Abstract: For a self-orthogonal module T, the relation between the quotient triangulated category D b (A)/K b (addT) and the stable category of the Frobenius category of T-Cohen-Macaulay modules is investigated. In particular, for a Gorenstein algebra, we get a relative version of the description of the singularity category due to Happel. Also, the derived category of a Gorenstein algebra is explicitly given, inside the stable category of the graded module category of the corresponding trivial extension algebra, via Happel’s functor $$F: D^b(A) \longrightarrow T(A)^{\mathbb{Z}}\mbox{-}\underline{\rm mod}$$ .