Graded Artin algebras, rational series, and bounds for homological dimensions
type: Item
instance of: scholarly article
title: Graded Artin algebras, rational series, and bounds for homological dimensions
zbMATH Open document ID: 0609.16001
DOI: 10.1016/0021-8693(87)90009-3
author: Dan Zacharia
published in: Journal of Algebra
publication date: 1987-01-01T00:00:00Z
review text: The author associates to every finitely generated graded module over a graded finite dimensional algebra a rational power series which is slightly different from the usual Poincaré-Betti series. Using rationality he is able to prove that the finitistic dimension conjecture is valid for a certain class of finitely generated graded modules (called special modules). As a consequence he obtains a proof of the Nakayama conjecture for graded artin algebras which is different from the one given by \textit{G. V. Wilson} [J. Algebra 85, 390-398 (1983; Zbl 0519.18012)]. He also uses rationality to give bounds for the global dimension of a graded artin algebra in terms of graded length and the number of nonisomorphic simple modules.
Mathematics Subject Classification ID: 16W50
Mathematics Subject Classification ID: 16P20
Mathematics Subject Classification ID: 16E10
Mathematics Subject Classification ID: 16Gxx
Mathematics Subject Classification ID: 16P10
zbMATH DE Number: 3985437
zbMATH Keywords: finitely generated graded module
zbMATH Keywords: graded finite dimensional algebra
zbMATH Keywords: rational power series
zbMATH Keywords: Poincaré-Betti series
zbMATH Keywords: rationality
zbMATH Keywords: finitistic dimension
zbMATH Keywords: Nakayama conjecture
zbMATH Keywords: graded artin algebras
zbMATH Keywords: global dimension
zbMATH Keywords: graded length
reviewed by: José L. Gómez-Pardo
MaRDI profile type: Publication
full work available at URL: 0021-8693(87)90009-3
OpenAlex ID: W2011876675
cites work: Graded Artin algebras
cites work: Projective resolutions over Artin algebras with zero relations
cites work: The Cartan map on categories of graded modules
cites work: Representation Theory of Artin Algebras II
cites work: Radical layers of representable functors
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description: scientific article; zbMATH DE number 3985437