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

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.

arXiv commentimproved introduction; minor notational changes and reorganization

References in corpus (8)

• Nilpotence and descent in equivariant stable homotopy theory
• The spectrum of the equivariant stable homotopy category of a finite group
• Models of G-spectra as presheaves of spectra
• Stratified categories, geometric fixed points and a generalized Arone-Ching theorem
• Equivariant Eilenberg-Mac Lane spectra in cyclic $p$-groups
• The Picard group in equivariant homotopy theory via stable module categories
• The $RO(C_4)$ integral homology of a point
• On the Coefficients of $(\mathbb{Z}/p{)}^n$-Equivariant Ordinary Cohomology with Coefficients in $\mathbb{Z}/p$

Cited by in corpus (1)

• On the parametrized Tate construction