Remarks
For certain purposes (see below) one uses bounded-below (An=0 for n<<0), bounded-above (An=0 for n>>0) or bounded (An=0 for |n|>>0) complexes instead of unbounded ones. The corresponding derived categories are usually denoted D+(A), D-(A) and Db(A), respectively.
If one adopts the classical point of view on categories, that morphisms have to be sets (not just classes), then one has to give an additional argument, why this is true. If, for example, the abelian category is small, i.e. has only a set of objects, then this issue will be no problem.
Composition of morphisms, i.e. roofs, in the derived category is accomplished by finding a third roof on top of the two roofs to be composed. It may be checked that this is possible and gives a well-defined, associative composition.
As the localization of K(A) (which is a triangulated category), the derived category is triangulated as well. Distinguished triangles are those quasi-isomorphic to triangles of the form for two complexes A and B and a map f between them. This includes in particular triangles of the form for a short exact sequence
in .
Read more about this topic: Derived Category
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