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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Ice quivers with potential associated with triangulations and Cohen-Macaulay modules over orders


Authors: Laurent Demonet and Xueyu Luo
Journal: Trans. Amer. Math. Soc. 368 (2016), 4257-4293
MSC (2010): Primary 16G20, 16H20, 13C14, 13F60
DOI: https://doi.org/10.1090/tran/6463
Published electronically: October 8, 2015
MathSciNet review: 3453371
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Abstract: Given a triangulation of a polygon $P$ with $n$ vertices, we associate an ice quiver with potential such that the frozen part of the associated Jacobian algebra has the structure of a Gorenstein tiled $K[x]$-order $\Lambda$. Then we show that the stable category of the category of Cohen-Macaulay $\Lambda$-modules is equivalent to the cluster category $\mathcal {C}$ of type $A_{n-3}$. It gives a natural interpretation of the usual indexation of cluster tilting objects of $\mathcal {C}$ by triangulations of $P$. Moreover, it extends naturally the triangulated categorification by $\mathcal {C}$ of the cluster algebra of type $A_{n-3}$ to an exact categorification by adding coefficients corresponding to the sides of $P$. Finally, we lift the previous equivalence of categories to an equivalence between the stable category of graded Cohen-Macaulay $\Lambda$-modules and the bounded derived category of modules over a path algebra of type $A_{n-3}$.


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Additional Information

Laurent Demonet
Affiliation: Department of Mathematics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464-8601, Japan

Xueyu Luo
Affiliation: Graduate School of Mathematics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464-8602, Japan

Received by editor(s): November 8, 2013
Received by editor(s) in revised form: April 22, 2014
Published electronically: October 8, 2015
Article copyright: © Copyright 2015 American Mathematical Society