On the construction of triangle equivalences
We give a self-contained account of an alternative proof of J. Rickard's Morita-theorem for derived categories  and his theorem on the realization of derived equivalences as derived functors . To this end, we first review the basic facts on unbounded derived categories (complexes unbounded to the right and to the left) and on derived functors between such categories (cf. , ). We then extend the formalism of derived categories to differential graded algebras (cf. ). This allows us to write down a formula for a bimodule complex given a tilting complex. We then deduce J. Rickard's results.
As a second application of the differential graded algebra techniques, we prove a structure theorem for stable categories admitting infinite sums and a small generator. This yields a natural construction of D. Happel's equivalence  between the derived category of a finite-dimensional algebra and the stable category of the associated repetitive algebra.
Finally, we use differential graded algebras to show that cyclic homology is preserved by derived equivalences (following ).