## 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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