On the density of fiber sum functors
type: Item
instance of: scholarly article
title: On the density of fiber sum functors
zbMATH Open document ID: 0812.16018
DOI: 10.1007/BF02572344
author: Peter Dräxler
published in: Mathematische Zeitschrift
publication date: 1995-05-09T00:00:00Z
full work available at URL: https://eudml.org/doc/174673
review text: Fiber sum functors furnish a technique to study the category \(A\)-mod of finitely generated left \(A\)-modules over a finite dimensional \(k\)-algebra where \(k\) is a field. Namely consider \(P \in A\text{-mod}\) satisfying \(\text{Ext}^ 1_ A(P,\text{fac }P) = 0 = \text{Ext}^ 1_ A(P,\text{ sub }P)\) and the full subcategory \({\mathcal K} = {\mathcal K}(A,P)\) of all \(V \in A\text{-mod}\) such that \(\text{Ext}^ 1_ A(V, \text{fac }P) = 0\). The fiber sum functor \(F\) with respect to \(P\) has as domain the category \(\text{mat}(P,{\mathcal K})\) of all \(A\)-homomorphisms \(\delta : U \to V\) where \(U \in \text{add }P\), \(V \in {\mathcal K}\) and as target the category \(A\)-mod. It maps a given object \(\delta \in \text{mat}(P,{\mathcal K})\) to its cokernel. A very classical tool to study the category \(A\)-mod is to describe the modules by their projective presentations. A more modern device for this study is the use of one-point (co)extensions. We explain that fiber sum functors give a common access to both techniques by choosing suitable modules \(P\). But our main interest lies in the situations not covered by these special cases. It was shown by the author [J. Algebra 113, 430-437 (1988; Zbl 0659.16020)] that \(F\) is always full and has a well-behaved, small kernel whereas the density of \(F\) was proved only for \(A\) of finite representation type. The first main result of our paper is to establish the density of \(F\) in general by observing that \(\text{fac }P\) is a covariantly finite subcategory and using results of \textit{M. Auslander} and \textit{I. Reiten} [Adv. Math. 86, 111-152 (1991; Zbl 0774.16006)]. Obviously \(F\) is only useful if \(\text{mat }(P,{\mathcal K})\) is better understood than \(A\)-mod. This is the case if \(k\) is algebraically closed and \(P\) is an indecomposable projective module with \(\dim_ k \text{End}_ A(P) = 1\). Under these assumptions \(\text{mat}(P,K)\) can be identified with the subspace category of the vectorsp…
Mathematics Subject Classification ID: 16G30
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 16G60
zbMATH DE Number: 703850
zbMATH Keywords: finitely generated left \(A\)-modules
zbMATH Keywords: finite dimensional \(k\)-algebra
zbMATH Keywords: fiber sum functor
zbMATH Keywords: projective presentations
zbMATH Keywords: finite representation type
zbMATH Keywords: indecomposable projective module
zbMATH Keywords: vectorspace category
zbMATH Keywords: representations of partially ordered sets
zbMATH Keywords: representation type
zbMATH Keywords: tame subspace type
zbMATH Keywords: tame representation type
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 703850