Lattices over subhereditary orders and socle-projective modules
type: Item
instance of: scholarly article
title: Lattices over subhereditary orders and socle-projective modules
zbMATH Open document ID: 0665.16004
DOI: 10.1016/0021-8693(89)90084-7
author: Klaus W. Roggenkamp
published in: Journal of Algebra
publication date: 1989-01-01T00:00:00Z
review text: A classical order \(\Lambda\) is called subhereditary if there is a hereditary overorder \(\Gamma\) with Rad \(\Gamma\subset \Lambda\). The \(\Lambda\)-lattices are then closely related to the socle-projective modules over the algebra \[ {\mathcal D}=\begin{pmatrix} \Gamma /Rad \Gamma & \Gamma /Rad \Gamma \\ 0& \Lambda /Rad \Gamma \end{pmatrix}. \] In the case of a generalized Bäckström order \(\Lambda\), \({\mathcal D}\) is a hereditary algebra. This case has been treated in previous work of the author [Mitt. Math. Semin. Gießen 159 (1983; Zbl 0513.16003)]. The theorems of equivalence concerning \(\Lambda\) and \({\mathcal D}\), existence of Auslander-Reiten sequences, and Brauer-Thrall 1 1/2 are presented in a very general context. In the classical situation where \(\Lambda\) is an order in a separable algebra over a local field, the Auslander-Reiten quiver \(A(\Lambda)\) of \(\Lambda\) is described. It is shown (Theorem IV) that indecomposable \(\Gamma\)-lattices have \(\Lambda\)- projective successors in A(\(\Lambda)\). The representation-finite subhereditary orders \(\Lambda\), for which \({\mathcal S}({\mathcal D})\) (the category of f.g. \({\mathcal D}\)-modules with projective socle) has preprojective components, are characterized in terms of A(\(\Lambda)\).
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 16Gxx
zbMATH DE Number: 4087853
zbMATH Keywords: classical order
zbMATH Keywords: \(\Lambda\)-lattices
zbMATH Keywords: socle-projective modules
zbMATH Keywords: generalized Bäckström order
zbMATH Keywords: hereditary algebra
zbMATH Keywords: Auslander-Reiten sequences
zbMATH Keywords: Brauer-Thrall
zbMATH Keywords: Auslander-Reiten quiver
zbMATH Keywords: representation-finite subhereditary orders
zbMATH Keywords: preprojective components
reviewed by: Wolfgang Rump
MaRDI profile type: Publication
full work available at URL: 0021-8693(89)90084-7
OpenAlex ID: W2094440485
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description: scientific article; zbMATH DE number 4087853