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

TL;DR

This paper provides a complete classification of perfect complexes over the Mackey ring for $C_2$, revealing new structural insights and computing key invariants, especially for the case when $ ext{l}=2$.

Contribution

It introduces a new splitting theorem for $ ext{l}=2$, enabling full descriptions of derived categories, Picard groups, and classifications of modules over the $C_2$-equivariant Eilenberg--MacLane spectrum.

Findings

Complete description of the derived category for odd $ ext{l}$

New splitting theorem for $ ext{l}=2$

Classification of finite modules over $H ext{Z}/2$

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.