The endomorphism ring of a localized coherent functor
type: Item
instance of: scholarly article
title: The endomorphism ring of a localized coherent functor
zbMATH Open document ID: 0873.16009
author: Ivo Herzog
published in: Journal of Algebra
publication date: 1997-07-08T00:00:00Z
review text: Let \(C\) be a commutative artinian ring and let \(\Lambda\) be an Artin \(C\)-algebra. The category of finitely presented objects in the category of additive functors from mod-\(\Lambda\) to \(Ab\) is denoted by \(Ab(\Lambda)\) (since the author identifies this with the free abelian category over the ring \(\Lambda)\). It is an open question as to what are the possible values for the Krull-Gabriel dimension of \(Ab(\Lambda)\) (\(0\), \(2\), \(\infty\) are known to occur). The author proves that the value 1 cannot occur. For the case that \(\Lambda\) is a finite-dimensional algebra over an algebraically closed field this has been shown by \textit{H. Krause} [Generic modules over Artin algebras, Proc. Lond. Math. Soc., III. Ser. (to appear)]. Herzog's proof involves showing that if \(S_0\) is the Serre subcategory of \(Ab(\Lambda)\) consisting of the finite length functors and if \(A\in Ab(\Lambda)\) then the endomorphism ring of the image of \(A\) in the localisation of \(Ab(\Lambda)\) at \(S_0\) is a locally Artin \(C\)-algebra. Then he finishes by using results on generic modules (in particular, results from \textit{W. W. Crawley-Boevey} [in ``Representations of algebras and related topics'', Lond. Math. Soc. Lect. Note Ser. 168, 127-184 (1992; Zbl 0805.16028)]).
Mathematics Subject Classification ID: 16G10
Mathematics Subject Classification ID: 03C60
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 16P60
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 18E35
Mathematics Subject Classification ID: 18A25
zbMATH DE Number: 1019701
zbMATH Keywords: Artin \(C\)-algebras
zbMATH Keywords: categories of finitely presented objects
zbMATH Keywords: categories of additive functors
zbMATH Keywords: free Abelian categories
zbMATH Keywords: Krull-Gabriel dimension
zbMATH Keywords: finite-dimensional algebras
zbMATH Keywords: Serre subcategories
zbMATH Keywords: finite length functors
zbMATH Keywords: endomorphism rings
zbMATH Keywords: localisations
zbMATH Keywords: generic modules
reviewed by: Mike Prest
MaRDI profile type: Publication
full work available at URL: jabr.1997.6920
OpenAlex ID: W2002221512
cites work: Q5586467
cites work: Q4097440
cites work: Q4028602
cites work: Preprojective modules over Artin algebras
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cites work: Q5599870
cites work: Des catégories abéliennes
cites work: The Ziegler Spectrum of a Locally Coherent Grothendieck Category
cites work: Q3999266
cites work: Functors on locally finitely presented additive categories
cites work: Generic Modules Over Artin Algebras
cites work: A proof of Saffe's conjecture
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DOI: 10.1006/JABR.1997.6920
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description: scientific article; zbMATH DE number 1019701