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

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