A geometric model for syzygies over 2-Calabi-Yau tilted algebras II

TL;DR

This paper establishes a geometric model linking the stable syzygy category of certain 2-Calabi-Yau tilted algebras, called dimer tree algebras, to a polygon with a checkerboard pattern, revealing finiteness and periodicity properties.

Contribution

It constructs a geometric category equivalent to the syzygy category of dimer tree algebras, confirming a conjecture and providing explicit periodic projective resolutions.

Findings

The syzygy category is equivalent to a 2-cluster category of type A.

The number of indecomposable syzygies is finite.

Projective resolutions are explicitly described and periodic.

Abstract

In this article, we continue the study of a certain family of 2-Calabi-Yau tilted algebras, called dimer tree algebras. The terminology comes from the fact that these algebras can also be realized as quotients of dimer algebras on a disc. They are defined by a quiver with potential whose dual graph is a tree, and they are generally of wild representation type. Given such an algebra $B$, we construct a polygon $\mathcal{S}$ with a checkerboard pattern in its interior, that defines a category $\text{Diag}(\mathcal{S})$. The indecomposable objects of $\text{Diag}(\mathcal{S})$ are the 2-diagonals in $\mathcal{S}$, and its morphisms are certain pivoting moves between the 2-diagonals. We prove that the category $\text{Diag}(\mathcal{S})$ is equivalent to the stable syzygy category of the algebra $B$. This result was conjectured by the authors in an earlier paper, where it was proved in the special case where every chordless cycle is of length three. As a consequence, we conclude that the number of indecomposable syzygies is finite, and moreover the syzygy category is equivalent to the 2-cluster category of type $\mathbb{A}$. In addition, we obtain an explicit description of the projective resolutions, which are periodic. Finally, the number of vertices of the polygon $\mathcal{S}$ is a derived invariant and a singular invariant for dimer tree algebras, which can be easily computed form the quiver.