Bernhard Keller

This is a contribution to the Tilting Handbook edited by L. Angeleri Hügel, D. Happel and H. Krause and conceived at the meeting Twenty years of tilting theory.

We review the basic definitions of derived categories and derived functors. We illustrate them on simple but non trivial examples. Then we explain Happel's theorem which states that each tilting triple yields an equivalence between derived categories. We establish its link with Rickard's theorem which characterizes derived equivalent algebras. We then examine invariants under derived equivalences. Using $t$-structures we compare two abelian categories having equivalent derived categories. Finally, we briefly sketch a generalization of the tilting setup to differential graded algebras.

Bernhard Keller, le 6 janvier, 2004.