On perpendicular categories of stones over quiver algebras
Perpendicular categories were introduced by Geigle, Lenzing and Schofield. Since then, they became a powerful tool in the representation theory of finite dimensional algebras, especially, when one deals with questions on quiver algebras. In general, to determine the perpendicular categories seems to be not an easy task, even in certain special cases, for example, to determine the perpendicular category of a stone (i.e., an indecomposable module without selfextension), one often meets difficulties. In the reviewed paper, the authors open a window to attack this problem for quiver algebras. They prove that for a quiver algebra there are (up to Auslander-Reiten translation \(\tau\)) finitely many stones which have a predescribed perpendicular category. This result has many interesting consequences, among them are the following: (1) There are finitely many \(\tau\)-orbits containing regular stones of quasi-length at least two. (2) There are finitely many elementary Kerner-bijections between the set of regular components of a quiver algebra \(H\) and the set of regular components of a quiver algebra \(A\) with one simple module less than \(H\) which are induced by tilting modules. (3) Given a wild quiver algebra and a nonnegative integer \(n\), there are only finitely many regular components containing bricks of quasi-length 2 having self-extensions of dimension \(n\). The tool used in the proof of the main result is based on a combination of developments of exceptional sequences, non-regular stones, tilting theory and perpendicular techniques.
scientific article; zbMATH DE number 949855