Auslander’s representation dimension measures how far a finite dimensional algebra is away from being of finite representation type. In [1], M. Auslander proved that a finite dimensional algebra A is of finite representation type if and only if the representation dimension of A is at most 2. Recently, R. Rouquier proved that there are finite dimensional algebras of an arbitrarily large finite representation dimension. One of the exciting open problems is to show that all finite dimensional algebras...

A ring Λ satisfies the Generalized Auslander-Reiten Condition ( ) if for each Λ-module M with $Ex{t}^i(M,M\oplus \Lambda )=0$ for all i > n the projective dimension of M is at most n. We prove that this condition is satisfied by all n-symmetric algebras of quasitilted type.

Using geometrical methods, Huisgen-Zimmermann showed that if M is a module with simple top, then M has no proper degeneration $M{<}_{deg}N$ such that ${}^{t}M{/}^{t+1}M{\simeq }^{t}N{/}^{t+1}N$ for all t. Given a module M with square-free top and a projective cover P, she showed that $di{m}_{k}Hom(M,M)=di{m}_{k}Hom(P,M)$ if and only if M has no proper degeneration $M{<}_{deg}N$ where M/M ≃ N/N. We prove here these results in a more general form, for hom-order instead of degeneration-order, and we prove them algebraically. The results of Huisgen-Zimmermann follow as consequences from our results....

Let A be a finite-dimensional algebra over a field k. We discuss the existence of trisections (mod₊ A,mod₀ A,mod₋ A) of the category of finitely generated modules mod A satisfying exactness, standardness, separation and adjustment conditions. Many important classes of algebras admit trisections. We describe a construction of algebras admitting a trisection of their module categories and, in special cases, we describe the structure of the components of the Auslander-Reiten quiver lying in mod₀ A.

We prove that for any representation-finite algebra A (in fact, finite locally bounded k-category), the universal covering F: Ã → A is either a Galois covering or an almost Galois covering of integral type, and F admits a degeneration to the standard Galois covering F̅: Ã→ Ã/G, where $G=\Pi \left({\Gamma }_A\right)$ is the fundamental group of ${\Gamma }_A$. It is shown that the class of almost Galois coverings F: R → R’ of integral type, containing the series of examples from our earlier paper [Bol. Soc. Mat. Mexicana 17 (2011)], behaves...