7 - The Derived Category D(A,M)

Summary

We begin by proving that if K is a triangulated category and S ⊆ K is a denominator set of cohomological origin, then the localized category KS is triangulated and the localization functor Q : K → KS is triangulated. In the case of the triangulated category K(A,M) and the set of quasi-isomorphisms S(A,M) in it, we get the derived category D(A,M) := K(A,M)S(A, M) and the triangulated localization functor Q : K(A,M) → D(A,M). We look at the full subcategories of K(A, M) corresponding to boundedness conditions and the corresponding derived categories. We prove that the obvious functor M → D(M) is fully faithful. The section ends with a study of the triangulated structure of the opposite derived category D(A,M)op .