Homological Algebra for Polynomial Mackey Rings over Prime Cyclic Groups

Abstract

Let $C_l$ denote the cyclic group of prime order $l$ and let $k$ be a field. We define a Mackey $\underline{k}$-algebra $\underline{k}[x_\theta]$ which is constructed by adjoining a free commutative variable to the free side of the constant Mackey functor $\underline{k}$. When $char(k)$ is relatively prime to $l$ we show that there is a an equivalence of categories between $\underline{k}[x_\theta]-\underline{Mod}$ and the category of modules over a certain twisted group ring. We calculate the free side of a certain Ext object $\underline{Ext}_{\underline{k}[x_\theta]}^*(\underline{k}, \underline{k})$ in the two cases when $char(k)$ is relatively prime to $l$ and when $char(k) = l = 2$.