On the construction of triangle equivalences

ON THE CONSTRUCTION OF TRIANGLE EQUIVALENCES

Bernhard Keller

Autumn 1994

Abstract. We give a self-contained account of an alternative proof of J. Rickard's Morita-theorem for derived categories (J. London Math. Soc. 39, 1989) and his theorem on the realization of derived equivalences as derived functors (J. London Math. Soc. 43, 1991) 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. We then extend the formalism of derived categories to differential graded algebras. 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.

http://www.math.jussieu.fr/~keller/publ/papabs.html
Bernhard Keller, le 2 Mars 1997.
keller@math.jussieu.fr

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