## 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 $A_{n-3}$.## References

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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