On algebras of $Ω^n$-finite and $Ω^{\infty}$-infinite representation type

Abstract

Co-Gorenstein algebras were introduced by A. Beligiannis in \cite{B}. In \cite{KM}, the authors propose the following conjecture (Co-GC): if $Ω^n (\mod A)$ is extension closed for all $n \leq 1$, then $A$ is right Co-Gorenstein, and they prove that the Generalized Nakayama Conjecture implies the Co-GC, also that the Co-GC implies the Nakayama Conjecture. In this article we characterize the subcategory $Ω^{\infty}(\mod A)$ for algebras of $Ω^{n}$-finite representation type. As a consequence, we characterize when a truncated path algebra is a Co-Gorenstein algebra in terms of its associated quiver. We also study the behaviour of Artin algebras of $Ω^{\infty}$-infinite representation type. Finally, it is presented an example of a non Gorenstein algebra of $Ω^{\infty}$-infinite representation type and an example of a finite dimensional algebra with infinite $ϕ$-dimension.