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 $D_n$.

50 pages. Several improvements after refereeing. arXiv admin note: text overlap with arXiv:1307.0676

References in corpus (3)

• Ice quivers with potentials associated with triangulations and Cohen-Macaulay modules over orders
• On Jacobian algebras from closed surfaces
• Tilting bundles on orders on $P^d$

Cited by in corpus (7)

• Tilting theory for Gorenstein rings in dimension one
• Algebras of partial triangulations
• Lifting preprojective algebras to orders and categorifying partial flag varieties
• From triangulated categories to module categories via homotopical algebra
• Tilting objects in singularity categories and levelled mutations
• On the boundary algebras of the Jacobian algebras of the bordered marked surfaces
• A geometric realization of socle-projective categories for posets of type $\mathbb{A}$