Derived Mackey functors and $C_{p^n}$-equivariant cohomology

TL;DR

This paper introduces a new algebraic approach to compute $G$-equivariant cohomology for finite groups, especially $C_{p^n}$, using a symmetric monoidal reconstruction theorem and derived Mackey functors, simplifying calculations.

Contribution

It develops a symmetric monoidal stratification framework for genuine $G$-modules and applies it to explicitly compute equivariant cohomology and Picard groups for $C_{p^n}$.

Findings

Reconstruction theorem for genuine $G$-$R$-modules in terms of fixed points and Tate cohomology.

Simplified algebraic description of genuine $G$-$bZ$-modules.

Explicit calculations of Picard groups and $RO(G)$-graded cohomology for $C_{p^n}$.

Abstract

We establish a novel approach to computing $G$-equivariant cohomology for a finite group $G$, and demonstrate it in the case that $G = C_{p^n}$. For any commutative ring spectrum $R$, we prove a symmetric monoidal reconstruction theorem for genuine $G$-$R$-modules, which records them in terms of their geometric fixedpoints as well as gluing maps involving their Tate cohomologies. This reconstruction theorem follows from a symmetric monoidal stratification (in the sense of \cite{AMR-strat}); here we identify the gluing functors of this stratification in terms of Tate cohomology. Passing from genuine $G$-spectra to genuine $G$-$\mathbb{Z}$-modules (a.k.a. derived Mackey functors) provides a convenient intermediate category for calculating equivariant cohomology. Indeed, as $\mathbb{Z}$-linear Tate cohomology is far simpler than $\mathbb{S}$-linear Tate cohomology, the above reconstruction theorem gives a particularly simple algebraic description of genuine $G$-$\mathbb{Z}$-modules. We apply this in the case that $G = C_{p^n}$ for an odd prime $p$, computing the Picard group of genuine $G$-$\mathbb{Z}$-modules (and therefore that of genuine $G$-spectra) as well as the $RO(G)$-graded and Picard-graded $G$-equivariant cohomology of a point.