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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.

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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