Generalized Z-homotopy fixed points of C_n spectra with applications to norms of MU_R

We introduce a computationally tractable way to describe the Z-homotopy fixed points of a C_n-spectrum E, producing a genuine Cn spectrum E^hnZ whose fixed and homotopy fixed points agree and are the Z-homotopy fixed points of E. These form the bottom piece of a contravariant functor from the divisor poset of n to genuine C_n-spectra, and when E is an N_∞-ring spectrum, this functor lifts to a functor of N_∞-ring spectra. For spectra like the Real Johnson-Wilson theories or the norms of Real bordism, the slice spectral sequence provides a way to easily compute the RO(G)-graded homotopy groups of the spectrum E^hnZ, giving the homotopy groups of the Z-homotopy fixed points. For the more general spectra in the contravariant functor, the slice spectral sequences interpolate between the one for the norm of Real bordism and the especially simple Z-homotopy fixed point case, giving us a family of new tools to simplify slice computations.