This is a joint work in progress with Khrystyna Serhiyenko.
The syzygy functor maps a module to the kernel of its projective cover. Thus a syzygy is a submodule of a projective. We want to study the stable category of syzygies over a 2-Calabi–Yau tilted algebra. For these algebras, this category is equivalent to the stable category of Cohen-Macauley modules, as well as to the singularity category of the algebra. It is a triangulated 3-Calabi–Yau category whose shift is given by the syzygy functor.
In this talk, I will present a geometric model for this category for a particular type of 2-CY tilted algebras. The algebra will be defined by specifying its quiver and relations. The syzygies, or rather their projective presentations, will be represented as diagonales (in fact 2-diagonals) in a regular polygon. This polygon is equipped with a checkerboard pattern defined by a fixed system of n diagonals, one for each vertex of the quiver. The projective presentation of the syzygy is given by the intersections of its diagonal with the fixed system of diagonals. This model also provides an interpretation of the irreducible morphisms, the Auslander-Reiten translation and the syzygy functor.