Appendix: Some remarks concerning tilting modules and tilted algebras. Oringin. Relevance. Future

A need was felt to make available surveys on the basic properties of tilting modules, tilting complexes and tilting functors, to collect outlines of the relationship to similar constructions in algebra and geometry, as well as reports on the growing number of generalizations. At the time the Handbook was conceived, there was a general consensus about the overall frame of tilting theory, with the tilted algebra as the core, surrounded by a lot of additional considerations and with many applications in algebra and geometry. One was still looking forward to further generalizations (say something like “pre-semi-tilting procedures for near-rings”), but the core of tilting theory seemed to be in a final shape. The Handbook was supposed to provide a full account of the theory as it was known at that time. The editors of this Handbook have to be highly praised for what they have achieved. But the omissions which were necessary in order to bound the size of the volume clearly indicate that there should be a second volume. Part 1 will provide an outline of this core of tilting theory. Part 2 will then be devoted to topics where tilting modules and tilted algebras have shown to be relevant. I have to apologize that these parts will repeat some of the considerations of various chapters of the Handbook, but such a condensed version may be helpful as a sort of guideline. Both Parts 1 and 2 contain historical annodations and reminiscences. The final Part 3 will be a short report on some striking recent developments which are motivated by the cluster theory of Fomin and Zelevinsky. In particular, we will guide the reader to the basic properties of cluster tilted algebras