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

Abstract

Given a triangulation of a polygon P P with n 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 ] K[x] -order Λ \Lambda . Then we show that the stable category of the category of Cohen-Macaulay Λ \Lambda -modules is equivalent to the cluster category C \mathcal {C} of type A n − 3 A_{n-3} . It gives a natural interpretation of the usual indexation of cluster tilting objects of C \mathcal {C} by triangulations of P P . Moreover, it extends naturally the triangulated categorification by C \mathcal {C} of the cluster algebra of type A n − 3 A_{n-3} to an exact categorification by adding coefficients corresponding to the sides of P P . Finally, we lift the previous equivalence of categories to an equivalence between the stable category of graded Cohen-Macaulay Λ \Lambda -modules and the bounded derived category of modules over a path algebra of type A n − 3 A_{n-3} .