A geometric model for syzygies over 2CalabiYau tilted algebras II
Abstract
In this article, we continue the study of a certain family of 2CalabiYau 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 2diagonals in $\mathcal{S}$, and its morphisms are certain pivoting moves between the 2diagonals. 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 2cluster 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.
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 arXiv:
 arXiv:2110.09976
 Bibcode:
 2021arXiv211009976S
 Keywords:

 Mathematics  Representation Theory;
 Mathematics  Combinatorics;
 16G20;
 13F60
 EPrint:
 32 pages, 17 figures