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

TL;DR

This paper characterizes the subcategory of modules with infinite syzygy chains in certain algebras, explores conditions for Co-Gorenstein algebras, and provides examples illustrating complex representation types.

Contribution

It offers a characterization of $ ext{Omega}^ ext{infinity}$ modules for algebras with finite $ ext{Omega}^n$-representation type and analyzes properties of algebras with infinite $ ext{Omega}^ ext{infinity}$-representation type.

Findings

Characterization of $ ext{Omega}^ ext{infinity}( ext{mod }A)$ for $ ext{Omega}^n$-finite algebras.

Criteria for truncated path algebras to be Co-Gorenstein.

Examples of non Gorenstein algebras with infinite $ ext{Omega}^ ext{infinity}$-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 $\Omega^n (\mod A)$ 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 $\Omega^{\infty}(\mod A)$ 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 $\phi$-dimension.