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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Ice quivers with potential associated with triangulations and Cohen-Macaulay modules over orders
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by Laurent Demonet and Xueyu Luo PDF
Trans. Amer. Math. Soc. 368 (2016), 4257-4293 Request permission

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
  • © Copyright 2015 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 368 (2016), 4257-4293
  • MSC (2010): Primary 16G20, 16H20, 13C14, 13F60
  • DOI: https://doi.org/10.1090/tran/6463
  • MathSciNet review: 3453371