The effective exchange interactions of the magnetic overlayer Fe/Cu(0 0 1) covered by a Cu-cap layer of varying thickness were calculated in real space from first principles. The effective two-dimensional Heisenberg Hamiltonian was constructed and used to estimate spin-wave stiffness constants and overlayer Curie temperatures within the random-phase approximation. Oscillatory behavior of the overlayer Curie temperature, spin-wave stiffness, and magnetic moment as a function of the cap-layer thickness was found and explained.