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

Co-Gorenstein algebras were introduced by A. Beligiannis [Commun. Algebra 28, No. 10, 4547–4596 (2000; Zbl 0964.18006)]. In [Appl. Categ. Struct. 27, No. 3, 277–287 (2019; Zbl 1470.16027)] S. Kvamme and R. Marczinzik proposed the co-Gorenstein conjecture: if the full subcategory Ωn (mod A) of the category modA of finitely generated right modules over an Artin algebra A is extension closed for all n ≥ 1, then A is right co-Gorenstein, and showed that the generalized Nakayama conjecture implies the co-Gorenstein conjecture, and the co-Gorenstein conjecture implies the Nakayama conjecture. In the paper under review, the authors prove that for an Artin algebra A of Ωn-finite representation type, a module is in the subcategory Ω∞ (mod A) if and only if it is either projective or Ω-periodic. Furthermore, they characterize a truncated path algebra to be co-Gorenstein in terms of its associated quiver. Moreover, they provide the examples of non-Gorenstein algebras of Ω∞-infinite representation type, and an example of a finite dimensional algebra with infinite ϕ-dimension, which is a counterexample of the ϕ-dimension conjecture. Note that the ϕ-dimension conjecture is also disproved independently by E. J. Hanson and K. Igusa [Math. Z. 300, No. 1, 807–826 (2022; Zbl 1489.16008)]. R