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Methods Citations. Citation Type. Has PDF. Publication Type. More Filters. We introduce a relative tilting theory in abelian categories and show that this work offers a unified framework of different previous notions of tilting, ranging from Auslander-Solberg relative … Expand.
View 2 excerpts, cites methods. Classification of representation-finite algebras and their modules T. A spectral sequence analysis of classical tilting functors S. Brenner and M. Butler; 5. Derived categories and tilting B. Keller; 6. Fourier-Mukai transforms L. Hille and M. Van den Bergh; 7. Tilting theory and homologically finite subcategories with applications to quasihereditary algebras I. Reiten; 8. Tilting modules for algebraic groups and finite dimensional algebras S. Donkin; 9.
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 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 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 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
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