On algebras of ${\Omega }^n$-finite and ${\Omega }^{\infty }$-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 ${\Omega }^n(\mathrm{mod}A)$ is extension closed for all $n\le 1$, 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 ${\Omega }^{\infty }(\mathrm{mod}A)$ for algebras of ${\Omega }^n$-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 ${\Omega }^{\infty }$-infinite representation type. Finally, it is presented an example of a non Gorenstein algebra of ${\Omega }^{\infty }$-infinite representation type and an example of a finite dimensional algebra with infinite $\varphi$-dimension.