Delooping level of Nakayama algebras

Abstract

We show that for a given Nakayama algebra Θ, there exist countably many cyclic Nakayama algebras Λ i , where i ∈ N, such that the syzygy filtered algebra of Λ i is isomorphic to Θ and we describe those algebras Λ i . We show, among these algebras, there exists a unique algebra Λ where the defects, representing the number of indecomposable injective but not projective modules, remain invariant for both Θ and Λ. As an application, we achieve the classification of cyclic Nakayama algebras that are minimal Auslander-Gorenstein and dominant Auslander-regular algebras of global dimension three. Specifically, by using the Auslander-Iyama correspondence, we obtain cluster-tilting objects for certain Nakayama algebras. Additionally, we introduce cosyzygy filteed algebras and show that it is dual of syzygy filtered algebra.