Derived Category

A derived category of an Abelian category is the category obtained from the homotopy category of cochain complexes by formally inverting quasi-isomorphisms. A quasi-isomorphism is a map of cochain complexes inducing isomorphisms on all cohomology objects. Thus distinct cochain complexes can represent the same derived object even when they are not isomorphic term by term (Stacks Project).

The bounded derived category restricts to bounded cochain complexes. It is a triangulated category, with shift of cochain complexes and distinguished triangles represented by mapping cones. For a projective variety , the notation uses the Abelian category of coherent sheaves. For example, a short exact sequence of coherent sheaves gives a distinguished triangle after viewing each coherent sheaf as a cochain complex concentrated in degree zero.