Fomin-Zelevinsky mutation and tilting modules over Calabi-Yau algebras
Abstract
We say that an algebra over a commutative noetherian ring is Calabi-Yau of dimension (-CY) if the shift functor gives a Serre functor on the bounded derived category of the finite length -modules. We show that when is -dimensional local Gorenstein the -CY algebras are exactly the symmetric -orders of global dimension . We give a complete description of all tilting modules of projective dimension at most one for 2-CY algebras, and show that they are in bijection with elements of affine Weyl groups, preserving various natural partial orders. We show that there is a close connection between tilting theory for 3-CY algebras and the Fomin-Zelevinsky mutation of quivers (or matrices). We prove a conjecture of Van den Bergh on derived equivalence of non-commutative crepant resolutions.
Cite
@article{arxiv.math/0605136,
title = {Fomin-Zelevinsky mutation and tilting modules over Calabi-Yau algebras},
author = {Osamu Iyama and Idun Reiten},
journal= {arXiv preprint arXiv:math/0605136},
year = {2010}
}
Comments
53 pages. To appear in Amer. J. Math. In 3rd version, abstract, 8.13 and 8.18 are added, and 2.3 is fixed