×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI arXiv