On perpendicular categories of stones over quiver algebras
type: Item
instance of: scholarly article
title: On perpendicular categories of stones over quiver algebras
zbMATH Open document ID: 0864.16010
DOI: 10.1007/BF02566431
author: Dieter Happel
author: Silke Hartlieb
author: Otto Kerner
author: Luise Unger
published in: Commentarii Mathematici Helvetici
publication date: 1997-06-12T00:00:00Z
full work available at URL: https://eudml.org/doc/140409
review text: 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.
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 16G10
Mathematics Subject Classification ID: 16G70
zbMATH DE Number: 949855
zbMATH Keywords: finite dimensional algebras
zbMATH Keywords: quiver algebras
zbMATH Keywords: perpendicular categories
zbMATH Keywords: Auslander-Reiten translations
zbMATH Keywords: regular stones
zbMATH Keywords: elementary Kerner-bijections
zbMATH Keywords: regular components
zbMATH Keywords: tilting modules
zbMATH Keywords: wild quiver algebras
zbMATH Keywords: bricks
zbMATH Keywords: self-extensions
zbMATH Keywords: exceptional sequences
zbMATH Keywords: tilting theory
reviewed by: Changchang Xi
MaRDI profile type: Publication
OpenAlex ID: W1976743783
Recommended article: Perpendicular categories with applications to representations and sheaves
Recommended article: Generalized quiver varieties and triangulated categories
Recommended article: Quiver Hecke algebras and categorification
Recommended article: Quivers, Perverse Sheaves, and Quantized Enveloping Algebras
Recommended article: Quiver algebras as Fukaya categories
Recommended article: Quasi‐Stone algebras
Recommended article: On the perpendicular category of a partial tilting module
Recommended article: Varieties of quasi-Stone algebras
Recommended article: Categorical Realizations of Quivers
Recommended article: Perverse sheaves and quivers
description: scientific article; zbMATH DE number 949855