Handbook of Tilting Theory
This is the long awaited handbook of tilting theory whose publication was first suggested at the Twenty Years of Tilting Theory Colloquium at Fraueninsel in November 2002. In a series of 15 survey chapters written by leading experts in the area, the handbook presents all major aspects and applications of tilting theory starting from the 'pre-history' in the 1973 works of Bernstein, Gelfand and Ponomarev up to the very recent cluster tilting approach. The key point of classical tilting theory is the close relationship between the module category of the original algebra and of the tilted (endomorphism) algebra End(T) where T is a tilting module. This goes back to the works of Brenner-Butler (1979) and Happel-Ringel (1982). The relationship is in terms of a pair of category equivalences generalizing Morita equivalence, and it is particularly strong in the case when the original algebra is hereditary.
This theory and its fundamental applications to algebras of finite representation type are presented in chapters 2 and 3. A further major step was Ricard's 1989 work making tilting theory part of Morita theory for derived categories. This approach is explained in chapters 4 and 5, while its recent applications to modular representation theory of finite groups is covered in chapter 14. Chapter 7 surveys the recent use of derived categories in non-commutative algebraic geometry, while Morita theory for ring spectra and its role in algebraic topology are presented in chapter 15. Happel's theorem characterizing hereditary categories with a tilting object and its applications appear in chapter 6. The simplicial complex of tilting modules is presented in chapter 10.
In 1991 Auslander and Reiten started to use tilting theory to study homologically finite subcategories of modules. These results and their applications to quasi-hereditary algebras and algebraic groups are surveyed in chapters 8 and 9. Infinite dimensional tilting theory and its relations to finitistic dimension conjectures are presented in chapters 11 and 12. The dual notions of infinite dimensional cotilting modules and the corresponding generalizations of Morita duality theory appear in chapter 13. Of particular interest is the appendix written by C. M. Ringel: it both ties up the individual chapters of the handbook and serves as an introduction to the new field of cluster tilting. The handbook presents a key and very active part of contemporary representation theory in a concise but complete way. It will be indispensable for a wide audience, from graduate students to active researchers in algebra, geometry and topology.