Equivariant cohomology for cyclic groups

In this paper, we compute the RO(Cₙ)-graded coefficient ring of equivariant cohomology for cyclic groups Cₙ, in the case of Burnside ring coefficients, and in the case of constant coefficients. We use the invertible Mackey functors under the box product to reduce the gradings in the computation from RO(Cₙ) to those expressable as combinations of λᵈ for divisors d of n, where λ is the inclusion of Cₙ in S¹ as the roots of unity. We make explicit computations for the geometric fixed points for Burnside ring coefficients, and in the positive cone for constant coefficients. The positive cone is also computed for the Burnside ring in the case of prime power order, and in the case of square free order. Finally, we also make computations at non-negative gradings for the constant coefficients.