I have to apologize for this post: while I have, as of late, been making efforts to make this blog more interesting and useful to outsiders (especially now that I have somewhat more readers than before), the present post will be a somewhat detailed walk-through of one of the first important results in BBD, and, as homological algebra, it is slightly on the technical side. Readers unfamiliar with the material may wish to skim the main result and skip the proof (or just read BBD for it).

**1. A cohomological functor**

We saw that a -structure on a triangulated category always implies that contains an *abelian* category , called the **heart** of . More is true, however:

Theorem 6The functor given by is a cohomological functor.

This is, of course, familiar from the case when is the derived category and then these are just the ordinary cohomology functors. In other words, it is a generalization of the long exact sequence in cohomology from a short exact sequence (triangle) of complexes. (more…)