On the representation type of locally bounded categories
type: Item
instance of: scholarly article
title: On the representation type of locally bounded categories
zbMATH Open document ID: 0604.16024
author: Piotr Dowbor
author: Andrzej Skowroński
published in: Tsukuba Journal of Mathematics
publication date: 1986-01-01T00:00:00Z
review text: Let K be an algebraically closed field. Given a K-algebra R we denote by mod(R) the category of finite dimensional right R-modules. We recall that R is representation-tame if for every \(d>0\) there are K[T]-R-bimodules \(_{K[T]}Q_ R^{(j)}\), \(j=1,...,m_ d\), such that: (i) \(_{K[T]}Q^{(j)}\) is finitely generated free, (ii) all but a finite number of indecomposable modules M in mod(R) with \(\dim_ kM=d\) are of the form \(M=S\otimes_{K[T]}Q_ R^{(j)}\) for some \(j\leq m_ d\) and a simple K[T]-module S. The paper contains interesting and useful properties equivalent to the tameness. The definition of representation-tame algebra is extended to locally bounded K-categories \(\tilde R\) and it is proved that \(\tilde R\) is representation-tame if and only if all finite full subcategories of \(\tilde R\) are representation-tame. The result is very useful in studying representation-tame algebras in terms of their covering categories (which are locally bounded).
Mathematics Subject Classification ID: 16Gxx
Mathematics Subject Classification ID: 16B50
Mathematics Subject Classification ID: 18E15
zbMATH DE Number: 3975152
zbMATH Keywords: category of finite dimensional right R-modules
zbMATH Keywords: indecomposable modules
zbMATH Keywords: locally bounded K-categories
zbMATH Keywords: representation-tame algebras
zbMATH Keywords: covering categories
reviewed by: Daniel Simson
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.21099/tkbjm/1496160389
OpenAlex ID: W403728970
DOI: 10.21099/TKBJM/1496160389
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description: scientific article; zbMATH DE number 3975152