A monoidal algebraic model for rational SO(2)-spectra

TL;DR

This paper constructs a monoidal model structure for the algebraic model of rational SO(2)-spectra, ensuring the algebraic and topological smash products are compatible, advancing the algebraic understanding of equivariant spectra.

Contribution

It develops a monoidal model structure on Greenlees' algebraic model for rational SO(2)-spectra, enabling better algebraic-topological correspondence.

Findings

Monoidal model structure on A(SO(2)) constructed

Derived product compatible with smash product

Facilitates monoidal Quillen equivalences

Abstract

The category of rational SO(2)-equivariant spectra admits an algebraic model. That is, there is an abelian category A(SO(2)) whose derived category is equivalent to the homotopy category of rational SO(2)-equivariant spectra. An important question is: does this algebraic model capture the smash product of spectra? The category A(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. Hence the standard techniques for constructing a monoidal model structure cannot be applied. In this paper we construct a monoidal model structure on A(SO(2)) and show that the derived product on the homotopy category is 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. We also provide a monoidal Quillen equivalence to a simpler monoidal model category R_*-modules that has explicit generating sets. Having monoidal model structures on A(SO(2)) and R_*-modules removes a serious obstruction to constructing a series of monoidal Quillen equivalences between the algebraic model and rational SO(2)-equivariant spectra.