  For the cyclic group $C_2$ we give a complete description of the derived
category of perfect complexes of modules over the constant Mackey ring
$\underline{\mathbb{Z}/\ell}$, for $\ell$ a prime. This is fairly simple for
$\ell$ odd, but for $\ell=2$ depends on a new splitting theorem. As corollaries
of the splitting theorem we compute the associated Picard group and the Balmer
spectrum for compact objects in the derived category, and we obtain a complete
classification of finite modules over the $C_2$-equivariant Eilenberg--MacLane
spectrum $H\underline{\mathbb{Z}/2}$. We also use the splitting theorem to give
new and illuminating proofs of some facts about $RO(C_2)$-graded Bredon
cohomology, namely Kronholm's freeness theorem and the structure theorem of C.
May.
