Silting modules and ring epimorphisms
type: Item
instance of: scholarly article
title: Silting modules and ring epimorphisms
zbMATH Open document ID: 1407.16006
author: Lidia Angeleri Hügel
author: Frederik Marks
author: Jorge Vitória
published in: Advances in Mathematics
publication date: 2016-10-10T00:00:00Z
full work available at URL: https://arxiv.org/abs/1504.07169
reviewed by: Nguyen Viet Dung
Mathematics Subject Classification ID: 16E60
Mathematics Subject Classification ID: 16G10
zbMATH DE Number: 6636662
zbMATH Keywords: tilting module
zbMATH Keywords: silting module
zbMATH Keywords: ring epimorphism
zbMATH Keywords: torsion pair
zbMATH Keywords: universal localisation
zbMATH Keywords: hereditary ring
MaRDI profile type: Publication
OpenAlex ID: W2963405217
arXiv ID: 1504.07169
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DOI: 10.1016/J.AIM.2016.08.035
review text: Silting modules were introduced and studied by the authors in their previous work [Int. Math. Res. Not. 2016, No. 4, 1251--1284 (2016; Zbl 1367.16005)], generalizing (possibly large) tilting modules, and also support \(\tau\)-tilting modules of [\textit{T. Adachi} et al., Compos. Math. 150, No. 3, 415--452 (2014; Zbl 1330.16004)].NEWLINENEWLINEIn this paper, the authors consider the notion of a partial silting module, and give an explicit description of the ring epimorphism arising from a partial silting module \(T_1\), as an idempotent quotient of the endomorphism ring of the Bongartz completion \(T\) of \(T_1\) (Theorem 3.5). This extends results proved in the context of tilting modules in [\textit{W. Geigle} and \textit{H. Lenzing}, J. Algebra 144, No. 2, 273--343 (1991; Zbl 0748.18007)] and [\textit{R. Colpi} et al., J. Pure Appl. Algebra 211, No. 1, 223--234 (2007; Zbl 1137.16009)], or support \(\tau\)-tilting modules in [\textit{G. Jasso}, Int. Math. Res. Not. 2015, No. 16, 7190--7237 (2015; Zbl 1357.16028)].NEWLINENEWLINEThe authors further investigate homological properties of the ring epimorphism associated with a partial silting module over a finite dimensional algebra. If the ring \(A\) is hereditary, it is shown as a main result that there is a bijection between minimal silting modules and homological ring epimorphisms, and under this assignment the minimal tilting modules correspond to injective homological ring epimorphisms (Theorem 5.8). Consequences are obtained for minimal tilting modules over hereditary rings (Corollary 5.9), and finite dimensional support tilting modules over finite dimensional hereditary algebras (Corollary 5.12).
description: scientific article; zbMATH DE number 6636662