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 EilenbergMacLane 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.
 Publication:

arXiv eprints
 Pub Date:
 March 2022
 DOI:
 10.48550/arXiv.2203.05287
 arXiv:
 arXiv:2203.05287
 Bibcode:
 2022arXiv220305287D
 Keywords:

 Mathematics  Algebraic Topology
 EPrint:
 42 pages, 15 figures, v2 accepted version to appear in Journal of Pure and Applied Algebra