On algebras of generalized standard semiregular type
This paper is part of the general theme of understanding algebras in terms of their Auslander-Reiten quivers. Briefly, the Auslander-Reiten quiver of an algebra \(A\) has vertices corresponding to the isomorphism classes of indecomposable \(A\)-modules, with arrows corresponding to irreducible homomorphisms (that is, non-isomorphisms which don't factor through an intermediate indecomposable module). The connected components of the Auslander-Reiten quiver can be divided into various types; a component \(\mathscr C\) is called \textit{generalized standard} if \(\mathrm{rad}^\infty_A(X,Y)=0\) for all modules \(X,Y\) in \(\mathscr C\), and \(A\) is of \textit{generalized standard type} if all components have this property. A component is \textit{semiregular} if it does not contain both a projective and an injective module, and \(A\) is semiregular if all components are semiregular. The principal aim of the paper is to consider algebras with either or both of these properties. The main results give equivalences between various different conditions in terms of the Auslander-Reiten quiver and constructions involving tame quasitilted algebras. This is a long and technical paper. In the section on preliminaries alone more then twenty terms are defined, and more than fifteen sources are cited. Nevertheless, the style of writing is accessible, and I am sure that experts will find this paper both interesting and useful. However, I think the learning curve is too steep, and the motivation too narrow, for this paper to attract outsiders to this research area.
scientific article; zbMATH DE number 6539248
scientific article