Contravariantly finite subcategories closed under predecessors.
type: Item
instance of: scholarly article
title: Contravariantly finite subcategories closed under predecessors.
zbMATH Open document ID: 1190.16013
author: Ibrahim Assem
author: Sonia Trepode
author: Flávio U. Coelho
published in: Journal of Algebra
publication date: 2009-10-09T00:00:00Z
review text: In this paper the following two main results are proved: Theorem A. Let \(A\) be an Artin algebra and \(\mathcal C\) be a full skeleton subcategory of the category of all indecomposable finitely generated right \(A\)-modules, closed under predecessors. Let \(E\) denote the direct sum of ext-injective modules in \(\mathcal C\). Then the following conditions are equivalent: 1) \(\mathrm{add\,}\mathcal C\) is contravariantly finite. 2) \(\mathcal C^\perp\) is contravariantly finite. 3) \(E\) is a cotilting module over the support algebra of \(\mathcal C\). 4) \(\mathrm{add\,}\mathcal C\) consists of all \(M\) such that \(\Hom_A(M,E)\neq 0\). 5) Any morphism \(f\colon L\to M\) where \(L\in\mathcal C\) and \(M\) is indecomposable not in \(\mathcal C\) factors through \(\mathrm{add\,}E\). Theorem B. Let \(A\) and \(\mathcal C\) be as in Theorem A and assume that \(\mathrm{pgd\,}\mathcal C<\infty\). Let \(F\) be the direct sum of all indecomposable projective \(A\)-modules not lying in \(\mathcal C\). Then the following conditions are equivalent: (i) \(\mathrm{add\,}\mathcal C\) is contravariantly finite. (ii) \(E\) is a tilting module over the support algebra of \(\mathcal C\). (iii) \(E\oplus F\) is a tilting \(A\)-module. Moreover, in this case \(\mathcal C^\perp=E^\perp\) and \(\mathcal C\) consists of all predecessors of \(E\).
reviewed by: Volodymyr Mazorchuk
Mathematics Subject Classification ID: 16E30
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 16G10
Mathematics Subject Classification ID: 18E30
zbMATH DE Number: 5611752
zbMATH Keywords: contravariantly finite subcategories
zbMATH Keywords: tilting modules
zbMATH Keywords: cotilting modules
zbMATH Keywords: predecessors
zbMATH Keywords: Artin algebras
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2009.05.012
OpenAlex ID: W1995875301
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DOI: 10.1016/J.JALGEBRA.2009.05.012
description: scientific article; zbMATH DE number 5611752