Edited by Lidia Angeleri Hügel
Edited by Dieter Happel
Edited by Henning Krause
Publisher: Cambridge University Press
Print Publication Year: 2007
Online Publication Date:May 2010
Online ISBN:9780511735134
Paperback ISBN:9780521680455
Chapter DOI: http://dx.doi.org/10.1017/CBO9780511735134.005
Subjects: Algebra, Geometry and topology
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Abstract
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.
Introduction
Motivation: Derived categories as higher invariants
Let k be a field and A a k-algebra (associative, with 1). We are especially interested in the case where A is a non commutative algebra. In order to study A, one often looks at various invariants associated with A, for example its Grothendieck group K0(A), its center Z(A), its higher K-groups Ki(A), its Hochschild cohomology groups HH * (A,A), its cyclic cohomology groups …. Of course, each isomorphism of algebras A → B induces an isomorphism in each of these invariants. More generally, for each of them, there is a fundamental theorem stating that the invariant is preserved not only under isomorphism but also under passage from A to a matrix ring Mn A, and, more generally, that it is preserved under Morita equivalence.
pp. i-iv
pp. v-viii
pp. 1-8
2 - Basic results of classical tilting theory: Read PDF
pp. 9-14
3 - Classification of representation-finite algebras and their modules: Read PDF
pp. 15-30
4 - A spectral sequence analysis of classical tilting functors: Read PDF
pp. 31-48
5 - Derived categories and tilting: Read PDF
pp. 49-104
6 - Hereditary categories: Read PDF
pp. 105-146
7 - Fourier-Mukai transforms: Read PDF
pp. 147-178
8 - Tilting theory and homologically finite subcategories with applications to quasihereditary algebras: Read PDF
pp. 179-214
9 - Tilting modules for algebraic groups and finite dimensional algebras: Read PDF
pp. 215-258
10 - Combinatorial aspects of the set of tilting modules: Read PDF
pp. 259-278
11 - Infinite dimensional tilting modules and cotorsion pairs: Read PDF
pp. 279-322
12 - Infinite dimensional tilting modules over finite dimensional algebras: Read PDF
pp. 323-344
13 - Cotilting dualities: Read PDF
pp. 345-358
14 - Representations of finite groups and tilting: Read PDF
pp. 359-392
15 - Morita theory in stable homotopy theory: Read PDF
pp. 393-412
Appendix: Some remarks concerning tilting modules and tilted algebras. Origin. Relevance. Future.: Read PDF
pp. 413-472
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