  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.
