Universal Spaces and Splittings of Equivariant Spectra

Let $G$ be a finite group. We re-analyze the splitting of rational $G$-spectra, which is discussed in Barne's thesis and traces back to Greenlees and May. We will study how the splitting behaves under a localization which is weaker than rationalization and discuss its application in computing equivariant cohomology of $G$-spectra. In particular, we will explicitly compute $\pi_\bigstar(H\underline{\mathbb{Z}})$ and $\pi_\bigstar(HA_G)$ for the dihedral group $G=D_{2p}$. The additive structures and partial multiplicative structures are already computed by Kriz, Lu, and Zou. Our new method will provide a systematic way to compute them as $RO(D_{2p})$-graded rings.