Projective resolutions and quasi-heredity
type: Item
instance of: scholarly article
title: Projective resolutions and quasi-heredity
zbMATH Open document ID: 0723.16004
DOI: 10.1007/BF01198636
author: Steffen Koenig
published in: Archiv der Mathematik
publication date: 1992-01-01T00:00:00Z
review text: Let A be an artinian ring. \textit{E. Green} defined [Representation Theory II, Lect. Notes Math. 832, 259-279 (1980; Zbl 0491.16015)] the type of A via the existence of a partial ordering of the set of isomorphism classes of indecomposable projective A-modules, such that the ordering satisfies conditions given by certain projective resolutions. In this note, the connection between a slightly modified version of Green's definition and the concept of quasi-heredity [\textit{E. Cline, B. Parshall} and \textit{L. L. Scott}: J. Reine Angew. Math. 391, 85-99 (1988; Zbl 0657.18005)] is studied. It is shown, that an artinian ring of type at most one is quasi- hereditary. This generalizes a result of \textit{V. Dlab} and \textit{C. M. Ringel} [Ill. J. Math. 33, 280-291 (1989; Zbl 0666.16014)] that global dimension two implies quasi-heredity. Moreover, endomorphism rings of certain projective modules over an algebra A of type at most 1 are studied and it is shown, that chains of such endomorphism rings can be chosen with all rings in the chain again being of type at most 1 and having global dimension less than or equal to gldim(A). To all definitions and results, analogues for orders over discrete valuation rings are given, using the author's concept of quasi-hereditary orders [Manuscr. Math. 68, 417-433 (1990; Zbl 0706.16008)].
reviewed by: Steffen Koenig
Mathematics Subject Classification ID: 16G10
Mathematics Subject Classification ID: 16D40
Mathematics Subject Classification ID: 16E10
Mathematics Subject Classification ID: 16G30
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 16P20
zbMATH DE Number: 4191914
zbMATH Keywords: artinian ring
zbMATH Keywords: indecomposable projective A-modules
zbMATH Keywords: projective resolutions
zbMATH Keywords: quasi-heredity
zbMATH Keywords: global dimension
zbMATH Keywords: endomorphism rings
zbMATH Keywords: quasi- hereditary orders
MaRDI profile type: Publication
cites work: Q3804882
cites work: Quasi-hereditary algebras
cites work: Q3952254
cites work: On hereditary orders
cites work: Quasi-hereditary orders
cites work: Global dimension two orders are quasi-hereditary
cites work: Q3495525
cites work: Lattices over orders. II
cites work: Lattices over orders I
cites work: A Characterization of Artinian Rings whose Endomorphism Rings have Finite Global Dimension
full work available at URL: https://doi.org/10.1007/bf01198636
OpenAlex ID: W2019062230
description: scientific article; zbMATH DE number 4191914