On syzygies over 2-Calabi-Yau tilted algebras

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.