Equivariant [formula omitted]-modules for the cyclic group C2

Abstract

For the cyclic group C2 we give a complete description of the derived category of perfect complexes of modules over the constant Mackey ring Z/ℓ_, for ℓ a prime. This is fairly simple for ℓ odd, but for ℓ=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 C2-equivariant Eilenberg–MacLane spectrum HZ/2_. We also use the splitting theorem to give new and illuminating proofs of some facts about RO(C2)-graded Bredon cohomology, namely Kronholm's freeness theorem [12,11] and the structure theorem of C. May [13].