A Geometric Model for Syzygies Over 2-Calabi–Yau Tilted Algebras II

Abstract

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 disk. 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, which 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.