Sincere silting modules and vanishing conditions

Abstract

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 $τ$ and Auslander-Reiten formula.