Piecewise hereditary one-point extensions
type: Item
instance of: scholarly article
title: Piecewise hereditary one-point extensions
zbMATH Open document ID: 0962.16013
author: Stefan Lache
published in: Journal of Algebra
publication date: 2001-03-19T00:00:00Z
review text: Throughout this review, by category is meant an Abelian locally finite \(k\)-category, where \(k\) is a field. If \(\mathcal A\) is such a category, we denote by \(D^b({\mathcal A})\) the bounded derived category of complexes over \(\mathcal A\). A finite-dimensional \(k\)-algebra \(A\) is said to be piecewise hereditary of type \(\mathcal H\) if there exists a hereditary category \(\mathcal H\) such that \(D^b(A)\) is triangle equivalent to \(D^b({\mathcal H})\). Examples of piecewise hereditary algebras are quasitilted algebras. While there are many results on quasitilted algebras, not much is known so far about the properties of piecewise hereditary algebras and their module categories. The author investigates those piecewise hereditary algebras which arise as one-point extensions of connected wild hereditary algebras. It is shown that if \(A\) is a wild hereditary algebra and \(M\) is an \(A\)-module such that \(A[M]\) is piecewise hereditary, then either (1) \(M\) is preprojective or preinjective and then \(A[M]\) is tilted, or (2) \(M\) is regular, indecomposable and quasisimple and then there exists \(r_0\in\mathbb{Z}\) such that \(A[\tau^r M]\) is quasitilted of type \(\mathcal H\) for all \(r\geq r_0\). The structure of the Auslander-Reiten quiver of piecewise hereditary algebras occurring in case (2) and for which \({\mathcal H}=H\text{-mod}\) for \(H\) a connected wild hereditary algebra is also investigated (note that case (1) is well-known). In fact it is shown that the connected components of the Auslander-Reiten quiver have the following structure: (1) There is one preprojective component which is the preprojective component of \(A\). (2) There is one preinjective component which is the preinjective component of a wild concealed factor algebra \(C\) of \(A[M]\). (3) The remaining components are stable of type \(\mathbb{Z} A_\infty\). This situation is similar to the one observed for tilted algebras, the only difference being the absence of a connectin…
reviewed by: Francois Huard
Mathematics Subject Classification ID: 16G70
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 18E30
Mathematics Subject Classification ID: 16G30
Mathematics Subject Classification ID: 16G60
zbMATH DE Number: 1441795
zbMATH Keywords: Abelian locally finite categories
zbMATH Keywords: bounded derived categories of complexes
zbMATH Keywords: finite-dimensional algebras
zbMATH Keywords: piecewise hereditary algebras
zbMATH Keywords: quasitilted algebras
zbMATH Keywords: one-point extensions
zbMATH Keywords: connected wild hereditary algebras
zbMATH Keywords: Auslander-Reiten quivers
zbMATH Keywords: connected components
zbMATH Keywords: preprojective components
zbMATH Keywords: preinjective components
MaRDI profile type: Publication
full work available at URL: jabr.1999.8126
OpenAlex ID: W1986913727
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DOI: 10.1006/JABR.1999.8126
description: scientific article; zbMATH DE number 1441795