On syzygies over 2-Calabi–Yau tilted algebras

Abstract

We characterize the syzygies and co-syzygies over 2-Calabi–Yau tilted algebras in terms of the Auslander–Reiten translation and the syzygy functor. We explore connections between the category of syzygies, the category of Cohen–Macaulay modules, the representation dimension of algebras and the Igusa–Todorov functions. In particular, we prove that the Igusa–Todorov dimensions of d-Gorenstein algebras are equal to d. For cluster-tilted algebras of Dynkin type D, we give a geometric description of the stable Cohen–Macaulay category in terms of tagged arcs in the punctured disc. We also describe the action of the syzygy functor in a geometric way. This description allows us to compute the Auslander–Reiten quiver of the stable Cohen–Macaulay category using tagged arcs and geometric moves.