The Krull-Gabriel dimension of the representation theory of a tame hereditary Artin algebra and applications to the structure of exact sequences
type: Item
instance of: scholarly article
title: The Krull-Gabriel dimension of the representation theory of a tame hereditary Artin algebra and applications to the structure of exact sequences
zbMATH Open document ID: 0593.16022
DOI: 10.1007/BF01171701
author: Werner Geigle
published in: Manuscripta Mathematica
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155103
review text: Let \(\Lambda\) be a tame hereditary Artin algebra and \({\mathcal F}=(\Lambda\)- mod\({}^{op},Ab)\) the category of all finitely presented contravariant functors from \(\Lambda\)-mod into the category of abelian groups. It is shown that \({\mathcal F}\) admits a filtration by Serre subcategories \(0={\mathcal F}_{-1}\subset {\mathcal F}_ 0\subset {\mathcal F}_ 1\subset {\mathcal F}_ 2={\mathcal F}\) where \({\mathcal F}_ i/{\mathcal F}_{i-1}\) is the full subcategory of \({\mathcal F}/{\mathcal F}_{i-1}\) consisting of all objects of finite length, \({\mathcal F}_ 1/{\mathcal F}_ 0\) is equivalent to the product category \({\mathcal R}\times {\mathcal R}\), for \({\mathcal R}\) the category of all regular \(\Lambda\)-modules, and \({\mathcal F}_ 2/{\mathcal F}_ 1\) to the category of all finite dimensional right vector spaces over the skew field \(End(_{\Lambda}Q)\), where \(_{\Lambda}Q\) denotes the unique indecomposable torsion-free divisible \(\Lambda\)-module. In particular \(\Lambda\)-mod has Krull-Gabriel dimension two and the above filtration leads to a hierarchy of exact sequences in \(\Lambda\)-mod. As an application, one obtains a description of morphisms between distinct Auslander-Reiten components of \(\Lambda\)-mod.
reviewed by: Andrzej Skowroński
Mathematics Subject Classification ID: 16Gxx
Mathematics Subject Classification ID: 16P60
Mathematics Subject Classification ID: 16P10
Mathematics Subject Classification ID: 16Exx
zbMATH DE Number: 3953001
zbMATH Keywords: tame hereditary Artin algebra
zbMATH Keywords: finitely presented contravariant functors
zbMATH Keywords: filtration by Serre subcategories
zbMATH Keywords: finite length
zbMATH Keywords: Krull-Gabriel dimension
zbMATH Keywords: exact sequences
zbMATH Keywords: Auslander-Reiten components
MaRDI profile type: Publication
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OpenAlex ID: W2053530858
description: scientific article; zbMATH DE number 3953001