Sincere silting modules and vanishing conditions

• Published in: arXiv.org 2022-11
• Main Authors: Liu, Jifen, Jiaqun Wei
• Format: Article
• Language: English
• Subjects: Modules; Rings (mathematics)
• ISSN: 2331-8422

Let \(R\) be a perfect ring and \(T\) be an \(R\)-module. We study characterizations of sincere modules, sincere silting modules and tilting modules in terms of various vanishing conditions. It is proved that \(T\) is sincere silting if and only if \(T\) is presilting satisfing the vanishing condition \(\mathrm{KerExt}^{0\le i\le 1}_R(T,-)=0\), and that \(T\) is tilting if and only if \(\mathrm{Ker}\mathrm{Ext}^{0\leqslant i\leqslant 1}_{R}(T,-)=0\) and \(\mathrm{Gen}T\subseteq \mathrm{Ker}\mathrm{Ext}^{1\leqslant i\leqslant 2}_{R}(T,-)\). As an application, we prove that a sincere silting \(R\)-module \(T\) of finite projective dimension is tilting if and only if \(\mathrm{Ext}^{i}_{R}(T,T^{(J)})=0\) for all sets \(J\) and all integer \(i\ge 1\). This not only extends a main result of Zhang [14]from finitely generated modules over Artin algebras to infinitely generated modules over more general rings, but also gives it a different proof without using the functor \(\tau\) and Auslander-Reiten formula.