Sincere silting modules and vanishing conditions

TL;DR

This paper characterizes sincere silting and tilting modules over perfect rings using vanishing Ext conditions, extending previous results to more general rings and providing new proofs.

Contribution

It provides new characterizations of sincere silting and tilting modules via Ext vanishing conditions, generalizing prior results to broader classes of rings and modules.

Findings

Sincere silting modules are characterized by specific Ext vanishing conditions.

Tilting modules are characterized by combined Ext vanishing and generation conditions.

Ext vanishing conditions can determine when sincere silting modules are tilting.

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 $\tau$ and Auslander-Reiten formula.