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 |$B$|⁠ , 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. [ABSTRACT FROM AUTHOR]