The derived category of complex periodic K-theory localized at an odd prime

We prove that for an odd prime $p$, the derived category $\mathcal{D}(KU_{(p)})$ of the $p$-local complex periodic $K$-theory spectrum $KU_{(p)}$ is triangulated equivalent to the derived category of its homotopy ring $\pi_*KU_{(p)}$. This implies that if $p$ is an odd prime, the triangulated category $\mathcal{D}(KU_{(p)})$ is algebraic.