Equivariant Z/ℓ-modules for the cyclic group C2

Abstract

For the cyclic group we give a complete description of the derived category of perfect complexes of modules over the constant Mackey ring , for ℓ a prime. This is fairly simple for ℓ odd, but for 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 -equivariant Eilenberg–MacLane spectrum . We also use the splitting theorem to give new and illuminating proofs of some facts about -graded Bredon cohomology, namely Kronholm's freeness theorem [12], [11] and the structure theorem of C. May