We investigate a one dimensional Bose-Hubbard model in a cavity with a cavity back-action. The back action create a long-range effective interaction between the particles. We consider the hard core boson limit. The hard core bosons can mapped to spins. In the spin language the effective long range interaction has an anti-ferromagnetic nature. The phase diagram was investigated by exact diagonalization and mean-field methods. The mean field equations are derived from variational principles, and give a good description of the phase diagram. There are a Mott-insulator (ferromagnetic) phase, an anti-ferromagnetic phase, a critical phase of the system. There might be also a supersolid phase.