra.
##### MSC:
 55P91 Equivariant homotopy theory in algebraic topology 55P42 Stable homotopy theory, spectra 55P62 Rational homotopy theory 55U35 Abstract and axiomatic homotopy theory in algebraic topology
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