A general Greenlees-May splitting principle

TL;DR

This paper extends the Greenlees-May splitting principle from finite groups to Mackey modules over arbitrary Green functors, providing an algebraic model for the rational equivariant Kasparov category of G-cell algebras.

Contribution

It generalizes the algebraic splitting principle to broader contexts and constructs an algebraic model for the rational equivariant Kasparov category.

Findings

Extended splitting principle to Mackey modules over any Green functor.

Provided an algebraic model for the rational equivariant Kasparov category.

Confirmed the algebraic part of the classical result in a more general setting.

Abstract

In equivariant topology, Greenlees and May used Mackey functors to show that, rationally, the stable homotopy category of $G$-spectra over a finite group $G$ splits as a product of simpler module categories. We extend the algebraic part (also independently proved by Th\'evenaz and Webb) of this classical result to Mackey modules over an arbitrary Green functor, and use the case of the complex representation ring Green functor to obtain an algebraic model of the rational equivariant Kasparov category of $G$-cell algebras.