Mackey homological algebra over cyclic groups

TL;DR

This paper explores modules and chain complexes over the Mackey functor for cyclic groups, developing foundational tools for homological calculations relevant to equivariant cohomology theories.

Contribution

It introduces new methods for homological algebra over Mackey functors for cyclic groups, aiding future studies in equivariant cohomology.

Findings

Developed foundational tools for homological calculations

Performed homological computations over Mackey functors

Laid groundwork for applications in $RO(C_n)$-graded cohomology

Abstract

Let $C_n$ denote a cyclic group of order $n$. In this paper we investigate modules and chain complexes over the constant integral Mackey functor $\underline{\mathbb{Z}}$ and perform some related homological calculations. Along the way we develop a number of foundational tools for working with these categories. These results are useful for the study of $RO(C_n)$-graded Bredon cohomology, though such applications are delegated to a sequel paper.