On the Cohomology Comparison Theorem

A relative derived category for the category of modules over a presheaf of algebras is constructed to identify the relative Yoneda and Hochschild cohomologies with its homomorphism groups. The properties of a functor between this category and the relative derived category of modules over the algebra associated to the presheaf are studied. We obtain a generalization of the $Special$ $ Cohomology$ $Comparison$ $Theorem$ of M. Gerstenhaber and S. D. Schack.