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

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

2006-05-04

arxiv.org/abs/math/0605136v3

Amer. J. Math. 130 (2008), no. 4, 1087--1149

Mathematics

Representation Theory

53 pages. To appear in Amer. J. Math. In 3rd version, abstract, 8.13 and 8.18 are added, and 2.3 is fixed

Scientific paper

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

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