On algebras of Ωn-finite and Ω∞-infinite representation type

Abstract

Co-Gorenstein algebras were introduced by Beligiannis in [A. Beligiannis, The homological theory of contravariantly finite subcategories: Auslander–Buchweitz contexts, Gorenstein categories and co-stabilization, Comm. Algebra 28(10) (2000) 4547–4596]. In [S. Kvamme and R. Marczinzik, Co-Gorenstein algebras, Appl. Categorical Struct. 27(3) (2019) 277–287], the authors propose the following conjecture (co-GC): if [Formula: see text] is extension closed for all [Formula: see text], then [Formula: see text] 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 paper, we characterize the subcategory [Formula: see text] for algebras of [Formula: see text]-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 behavior of Artin algebras of [Formula: see text]-infinite representation type. Finally, an example of a non-Gorenstein algebra of [Formula: see text]-infinite representation type and an example of a finite dimensional algebra with infinite [Formula: see text]-dimension are presented.