On the construction of triangle equivalences

Abstract

We give a self-contained account of an alternative proof of J. Rickard's Morita-theorem for derived categories [135] and his theorem on the realization of derived equivalences as derived functors [139]. 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. [159], [19]). We then extend the formalism of derived categories to differential graded algebras (cf. [77]). 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 [63] 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 [93]).