Artin algebras with directing indecomposable projective modules
Let \(A\) be an Artin algebra. We denote by \(\text{mod }A\) the category of its finitely generated right modules. An indecomposable projective \(P\) in \(\text{mod }A\) is called directing if it belongs to no cycle \(P = M_0 @>f_1>> M_1 @>f_2>> \dots @>f_n>> M_n = P\) where the \(M_i\) are indecomposable objects in \(\text{mod }A\) and the \(f_i\) are non-zero and non isomorphisms. The aim of this paper is to study the structure and the representation type of the Artin algebras such that all indecomposable projectives are directing. For this purpose, the authors define the notion of a special directing one-point extension, and their first main theorem states that an Artin algebra is such that all its indecomposable projectives are directing if and only if it is obtained from a semisimple Artin algebra by an iteration of special directing one-point extensions. They also characterize these algebras by the structure of the components of their Auslander-Reiten quivers containing a non-directing module. Among the consequences of this theorem is that, if \(A\) is an Artin algebra whose Auslander-Reiten quiver has a regular component which is not quasi-serial, then \(A\) has both a non-directing indecomposable projective and a non-directing indecomposable injective module. The authors then study the representation type of an Artin algebra \(A\) such that every indecomposable projective module is directing and show that such an algebra is not wild if and only if it is not strictly wild, if and only if it is cycle-finite, if and only if \(\text{rad}^{\infty}(\text{mod }A)\) is nilpotent, and if and only if \(A\) is a multicoil algebra of a special type. If, moreover, \(A\) is a finite dimensional algebra over an algebraically closed field, then these conditions are equivalent to the fact that \(A\) is tame, and even domestic. As a consequence, if \(A\) is a tame non-domestic algebra over an algebraically closed field, then \(A\) has a non-directing indecomposable projective module and a non-directing indecomposable injective module.
scientific article; zbMATH DE number 589956