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Let $R$ be a perfect ring and $T$ be an $R$-module. We studycharacterizations of sincere modules, sincere silting modules and tiltingmodules in terms of various vanishing conditions. It is proved that $T$ issincere silting if and only if $T$ is presilting satisfing the vanishingcondition $\mathrm{KerExt}^{0\le i\le 1}_R(T,-)=0$, and that $T$ is tilting ifand only if $\mathrm{Ker}\mathrm{Ext}^{0\leqslant i\leqslant 1}_{R}(T,-)=0$ and$\mathrm{Gen}T\subseteq \mathrm{Ker}\mathrm{Ext}^{1\leqslant i\leqslant2}_{R}(T,-)$. 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 generatedmodules over Artin algebras to infinitely generated modules over more generalrings, but also gives it a different proof without using the functor $\tau$ andAuslander-Reiten formula.

Sincere silting modules and vanishing conditions