On syzygies over 2-Calabi–Yau tilted algebras

Abstract

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.

Syzygy Filtrations of Cyclic Nakayama Algebras

“…The second result appears in the literature [ES17], [LM18] etc. However we give an elementary proof in the case of Nakayama algebras by new methods.…”

τ-tilting finite gentle algebras are representation-finite

“…Despite their simple definition, gentle algebras are encountered in many different contexts. They were introduced in [AS87] in the study of iterated tilted algebras of type A m , but have recently appeared in connection with dimer models [Boc12,Bro12], enveloping algebras of some Lie algebras [HK06], cluster algebras and categories arising from triangulated surfaces [LF09,ABCJP10], m-Calabi-Yau tilted algebras [GE17,GE18], non-kissing complexes of grids and associated objects [McC17, GM18, PPP17, BDM + 17], non-commutative nodal curves [BD18], and partially wrapped Fukaya categories [HKK17,LP18]. Surface models have been introduced to study the category representations of a gentle algebra and associated categories [BCS18,OPS18,PPP18].…”

On representation dimension of tame cluster tilted algebras

“…Recently, García Elsener and Schiffler in [10] proved that the representation dimension of tame cluster tilted algebras is equal to three. The proof is based on a result of Bergh and Oppermann, that the category of Cohen Macaulay modules of this algebras is finite.…”