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For the cyclic group $C_2$ we give a complete description of the derivedcategory 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 corollariesof the splitting theorem we compute the associated Picard group and the Balmerspectrum for compact objects in the derived category, and we obtain a completeclassification of finite modules over the $C_2$-equivariant Eilenberg--MacLanespectrum $H\underline{\mathbb{Z}/2}$. We also use the splitting theorem to givenew and illuminating proofs of some facts about $RO(C_2)$-graded Bredoncohomology, namely Kronholm's freeness theorem and the structure theorem of C.May.

Equivariant $\underline{\mathbb{Z}/\ell}$-modules for the cyclic group $C_2$