English

Fomin-Zelevinsky mutation and tilting modules over Calabi-Yau algebras

Representation Theory 2010-11-01 v3 Commutative Algebra

Abstract

We say that an algebra Λ\Lambda over a commutative noetherian ring RR is Calabi-Yau of dimension dd (dd-CY) if the shift functor [d][d] gives a Serre functor on the bounded derived category of the finite length Λ\Lambda-modules. We show that when RR is dd-dimensional local Gorenstein the dd-CY algebras are exactly the symmetric RR-orders of global dimension dd. 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.

Keywords

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

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