Derived Mackey functors and Cpn-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 = Cpn. 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 AMR-strat); here we identify the gluing functors of this stratification in terms of Tate cohomology. Passing from genuine G-spectra to genuine G-Z-modules (a.k.a. derived Mackey functors) provides a convenient intermediate category for calculating equivariant cohomology. Indeed, as Z-linear Tate cohomology is far simpler than S-linear Tate cohomology, the above reconstruction theorem gives a particularly simple algebraic description of genuine G-Z-modules. We apply this in the case that G = Cpn for an odd prime p, computing the Picard group of genuine G-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.