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

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, we construct a polygon with a checkerboard pattern in its interior, which defines a category. The indecomposable objects of are the 2-diagonals in, and its morphisms are certain pivoting moves between the 2-diagonals. We prove that the category is equivalent to the stable syzygy category of the algebra. 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. In addition, we obtain an explicit description of the projective resolutions, which are periodic. Finally, the number of vertices of the polygon is a derived invariant and a singular invariant for dimer tree algebras, which can be easily computed form the quiver.