Pure injective envelopes of finite length modules over a generalized Weyl algebra
type: Item
instance of: scholarly article
title: Pure injective envelopes of finite length modules over a generalized Weyl algebra
zbMATH Open document ID: 1023.16003
author: Gennadi Puninski
author: Mike Prest
published in: Journal of Algebra
publication date: 2002-09-08T00:00:00Z
review text: Generalised Weyl algebras have proved to be a fruitful and important domain of research and a significant source of examples. The Ziegler spectrum of a ring, the space of all isomorphism types of indecomposable pure-injective modules, is coming to be recognized as an important tool in describing the complexity of the category of modules over the ring. The inspiration for the Ziegler spectrum comes from mathematical logic (complexity of first-order definable sets). In this paper the authors investigate the structure of some of the pure-injective modules over generalised Weyl algebras. From the abstract to the paper: ``We consider the pure injective hulls of finite length modules, the elementary duals of these, torsionfree pure injective modules, and the closure in the Ziegler spectrum of the category of finite length modules supported on a non-degenerate orbit of a generalised Weyl algebra. We also show that this category is a direct sum of uniserial categories and admits almost split sequences. We find parallels to but also marked contrast with the behaviour of pure injective modules over finite dimensional algebras and hereditary orders.'' Extensive use is made of the known structure of generalised Weyl algebras and the classification of the simple modules as developed by \textit{V. V. Bavula} [Algebra Anal. 4, No. 1, 75-97 (1992; Zbl 0807.16027), English translation in St. Petersb. Math. J. 4, No. 1, 71-92 (1993)] and by \textit{V. Bavula} and \textit{F. Van Oystaeyen} [J. Algebra 194, No. 2, 521-566 (1997; Zbl 0927.16002)]. The structure theory developed by the authors is quite interesting and I will summarize some of the main results below. The problem of classifying all the indecomposable pure-injective modules over a generalized Weyl algebra \(A\) is unsolvable, so the authors look for likely subfamilies that might be classifiable. One starting point is suggested by the classification of the simple modules: the family of finite length modules supported on a no…
reviewed by: Thomas Kucera
Mathematics Subject Classification ID: 16D50
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 16D60
Mathematics Subject Classification ID: 16D80
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 16B70
Mathematics Subject Classification ID: 03C60
Mathematics Subject Classification ID: 16S36
Mathematics Subject Classification ID: 16G70
zbMATH DE Number: 1797467
zbMATH Keywords: pure-injective modules
zbMATH Keywords: pure-injective envelopes
zbMATH Keywords: generalised Weyl algebras
zbMATH Keywords: Ziegler spectra
zbMATH Keywords: categories of modules
zbMATH Keywords: finite length modules
zbMATH Keywords: almost split sequences
zbMATH Keywords: simple modules
zbMATH Keywords: direct sums
zbMATH Keywords: uniserial homogeneous modules
zbMATH Keywords: dualities
Wikidata QID: Q122689207
MaRDI profile type: Publication
OpenAlex ID: W2072557515
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DOI: 10.1006/JABR.2001.9116
description: scientific article; zbMATH DE number 1797467