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 ${\text{KerExt}}_R^{0\le i\le 1}(T,-)=0$, and that T is tilting if and only if ${\text{KerExt}}_R^{0⩽i⩽1}(T,-)=0$ and $\text{Gen}T\subseteq {\text{KerExt}}_R^{1⩽i⩽2}(T,-)$. As an application, we prove that a sincere silting R-module T of finite projective dimension is tilting if and only if ${\text{Ext}}_R^i(T,T^{(J)})=0$ for all sets J and all integer $i\ge 1$. 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.

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 16L30
• 16D10

KEYWORDS:

• Silting modules
• sincere modules
• tilting modules
• vanishing conditions

Authors’ contributions

All authors reviewed the manuscript.

Disclosure statement

The authors have no competing interests as defined by Springer, or other interests that might be perceived to influence the results and/or discussion reported in this paper.

Additional information

Funding

Supported by the National Natural Science Foundation of China (Grant No. 11771212) and a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.