DERIVED CATEGORIES AND TILTING
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.