K-theory Greenlees–May splitting for C_{p^n}-Mackey functors (n>1)

Determine whether a Greenlees–May-type splitting holds for the algebraic K-theory of the Burnside Green functor A_{C_{p^n}} (equivalently, for the category of C_{p^n}-Mackey functors) when n>1; specifically, ascertain whether K(A_{C_{p^n}}) admits a canonical decomposition analogous to the Greenlees–May splitting known for G-theory and for K-theory in the case n=1.

Background

The paper proves a Greenlees–May-type splitting for G-theory under mild hypotheses and applies it to obtain a direct-sum decomposition for the G-theory of Mackey functors over cyclic p-groups.

Greenlees previously established a similar splitting for the K-theory of C_p-Mackey functors. Extending this splitting to K-theory for higher cyclic p-groups would parallel the G-theory phenomenon, but the authors note their current methods are insufficient, partly due to constraints arising from their computation of K(\underline{F_2}).