This is more or less standard material on Grothendieck and Serre Duality Theory, and there are several references available.

For instance, the stetement that you want is a consequence of the results in Chapter 5 of S.Ishii's book *Introduction to Singularities*.  

Look in particular at Theorem 5.3.6, Theorem 5.3.8 and Corollary 5.3.9.