Mackey homological algebra over cyclic groups

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.