Derived Functor Cohomology Groups with Yoneda Product

This work presents an exposition of both the internal structure of derived category of an abelian category D*(A) and its contribution in solving problems, particularly in algebraic geometry. Calculation of some morphisms will be presented between objects in D*(A) as elements in appropriate cohomology groups along with their compositions with the help of Yoneda construction under the assumption that the homological dimension of D*(A) is greater than or equal to 2. These computational settings will then be considered under sheaf cohomological context with a particular case from projective geometry.