  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.

Algebraic Topology;Group Theory