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

Abstract

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.

42 pages, 15 figures, v2 accepted version to appear in Journal of Pure and Applied Algebra