Derived Category - Definition

Definition

Let be an abelian category. We obtain the derived category in several steps:

• The basic object is the category of chain complexes in . Its objects will be the objects of the derived category but its morphisms will be altered.
• Pass to the homotopy category of chain complexes by identifying morphisms which are chain homotopic.
• Pass to the derived category by localizing at the set of quasi-isomorphisms. Morphisms in the derived category may be explicitly described as roofs, where s is a quasi-isomorphism and f is any morphism of chain complexes.

The second step may be bypassed since a homotopy equivalence is in particular a quasi-isomorphism. But then the simple roof definition of morphisms must be replaced by a more complicated one using finite strings of morphisms (technically, it is no longer a calculus of fractions), and the triangulated category structure of arises in the homotopy category. So the one step construction is more efficient in a way but more complicated and the result is less powerful.