Triads.
type: Item
instance of: scholarly article
title: Triads.
zbMATH Open document ID: 1079.16008
author: Wolfgang Rump
published in: Journal of Algebra
publication date: 2005-02-22T00:00:00Z
review text: The present paper is a following up of the author's previous paper [\textit{W. Rump}, The category of lattices over a lattice finite ring, to appear in Algebr. Represent. Theory], where the author develops a theory of so-called \(L\)-functors, which is in some sense a functorial replacement of the Auslander-Reiten translation. In the paper under review the author is concerned with Krull-Schmidt categories with enough Ext-injective and enough Ext-projective objects, so that every split epimorphism has a kernel. The main result of the paper is that such a category has almost split sequences if and only if some category of short exact sequences is semilocal, cosemilocal and has enough semisimple objects. This in turn is equivalent to the fact that the category of two-term complexes with radical morphism up to homotopy has \(L\)-functors. On the way the author has to develop a theory that mimics the construction of a derived category in this more general setup. The author calls these structures triads and shows that they have interesting properties.
reviewed by: Alexander Zimmermann
Mathematics Subject Classification ID: 16G70
Mathematics Subject Classification ID: 18E30
Mathematics Subject Classification ID: 16G30
zbMATH DE Number: 2136729
zbMATH Keywords: \(L\)-functors
zbMATH Keywords: Auslander-Reiten triangles
zbMATH Keywords: triadic categories
zbMATH Keywords: almost split sequences
zbMATH Keywords: triangulated categories
zbMATH Keywords: Auslander-Reiten translations
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2004.07.007
OpenAlex ID: W4211198449
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DOI: 10.1016/J.JALGEBRA.2004.07.007
description: scientific article; zbMATH DE number 2136729