Ice Quivers with Potential Arising from Once-punctured Polygons and Cohen–Macaulay Modules

Abstract

Given a tagged triangulation of a once-punctured polygon $P^{\ast }$ with $n$ vertices, we associate an ice quiver with potential such that the frozen part of the associated frozen Jacobian algebra has the structure of a Gorenstein $K[X]$-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 $D_n$. It gives a natural interpretation of the usual indexation of cluster tilting objects of $\mathcal{C}$ by tagged triangulations of $P^{\ast }$. Moreover, it extends naturally the triangulated categorification by $\mathcal{C}$ of the cluster algebra of type $D_n$ 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 $D_n$.