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Fomin-Zelevinsky mutation and tilting modules over Calabi-Yau algebras. (English) Zbl 1162.16007
Let $$\Lambda$$ be an algebra over a commutative Noetherian ring $$R$$. Then $$\Lambda$$ is said to be Calabi-Yau of dimension $$d$$ (or $$d$$-Calabi-Yau) provided the $$d$$-th power of the shift functor on the bounded derived category of the category of finite length $$\Lambda$$-modules gives a Serre functor.
The first main result of this article is to show that if $$R$$ is $$d$$-dimensional local Gorenstein, an $$R$$-algebra $$\Lambda$$ is $$d$$-Calabi-Yau if and only if it is a symmetric $$R$$-order of global dimension $$d$$.
The authors also describe the tilting modules (of projective dimension less than or equal to $$1$$) for $$2$$-Calabi-Yau algebras in terms of affine Weyl groups. They go on to show that the change in the quiver of the endomorphism ring of a tilting module over a $$3$$-Calabi-Yau algebra is given by Fomin-Zelevinsky cluster mutation. Finally, they prove a conjecture of M. Van den Bergh [Duke Math. J. 122, No. 3, 423-455 (2004; Zbl 1074.14013) and in The legacy of Niels Henrik Abel. Papers from the Abel bicentennial conference, Univ. Oslo, Oslo, Norway 2002. Berlin: Springer. 749-770 (2004; Zbl 1082.14005)] concerning non-commutative crepant resolutions.

##### MSC:
 16G10 Representations of associative Artinian rings 16G20 Representations of quivers and partially ordered sets 16G50 Cohen-Macaulay modules in associative algebras 18E30 Derived categories, triangulated categories (MSC2010) 14A22 Noncommutative algebraic geometry 16S38 Rings arising from noncommutative algebraic geometry 16D90 Module categories in associative algebras
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