Sincere silting modules and vanishing conditions

Abstract

We study characterizations of sincere modules, sincere silting modules and tilting modules in terms of various vanishing conditions. Let R be a perfect ring and T be an R-module. It is proved that T is sincere silting if and only if T is presilting satisfing the vanishing condition KerExt R 0 ≤ i ≤ 1 ( T , − ) = 0 , and that T is tilting if and only if KerExt R 0 ⩽ i ⩽ 1 ( T , − ) = 0 and Gen T ⊆ KerExt R 1 ⩽ i ⩽ 2 ( T , − ) . As an application, we prove that a sincere silting R-module T of finite projective dimension is tilting if and only if Ext R i ( T , T ( J ) ) = 0 for all sets J and all integer i ≥ 1 . This not only extends a main result of Zhang’s paper [Self-orthogonal τ-tilting modules and tilting modules, J Pure Appl Algebra, 2022, 226: 106860] from finitely generated modules over Artin algebras to infinitely generated modules over more general rings, but also gives it a different proof without using Auslander-Reiten translations.