Relative singularity categories

We study the properties of the relative derived category DCb(A) of an abelian category AA relative to a full and additive subcategory CC. In particular, when A=AA=A-mod for a finite-dimensional algebra A   over a field and CC is a contravariantly finite subcategory of A  -mod which is admissible and closed under direct summands, the CC-singularity category DC-sg(A)=DCb(A)/Kb(C) is studied. We give a sufficient condition when this category is triangulated equivalent to the stable category of the Gorenstein category G(C)G(C) of CC.