Wakamatsu tilting modules.
type: Item
instance of: scholarly article
title: Wakamatsu tilting modules.
zbMATH Open document ID: 1075.16006
author: Francesca Mantese
author: Idun Reiten
published in: Journal of Algebra
publication date: 2004-10-01T00:00:00Z
review text: Let \(A\) be an Artin algebra. A self-orthogonal \(A\)-module, \(T\), is called a `Wakamatsu tilting module' if \(_AA\) admits a (possibly infinite) coresolution by modules from \(\text{add}(T)\). A full subcategory in \(A\text{-mod}\) is called `resolving' if it is closed under direct sums, direct summands, isomorphisms, extensions and kernels of epimorphisms, and contains all projective modules. A `cotorsion theory' is a pair \((\mathcal{C,D})\) of subcategories of \(A\)-modules such that \(\mathcal C\) is the left orthogonal complement of \(\mathcal D\) with respect to \(\text{Ext}^1\), and \(\mathcal D\) is the right orthogonal complement of \(\mathcal C\) with respect to \(\text{Ext}^1\). The class \({\mathcal C}\cap{\mathcal D}\) is called the `kernel' of \((\mathcal{C,D})\). In the paper under review the main results are the following two facts about the Wakamatsu tilting modules: Theorem. There is a natural bijection between the isomorphism classes of basic Wakamatsu tilting modules and coresolving subcategories in \(A\text{-mod}\) with an \(\text{Ext}\)-projective generator, maximal among those with the same \(\text{Ext}\)-projective generator. Theorem. There is a natural bijection between the isomorphism classes of basic Wakamatsu tilting modules and cotorsion theories, minimal among those with the same kernel and the first category having an \(\text{Ext}\)-injective cogenerator. Both results have natural dual versions, which are also included in the paper. The paper is finished with a discussion of properties of Wakamatsu tilting modules of finite projective dimension and complements of Wakamatsu tilting modules.
reviewed by: Volodymyr Mazorchuk
Mathematics Subject Classification ID: 16E30
Mathematics Subject Classification ID: 16G10
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 16S90
zbMATH DE Number: 2105342
zbMATH Keywords: tilting modules
zbMATH Keywords: resolving categories
zbMATH Keywords: torsion theories
zbMATH Keywords: contravariantly finite categories
zbMATH Keywords: Artin algebras
zbMATH Keywords: cogenerators
Wikidata QID: Q105553853
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2004.03.023
OpenAlex ID: W1980097867
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DOI: 10.1016/J.JALGEBRA.2004.03.023
description: scientific article; zbMATH DE number 2105342