On syzygies over 2-Calabi-Yau tilted algebras

TL;DR

This paper characterizes syzygies over 2-Calabi-Yau tilted algebras, explores their connections with Cohen-Macaulay modules and Igusa-Todorov functions, and provides geometric descriptions for cluster-tilted algebras of Dynkin type D.

Contribution

It offers a new characterization of syzygies in 2-Calabi-Yau tilted algebras and links algebraic properties with geometric models for Dynkin type D.

Findings

Igusa-Todorov dimensions of d-Gorenstein algebras equal d

Geometric description of the stable Cohen-Macaulay category for Dynkin type D

Computed the Auslander-Reiten quiver using tagged arcs and geometric moves

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.