Fomin-Zelevinsky Mutation and Tilting Modules over Calabi-Yau Algebras

We say that an algebra Λ 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 Λ-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 noncommutative crepant resolutions.