Ice quivers with potential arising from once-punctured polygons and Cohen-Macaulay modules

TL;DR

This paper constructs a new algebraic framework linking tagged triangulations of once-punctured polygons to Cohen-Macaulay modules and cluster categories of type D, enriching the categorification of type D cluster algebras.

Contribution

It introduces an ice quiver with potential from triangulations, establishing an equivalence between Cohen-Macaulay modules and cluster categories, and extends categorification to include coefficients.

Findings

Stable category of Cohen-Macaulay modules is equivalent to cluster category of type D_n.

Provides a natural indexing of cluster tilting objects via tagged triangulations.

Extends cluster algebra categorification to include coefficients and graded modules.

Abstract

Given a tagged triangulation of a once-punctured polygon $P^*$ 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 $P^*$. 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 $D_n$.