Mackey homological algebra over cyclic groups

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.

Figures (5)

• Figure 1: The $\underline{{\mathbb Z}}$-modules $\underline{{\mathbb Z}}$, $I_{\EuScript F}$, and $\underline{{\mathbb Z}}/{\EuScript F}$
• Figure 2: The $\underline{{\mathbb Z}}$-module $\underline{{\mathbb Z}}(e;d)$ and its restriction maps.
• Figure 3: Künneth spectral sequence for $\underline{H}_*(S^{\lambda_a}\Box S^{\lambda_b})$
• Figure 4: Universal Coefficient spectral sequence for $\underline{H}_*(\mathop{\mathrm{\underline{Hom}}}\nolimits(S^{\lambda_a}, S^{\lambda_b}))$
• Figure 5: Künneth spectral sequence for $\underline{H}_*\bigl (\mathop{\mathrm{\underline{Hom}}}\nolimits(S^{\lambda_a},\underline{{\mathbb Z}}) \Box S^{\lambda_b}\bigr )$

Theorems & Definitions (139)

• Proposition 1.1
• Remark 1.4
• Proposition 2.1
• proof
• Definition 2.2
• Example 2.3
• Definition 2.4
• Remark 2.5
• Proposition 2.6
• proof