We study the quantum coherence and monogamy relationship of a tripartite W-state entangled system for Dirac fields in the background of a Schwarzschild black hole. We find that quantum coherence first decreases and then shows the phenomenon of freezing with the growth of the Hawking temperature. We also find that the l1 norm of coherence is always equal to the sum of coherence of all bipartite systems for any Hawking temperature, while a similar monogamy relationship for the relative entropy of coherence is absent. Moreover, we extend the related investigations to the N-partite W-state systems. It is shown that a similar monogamy relationship for the l1 norm of coherence is still satisfied, and the phenomenon of coherence freezing also exists.